EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM

In this paper, the author considers the two-dimensional delay systems $$\[\mathop x\limits^ \cdot (t) = Ax(t) + Bx(t - r),A,B \in {R^{2 \times 2}},x \in {R^2},r = const \ge 0\]$$ and gives the necessary and sifficient conditions under which where exists a simple type of positive definite Liapunov functional $$\[V(\varphi ) \buildrel \Delta \over = {\varphi ^''}(0){T_\varphi }(0) + \int_{ - \tau }^0 {{\varphi ^''}(\theta )E\varphi (\theta )d\theta } \]$$ and $\[\alpha (s)\]$(where T , E are positive definite 2x2 matrices, $\[\varphi \in C([ - \tau ,0],{R^n})\]$, "." stands for transpose, $\[\alpha (s)\]$ is continuous and $\[\alpha (0) = 0,\alpha (s) > 0,s > 0\]$. such that $\[{V_{(*)}}(\varphi ) \le - \alpha (\left| {\varphi (0)} \right|).\]$.     

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Consider the two-sided truncation distrbution families written in the formf(x,θ)dx=w(θ_1, θ_2)h(x)I_([θ_1,θ_2])(x)dx, where θ=(θ_1,θ_2).T(x)=(t_1(x), t_2(x))=(min(x_1,…,x_m), max(x_1, …,x_m))is a sufficient statistic and its marginal density is denoted by f(t)dμ~T. The prior distribution of θ belongs to the familyF={G:∫‖θ‖~2dG(θ)<∞}.In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ_n (t), by using the kernel estimation of f(t). Under a quite general assumption imposed upon f(t) and h(x), it is shown that φ_n(t) is an asymptotically optimal EBE of θ.     

UNUNIQUENESS OF SOME HOLOMORPHIC FUNCTIONS

In the present paper, we show that there exist a bounded, holomorphic function $\[f(z) \ne 0\]$ in the domain $\[\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\[f(z)\]$ has a Dirichlet expansion $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\[x > {x_f}\]$ if and only if $\[\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\[[1, + \infty )\]$, where $\[\alpha \]$ is a positive constant,$\[{x_f}( < + \infty )\]$ is the abscissa of convergence of $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\[\{ {u_n}\} \]$ satisfies $\[\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in [4].     

AUTOMORPHISMS OF SL (2, K) OVER SKEW FIELDS

In this paper, the author proves the following resu: It Let K be a skew field and A be an automorphism of SL(2, K). Then there exists A∈GL(2, K), an automorphism σ or an anti-automorphism τ of K, such that A is of theform AX=AX~σA~(-1) for all X∈SL(2, K)or AX=A(X~τ~2)~(-1)A~(-1) for all X∈SL(2, K),where X~σ, X~τ are the matrices obtained by applying σ, τ on X respee tively and X' is thetranspose of X.     

ON THE EXPECTED SAMPLE SIZES OF SOME POWER ONE TESTS FOR NORMAL MEAN WITH UNKNOWN VARIANCE

Suppose that $\[{x_1},{x_2}, \cdots \]$ are i i d. random variables on a probability space $\[(\Omega ,F,P)\]$ and $\[{x_1}\]$ is normally distributed with mean $\[\theta \]$ and variance $\[{\sigma ^2}\]$, both of which are unknown. Given $\[{\theta _0}\]$ and $\[0 < \alpha < 1\]$, we propose a concrete stopping rule T w. r. e.the $\[\{ {x_n},n \ge 1\} \]$ such that $$\[{P_{\theta \sigma }}(T < \infty ) \le \alpha \begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta \le {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[{P_{\theta \sigma }}(T < \infty ) = 1\begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta > {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[\mathop {\lim }\limits_{\theta \downarrow {\theta _0}} {(\theta - {\theta _0})^2}{({\ln _2}\frac{1}{{\theta - {\theta _0}}})^{ - 1}}{E_{\theta \sigma }}T = 2{\sigma ^2}{P_{{\theta _0}\sigma }}(T = \infty )\]$$ where $\[{\ln _2}x = \ln (\ln x)\]$.     

UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.     

ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

Let X_1,…,X_n be iid samples drawn from an m-dimensional population with a probabilitydensity f,belonging to the family C_(ka),i.e.the family of all densities whose partialderivatives of order k are bounded by a.It is desired to estimate the value of f at somepredetermined point a,for example a=0.Farrell obtained some results concerning the bestpossible convergence rates for all estimator sequence,from which it follows,for example,thatthere exists no estimator sequence{γ_n(0)=γ_n(X_1,…,X_n,0)}such that(?)E_f[γ_n(0)-f(0)]~2=o(n~(-2k/(2k m))).This article pursues this problem further and proves that there existsno estimator sequence{γ_n(0)}such thatn~(-k/(2k m))(γ_n(0)-f(0))(?)0,for each f∈C_(ka),where(?)denotes convergence in probability.     

ON ADMISSffilLITY OF VARIANCE COMPONENTS ESTIMATES

Suppose that there is a variance components model $$\[\left\{ {\begin{array}{*{20}{c}} {E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}} \end{array}} \right.\]$$ where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$. This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction.     

THE OPTIMAL RATE OF CONVERGENCE OF ERROR FOR kNN MEDIAN REGRESSION ESTIMATES

Let(X,Y),(X_1,Y_1),…,(X_n,Y_n)be iid.random vectors,where Y is one-dimensional.It is desired to estimate the conditional median(X)of Y,by use of Z_n={(X_i,Y_i),i=1,…,n}and X.Denote by(X,Z_n)the kNN estimate of(X),and putH_(nk)(Z_n)=E{|(X,Z_n)-(X)||Z_n},the conditional mean absolute error.This articalestablishes the optimal convergence rate of P(H_(nk_n)(Z_n)>ε),under fairly generalassumptions on(X,Y)and k_n,which tends to ∞ in some suitable way.     

EXPONENTIAL STABILITY OF LINEAR SYSTEMS IN BANACH SPACES

In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class $\[{H^2}(\sigma )\]$ and by this lemma obtains some fundamental results of exponential stability of $\[{C_0}\]$-semigroup of bounded linear operators in Banach spaces. Specially, if $\[{\omega _s} = \sup \{ {\mathop{\rm Re}\nolimits} \lambda ;\lambda \in \sigma (A) < 0\} \]$ and $\[\sup \{ \left\| {{{(\lambda - A)}^{ - 1}}} \right\|;{\mathop{\rm Re}\nolimits} \lambda \ge \sigma \} < \infty \]$ , where \[\sigma \in ({\omega _s},0)\]) and A is the infinitesimal generator of a $\[{C_0}\]$-semigroup in a Banach space $X$, then $\[(a)\int_0^\infty {{e^{ - \sigma t}}\left| {f({e^{tA}}x)} \right|} dt < \infty \]$, $\[\forall f \in {X^*},x \in X\]$; (b) there exists $\[M > 0\]$ such that $\[\left\| {{e^{tA}}x} \right\| \le N{e^{\sigma t}}\left\| {Ax} \right\|\]$, $\[\forall x \in D(A)\]$; (c) there exists a Banach space $\[\hat X \supset X\]$ such that $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$.     

A NOTE ON A PROBLEM OF BOAS R.P.

To answer the rest part of the problem of Boas R. P. on derivative of polynomial, it is shown that if $\[p(z)\]$ is a polynomial of degree n such that $\[\mathop {\max }\limits_{\left| z \right| \le 1} \left| {p(z)} \right| \le 1\]$ and $\[{p(z) \ne 0}\]$ in $\[\left| z \right| \le k,0 < k \le 1\]$, then $\[\left| {{p^''}(z)} \right| \le n/(1 + {k^n})\]$ for $\[\left| z \right| \le 1\]$. The above estimate is sharp and the equation holds for $\[p(z) = ({z^n} + {k^n})/(1 + {k^n})\]$.     

UNBOUNDED SOLUTIONS OF CONSERVATIVE OSCILLATORS UNDER ROUGHLY PERIODIC PERTURBATIONS

This note is concerned with the equation $$\[\frac{{{d^2}x}}{{d{t^2}}} + g(x) = p(t)\begin{array}{*{20}{c}} {}&{(1)} \end{array}\]$$ where g(x) is a continuously differentiable function of a $\[x \in R\]$, $\[xg(x) > 0\]$ whenever $\[x \ne 0\]$, and $\[g(x)/x\]$ tends to $\[\infty \]$ as \[\left| x \right| \to \infty \]. Let p(t) be a bounded function of $\[t \in R\]$. Define its norm by $\[\left\| p \right\| = {\sup _{t \in R}}\left| {p(t)} \right|\]$ The study of this note leads to the following conclusion which improves a result due to J. E. Littlewood, For any given small constants $\[\alpha > 0,s > 0\]$, there is a continuous and roughly periodic(with respect to $\[\Omega (\alpha )\]$) function p(t) with $\[\left\| p \right\| < s\]$ such that the corresponding equation (1) has at least one unbounded solution.     

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETER OF ONE-DIMENSIONAL DISCRETE EXPONENTIAL FAMILIES

Consider the discrete exponential family written in the form P_θ(X=x)=h(x)β(θ)θ~x,x=0,1,2,…,where h(x)>0,x=0,1,2,…,The prior distribution of θ belongs to thefa     

ESTIMATE OF d_0/d~* FOR STARLIKE FUNCTIONS

Let S~* be the class of functionsf(z)analytic,univalent in the unit disk|z|<1 andmap|z|<1 onto a region which is starlike with respect to w=0 and is denoted as D_f.Letr_0=r_0(f)be the radius of convexity of f(2).In this note,the author proves the following result:(d_0/d~*)≥0.4101492,where d_0= f(z),d~*=|β|.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

COMPLETELY POSITIVE MAPS AND $*$-OMORPHISM OF $\[{C^*}\]$-ALGEBRAS

Let $A$, $B$ be unital $\[{C^*}\]$-algebras. $\[{\chi _A} = \{ \varphi |\varphi \]$ are all completely postive linear maps from $\[{M_n}(C)\]$ to $A$ with $\[\left\| {a(\varphi )} \right\| \le 1\]$ $}$. $\[(a(\varphi ) = \left( {\begin{array}{*{20}{c}} {\varphi ({e_{11}})}& \cdots &{\varphi ({e_{1n}})}\{}& \cdots &{}\{\varphi ({e_{n1}})}& \cdots &{\varphi ({e_{nn}})} \end{array}} \right),\]$ where $\[\{ {e_{ij}}\} \]$ is the matrix unit of $\[{M_n}(C)\]$. Let $\[\alpha \]$ be the natural action of $\[SU(n)\]$ on $\[{M_n}(C)\]$ For $\[n \ge 3\]$, if $\[\Phi \]$ is an $\[\alpha \]$-invariant affine isomorphism between $\[{\chi _A}\]$ and $\[{\chi _B}\]$, $\[\Phi (0) = 0\]$, then $A$ and $B$ are $\[^*\]$-isomorphic In this paper a counter example is given for the case $\[n = 2\]$.     

GLOBAL EXISTENCE OF THE SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear parabolic equation: $$\[\left\{ {\begin{array}{*{20}{c}} {{u_t} = \Delta u + F(u,{D_x}u,D_x^2u),(t,x) \in {B^ + } \times \Omega ,}\{u(0,x) = \varphi (x),x \in \Omega }\{u{|_{\partial \Omega }} = 0} \end{array}} \right.\]$$ where $\[\Omega \]$ is the exterior domain of a compact set in $\[{R^n}\]$ with smooth boundary and F satisfies $\[\left| {F(\lambda )} \right| = o({\left| \lambda \right|^2})\]$, near $\[\lambda = 0\]$. It is proved that when $\[n \ge 3\]$, under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, It is also proved that the solution has the decay property $\[{\left\| {u(t)} \right\|_{{L^\infty }(\Omega )}} = o({t^{ - \frac{n}{2}}})\]$, as $\[t \to + \infty \]$.     

ON THE RELATIVE POSITION OF LIMIT CYCLES FOR THE EQUATION OF TYPE $\[{(II)_{l = 0}}\]$

In this paper, we consider the relative position of limit cycles for the system $$\[\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}} \end{array}\]$$ under the condition $$\[a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$ The main result is as follows: (i)Under Condition (2), if $\[\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\[{(1)_{{\delta _0}}}\] $ has no limit cycles and on singular closed trajectory through a saddle point in the whole plane, (ii)Under condition (2), the foci 0 and R'' cannot be surrounded by the limit cycles of system (1) simultaneously.     

ON THE DIOPHANTINE EQUATION $\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$

In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.