关于富里埃及数和幂级数的蔡查罗平均

Let f(x) be a periodic function of period 2π,|f(x)|and |f(x)|^p integrable on [0,2π].It is said to be f∈H_X¹,if (∫₀^2π|f(x+h)-f(x)|^pdx)^1/p≤|h|.It is said to be f∈H_X²,if (∫₀^2π|f(x+h)+f(x-h)-2f(x)|^pdx)^1/p≤2|h|.     

一类奇异积分的计算方法

By means of residues and special integral loops, we give a general evaluation of the following kind of singular integrals:I(m,p,n)=∫₀^∞ (sin^mxcos^px) /xⁿdx,where m,n,p are all non-negative integers，m≥n> 0, and m，n are simultaneously odd or even.     

Fourier级数子列的可和性

Let H_α0^ω denote the set of functions f(x)∈L_2π such that ω(f,x₀;t)≤ω(t),ω(t) being a given modulus of continuity. Let {n_k} be a set of natural numbers satisfying the condition n_i+1/n_k>q>1, and let A= (α_πk) be a regular summation matrix.     

用Hermite-Fejér型插值多项式逼近连续函数

Let f(x)∈C_[-1,1],T_n(x)=cos (n arccos x),U_n(x)=(sin((n+1)arccosx))/(1-x²)^1/2,P_n(x) be the Legendre polynomials of degree n. And let ω(t ) be a given modulus of continuity, H_ω={f|ω(f,t)≤ω(t)}.A. K. Sharma and J. Tzimbalario(J. Appro. Th., 13(1975), 431-442) considered the operators L_n,p (f, x) (p= 0, 1, 2,3) and obtained some theorems.In this paper, we prove the following theorems.     

Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification*

Let M be an open Riemann surface and G : M → Pⁿ(C) be a holomorphic map. Consider the conformal metric on M which is given by ds² = k e Gk^2m|ω|², where eG is a reduced representation of G, ω is a holomorphic 1-form on M and m is a positive integer. Assume that ds² is complete and G is k-nondegenerate(0 ≤ k ≤ n). If there are q hyperplanes H₁, H₂, · · · , H_q Pⁿ(C) located in general position such that G is ramified over H_j with multiplicity at least γ_j (> k) for each j ∈ {1, 2, · · · , q}, and it holds that Xj=1 1 -k/γ_j> (2n - k + 1) （mk/2+ 1）,then M is flat, or equivalently, G is a constant map. Moreover, one further give a curvature estimate on M without assuming the completeness of the surface.     

关于集值系统x∈F(x,y),y∈G(x,y)解的存在性

Let (E, ‖·‖) be a uniformly convex Banach space, X a nonempty compact and convex subset of E. Let F be a closed mapping of X×X into 2^X, G a mapping of X×X into C(X). It is shown that if for any f∈C(X), x∈X, F(x,f(x)) is a closed and convex subset of X, and G(x,f(x)) is a continuous function and for any x,y₁,y₂∈X,H(x,G(x,y₁),G(x,y₂))≤‖y₁-y₂‖ then there exist X₀,y₀∈X such that X₀∈F(x₀,y₀) and y₀∈G(x₀, y₀).     

关于M/G/l随机服务系统中L.Takacs引理的证明

L. Takacs' Lemma (ef [1] PP47-48) is a very important tool for the investigation of the transient behavior of the M/G/1 System. But the original proof contains a gap when γ(s, w) is expanded into Lagrange series by means of Lagrange theorem as given in the appendix of [1]. This is due to the fact that φ(s)=∫₀^∞e^-sxdH(x), the Laplace transform of the service time distribution , may not be analytic at s=0. An example of such a distrbution function H(x) is gives and a specific Lagrange theorem is proved By using this specific theorem, the gap in the original proof of Takacs′ Lemms is eliminated.     

二值回归模型中自变量误差的处理

In this paper the binary regression model, in which the response Y is binary, i.e. Y takes the value 0 or 1 only, is considered. The conditional probability P{Y=1|X=x} is assumed to be of the form G(a₀+a₁x), where x is the measurement of the factor, a= (a₀,a₁)′ is the vector of the unknown parameter being estimated and G(a) = (1+e^a)^-1. The measurement x is assu-med to be contaminated when the value of x is out of certain limit. M-estimation is employed to estimate the unknown a of the model. In this paper, the consistent M-estimation is proved to be asymptotically normal and the optimal M-equation is obtained. Finally, a computation method is introduced to solve the M-equation and to get the optimal solution.     

关于有限域中原根分布的注记

Let f₁(x) , …,f_r(x) denote quadratfree polynomials over GF(q) that are relatively prime in pairs and of degree≥1. We have proved that if q≥(k-1)²(2^rω(q-1)-1)², then all the f_i(ξ) are primitive roots of GF(q) for some ξ∈GF(q), where k =degf₁+…+degf_r . With extending the results obtained by Wang Juping, Sun Qi and Han Wenbao respectively, we have also given a quantitative analysis of Carlitz's result.     

位置刻度参数最优同变区间估计的minimax性质

Let X be a p-dimensional random vector having a density function σ^-p((x-θ1)/σ),where p≥2, x∈R^p, θ∈R¹, σ>0, 1= (1, …, 1)_p×1¹. The minimax property of best equivariant confidence intervals for the location parameter θ and scale parameter σ are proved under followiny loss function     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

Refinement of the asymptotics of the power of the quantile test

One investigates the asymptotic properties of the quantile test, similar to the properties of the Pearson's chi-square test of fit.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 5–15, 1986.The author is grateful to D. M. Chibisov for useful remarks.