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|.     

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 $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$.     

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 GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES

Let X_1,…,X,be a sequence of p-dimensional iid.random vectors with a commondistribution F(x).Denote the kernel estimate of the probability density of F(if it exists)by_n(x)=n~(-1)h~_n(-p)K((x-X_i)/h_n)Suppose that there exists a measurable function g(x)and h_n>0,h_n→0 such thatlim sup丨f_n(x)-g(x)丨=0 a.s.Does F(x)have a uniformly continuous density function f(x)and f(x)=g(x)?This paperdeals with the problem and gives a sufficient and necessary condition for generalp-dimensional case.     

STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA

Let X_1,…,X_n be a sequence of independent identically distributed random variableswith distribution function F and density function f.The X_are censored on the right byY_i,where the Y_i are i.i.d.r.v.s with distribution function G and also independent of theX_i.One only observesLet S=1-F be survival function and S be the Kaplan-Meier estimator,i.e.,where Z_are the order statistics of Z_i and δ_((i))are the corresponping censoring indicatorfunctions.Define the density estimator of X_i by where =1-and h_n(>0)↓0.     

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 $\[p(z) = ({z^n} + {k^n})/(1 + {k^n})\]$.     

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.     

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].     

ON A THEOREM CONCERNING CARLESON MEASURE AND ITS APPLICATIONS

A measure μ is called Carleson measure,iff the condition of Carleson type μ(Q~*)≤C|Q|~α(a≥1)is satisfied,where C is a constant independent of the cube Q with edge lengthq>0 in R~n and Q~*={(y,t)∈R_+~(+1)|y∈Q,0相似文献   

ON A CONJECTURE OF F. NEVANLINNA CONCERNING DEFICIENT FUNCTION (II)

The paper proves on the basis of [1] the following theorem: Let $\[f(z)\]$ be an entire function of lower order $\[\mu < \infty \]$, and $\[{a_i}(z)(l = 1,2, \cdots ,k)\]$ be meromorphic functions which satisfy $\[T(r,{a_i}(z)) = o\{ T(r,f)\} \]$. If $$\[\sum\limits_{i = 1}^k {\delta ({a_i}(z),f) = 1\begin{array}{*{20}{c}} {({a_i}(z) \ne \infty )}&{(1)} \end{array}} \]$$ then the deficiencies $\[\delta ({a_i}(z),f)\]$ are equal to $\[\frac{{{n_1}}}{\mu }\]$, where $\[{n_i}\]$ is an integer,$\[l = 1,2, \cdots ,k\]$.     

ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS

Letf_v(z)=∑a_(v,,k)z~(λ_(v,k))(v=1,…,s)be s power series with algebraic coefficients a_(v,k),convergence radii R_v>0 and sufficientlyrapidly increasing integers λ_(v,k).It is shown that under certain conditions depending only ona_(v,k) and λ_(v,k),(i)f_1(θ_1),…,f_s(θ_s)are algebraically independent for arbitrary algebraicnumbers θ_1,…,θ_s with θ<丨θ_v丨相似文献   

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 $\[{\ln _2}x = \ln (\ln x)\]$.     

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 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.     

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.     

THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.     

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 $\[n = 2\]$.     

TIME OPTIMAL BOUNDARY CONTROL FOR SYSTEMS GOVERNED BY PARABOLIC EQUATIONS

In this paper we consider the systems governed, by parabolioc equations \[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\] subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\] We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control \[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time. The following maximum principle is proved: Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a solution to the equation \[\left\{ {\begin{array}{*{20}{c}} { - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0} \end{array}} \right.\] with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that \[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \]