RESEARCH ANNOUGCEMENTS—Kolmogorov Width of Classes of Smooth Functions on the Sphere s^d—1

Let Rd be the d-dimensional Euclidean space and Sd-1 = {(x1,…,xd): x21 + … + x2d = 1} be the unit sphere of Rd equipped with the normal Lebesgue measure dσ(x).For r ＞ 0,denotc by Brp := Brp(Sd-1) (1 ≤ p ≤∞) the class of functions f on the sphere Sd-1 representable in the form     

BAHADUR ASYMPTOTIC EFFICIENCY FOR PARAMETER ESTIMATE IN MULTIPLE REGRESSION

1. INTRODUCTION Consider the multiple regression model xi=ciβ+ei,i=1,2,…,(1) where c_i=(c_(il),…,c_(ip)), i=1, 2,…are given row vectors; β∈R~p is an unknown parameter vector; e_1, e_2,…are i.i.d. random variables with a common probability density function f(x) with respect to the Lebesgue measure. Let x_1, x_2,…be a sequence of observa-     

Strong Consistency of Non-parametric Regression Estimates with Censored Data

Let (X,Y) be an R~d×R valued random vector with E|Y|<∞ and(X_1,Y_1) (X_2,Y_2), …, (X_n,Y_n) be i.i.d.observations of (X,Y). To estimate the regression function m(x)=E(Y|X=x), Stone suggested m_n(x)=sum from i=1 to n(W_(ni)(x)Y_i), where W_(ni)(x)=W_(ni)(x,X_1,X_2,…,X_n)(i=1,2,…,n) are weight functions. Devroye and Chen Xiru established the strong consistency of m_n(x). In this paper, we discuss the case that{Y_i} are censored by {t_i}, where{t_i} are i.i.d. random variables and also independent of{Y_i}. Under certainconditions we still obtain the strong consistency of m_n(x).     

PERIODIC SOLUTIONS FOR GENERALIZED PREDATOR-PREY SYSTEMS WITH TIME DELAY AND DIFFUSION

A set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions for delayed generalized predator-prey dispersion system x'1 (t) = x1 (t)g1 (t, x1 (t) ) - a1 (t)y(t)p1 (x1 (t) ) D1 (t)(x2(t) - x1 (t) ),x'2 (t) = x2 (t)g2 (t, x2 (t) ) - a2 (t)y(t)p2 (x2 (t) ) D2(t)(x1 (t) - x2 (t)),y' (t) = y(t) {-h(t, y(t) ) b1 (t)p1 (x1 (t - τ1 ) ) b2(t)p2(x2(t - τ2))],where ai(t), bi(t) and Di(t)(i = 1, 2) are positive continuous T-periodic functions, gi(t, xi)(i = 1,2) and h(t,y) are continuous and T-periodic with respect to t and h(t,y) ＞ 0 for y ＞ 0, t, y ∈ R, pi(x)(i = 1, 2) are continuous and monotonously increasing functions, and pi(xi) ＞ 0 for xi ＞ 0.     

APPROXIMATION OF A CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES＊

Using the fixed point and direct methods,we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation 2f p i=1 xi + q j=1 yj + 2 d k=1 zk 2 = p i=1 f(xi) + q j=1 f(yj) + 2 d k=1 f(zk),where p,q,d are integers greater than 1,in non-Archimedean normed spaces.     

The definite integral

<正>1.What does the definite integral mean?The definite integral of f(x)fromato b is defined the limit of the sum as n→∞.That is limn→∞∑n i=1f(ξi)·Δxi.We divide the interval[a,b]into n subintervals of equal widthΔx=(b-a)n.Let x0=a,x1,x2,…,xn=b be the endpoints of these subintervals and we chooseξiis any point in the ith subinterval,that is,xi-1≤ξi≤xi,then,the sum∑n f(ξi)·Δxiis called a Rie-     

Diophantine inequality involving binary forms

Let r =2^d-1 ＋ 1. We investigate the diophantine inequality
｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,
where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.     

ON THE UNIFORM CONSISTENCY OF THEKAPLAN-MEIER ESTIMATOR UNDER HEAVY CENSORING

Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)＜1)＞to=sup(t2G(t)＜1) The sufficient condition is discussed.     

Maximum likelihood character of distributions

In this paper,some distributions in the family of those with invariance under orthogonaltransformations within an s-dimensional linear subspace are characterized by maximun likelihoodcriteria.Specially,the main result is:suppose P_v is a projection matrix of a given s-dimensionalsubspace V,and x_1,…,x_n are i.i.d.samples drawn from a population with a pdf f(x′P_vx),wheref(·) is a positive and continuously differentiable function.Then P_v(M_n) is the maximum likelihoodestimator of P_v ifff(x)=c_kexp(kx) (k>0),where M_n=sum from i=1 to n x_ix′_i,P_v(M_n)=sum from i=1 to (?) (?)_i(?)′_t,λ_1,…,λ_(?) are the first s largest eigenvalues of matrix M_n,and(?)_1,…,(?)_(?) are their associated eigenvectors.     

On Steady States and Its Capturing Schemes for Shallow-Water Equations with Source Terms

Consider the shallow fiuid fiow in a ehannel with wavy bottom,whieh 15 governed by thefollowing equations:h‘+(h。)二=0,(h。)‘+(hoZ+夕hZ/2)二=一夕hB‘(x),(1)where h denotes the hight of the fluid above the wavy bottom eharaeterized by a funetionB(二)and 0 15 the horizontal veloeity of the fluid.It 15 not diffieult to show that every steadystate(i.e.,asympototie solution)of(1)obeys the following balanee eonditions: h(x)。(x)=eonst.,uZ(x)/2+夕h(x)+夕刀(x)=eonst.(2)and that when h and 0 are smo…