Distribution of Primitive A-Roots of Composite Moduli Ⅱ

Abstract We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length ∣I∣ ≫ q ^1/2+∈. Similar results are obtained for the distribution of ordered pairs (x, x ⁻¹) with x a primitive λ-root, and for the number of primitive λ-roots satisfying inequalities such as |x − x ⁻¹| ≤ B. (Dedicated to Professor Wang Yuan on the occasion of his 75th birthday) *Project supported by the National Natural Science Foundation of China (No.19625102) and the 973 Project of the Ministry of Science and Technology of China.     

The green function for the weighted biharmonic operator Δ(1−∣z∣2)−αΔ, and factorization of analytic functions, and factorization of analytic functions

An explicit integral formula is obtained for the Green function of the weighted biharmonic operator Δ(1−∣z∣²)^−αΔ in the unit disk of the complex plane for the case α ∈ (−1, 0). The formula shows the positivity of the Green function. This is a basis for a theorem on factorization of analytic functions in the weighted Bergman spaces with the weights ω(z)=(1−∣z∣²)^α as products of a nonvanishing function and a function of special form responsible for the zeros. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 203–221.     

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory.     

Asymptotic behaviour of a Brownian motion on exterior domains

We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ? ^c is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p ^? _t (x, y) behaves, for large t, as u(x)u(y)(t(log t)²)⁻¹ for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?) ^r , such that u(x) ∼ log ∣x∣. Received: 29 January 1999 / Revised version: 11 May 1999     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory. Received October 18, 2001; in final form April 11, 2002     

On the irrationality measure for certain series

Let K be either the rational number field \Bbb Q{\Bbb Q} or an imaginary quadratic field. We give irrationality results for the number q = ?_n=1^￥rⁿ/(qⁿ-r^l)\theta=\sum_{n=1}^{\infty}{r^n}/(q^n-r^l), where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and 1 ￡ l ? \Bbb Z1\le l\in{\Bbb Z} with q ⁿ ≠ r ^l (n ≥ 1).     

Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment [–1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis [0, +∞).     

On the irrationality measure for certain series

Let K be either the rational number field or an imaginary quadratic field. We give irrationality results for the number , where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and with q ⁿ ≠ r ^l (n ≥ 1). Authors’ addresses: Kenji Amano, NS solutions Corporation, 2-27-1 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan; Yohei Tachiya, Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C _β,∞^ψ by Poisson integrals in the metric of the space C.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.     

Construction of Rank 2 Semistable Torsion Free Sheaves Which Are Not Limit of Vector Bundles

We show the following theorem of compensated compactness type: Ifu _n ⇁u weakly in the spaceH ^1,p (Ω, ℝ ^k ) and if also in the sense of distributions then ∂_α(∣∇u∣ ^p-2∂_α u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S ^k −1.     

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j Symbols, and Character Localization

In this paper, we employ a technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index J) of generic Wigner matrix elements D^J_MM￠(g){D^{J}_{MM'}(g)} . We use this result to derive asymptotic formulae for the character χ ^J (g) of an SU(2) group element and for Wigner’s 3j symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for χ ^J (g) is in fact exact. The result hints at a “Duistermaat-Heckman like” localization property for discrete sums.     

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution u _ɛ does not converge, in the H ¹-norm on the whole domain, towards some u ₀. In this paper we construct correctors to have good approximations of u _ɛ in the H ¹-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.     

On some nonlinear boundary-value problems in cylindrical domains

The paper deals with a linear elliptic second-order equation in divergent form in a semi-infinite cylinder Ω=Ω×[0,∞), where Ω is a bounded domain, Ω⊂ℝⁿ⁻¹, 0≤x_n<∞. The solution under consideration satisfies the nonlinear boundary condition, ∂u/∂v+a₀∣u∣^q−1u=0, where ϖ/ϖν is the differentiation along the external conormal a₀=const>0, q=const. The asymptotic behavior of the solution for x_n→+∞ is studied. Bibliography: 15 titles. Dedicated to O. A. Oleinik The investigations described in this paper were supported in part by the Russian Foundation for Basic Research, project No. 93-011-16035, and by the International Science Foundation, grant MIE000. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

A method for approximations to the pdf'S and cdf'S of GLSE's and its application to the seemingly unrelated regression model

This paper first develops a valid method for approximations to the pdf's and cdf's of GLSE in linear models and, applying this method to the Zellner estimator with an unrestricted sample covariance in the seemingly unrelated regression model, obtains an approximate pdf with a bound of ordern ⁻² and an approximate covariance matrix with a bound of ordern ⁻³ This research was done at the London School of Economics while the authors were British Council scholars. Kariya is grateful to Professor J. Durbin for a general discussion on asymptotic expansions. Further the authors deeply appreciate Professor Y. Kataoka and anonymous referee for their invaluable comments and suggestions.     

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.     

Lower semicontinuity of quasiconvex integrals

Summary New lower semicontinuity results for quasiconvex integrals. In particular, under certain structure conditions and growth 0≤F(ξ)≤C(∣≤∣ ^q ) the functional ∫ _Ω F(∇u)dx is proved to be lower semicontinuous onW ^1,q with respect to the weak convergence inW ^1,p, p≥q−1. Research supported by the grant No. 201/93/2171 of Czech Grant Agency (GAČR) and by the grant No. 364 of Charles University (GAUK). This article was processed by the author using the Springer-VerlagTex P Jourlg marco package 1991.     

On the Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

The main aim of this paper is to prove that the maximal operator σ ₀ ^k*:= sup _n ∣σ _n,n ^k ∣ of the Fejér means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H _1/2 to the space weak-L _1/2. Supported by the Hungarian NFSR (OTKA), grant no. M 36511/2001, T 048780.