Almost Sure Convergence of Nonparametric Begression Estimates

Let (X, Y), (X_i,Y_i), i=1,\cdots, n, be iid.R^d*R^1-ralued vandom vectors with E(|Y|) <\infinity and m(x) =E(Y|X=x) be the regression function. Select the weight functions W_ni(x) =W_ni(x; X_1,\cdots, X_n), and use m_n(x) =[\sum\limits_{i = 1}^n {{W_{ni}}(x){Y_i}} \] an estimator of m(x). This paper shows that [\mathop {\lim }\limits_{n \to \infty } \]m_n(X) =m(X), a. s., under weaker conditions.     

Oscillatory and Asymptotic Behavior of First Order Functional Differential Equations

In this paper the author discusses the following first order functional differential equations: $x''(t)+[\int_a^b {p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (1)$ $x''(t)+[\int_a^b {f(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (2)$ Some sufficient conditions of oscillation and nonoscillation are obtained, and two asymptotic properties and their criteria are given. These criteria are better than those in [1, 2], and can be used to the following equations: $x''(t)+[\sum\limits_{i = 1}^n {{p_i}(t)x[{g_i}(t)] = 0} \] (3)$ $x''(t)+[\sum\limits_{i = 1}^n {{f_i}(t)x[{g_i}(t)] = 0} \] (4)$     

Necessary Conditions of L_1-Convergenoe of Kernel Regression Estimators

Let (X_1,Y_1),\cdots,(X_n,Y_n) be iid. and R^d *R-valued samples of (X,Y). The kernel estimator of the regression function m(x)\triangleq E(Y|X=x) (if it exists), with kernel K, is denoted by $\[{m_n}(x) = \sum\limits_{i = 1}^n {{Y_i}K(\frac{{{X_i} - x}}{{{h_n}}})/\sum\limits_{j = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} } \]$ Many authors discussed the convergence of m_n(x) in various senses, under the conditions h_n\rightarrow 0 and nh_u^d\rightarrow \infinity asn\rightarrow \infinity. Are these conditions necessary? This paper gives an affirmative answer to this bprolemuithe case of L_1-conversence, when K satisfies (1.3) and E(|Y|log^+|Y|)<\infinity.     

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

ON THE DIOPHANUNE EQUATION $\[\sum\limits_{i = 0}^N {\frac{{{x_i}}}{{{d_i}}}} \equiv 0\]$ (mod 1) AND ITS APPLICATIONS

The number $\[A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\[\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\[{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\[N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\[{F_q}\]$ of the equation $$\[\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\[a_i^''s\]$ belong to $\[F_q^*\]$. the multiplication group of $\[F_q^{[1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\[A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\[A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\[{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\[A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$     

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     

Strong Convergence of the Kernel Estimates of Nonparametric Regression Functions

Let (X, Y), (X_1, Y_1),\cdots, (X_n, Y_n) be i. i. d. random vectors taking values in R_d\times R with E(|Y|)<\infinity, To estimate the regression function m(x)=E(Y|X=x), we use the kernel estimate $m_n(x)=[\sum\limits_{i = 1}^n {K(\frac{{{X_i} - x}}{{{h_n}}}){Y_i}/} \sum\limits_{i = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} \]$ where K(x) is a kernel function and h_n a window width. In this paper, we establish the strong consistency of m_n(x) when E(|Y|^p)<\infinity for some p>l or E{exp(t|Y|^\lambda)}<\infinity for some \lambda>0 and t>0. It is remakable that other conditions imposed here are independent of the distribution of (X, Y).     

A NECESSARY AND SUFFICIENT CONDITION THAT BIHOLOMORPHIC MAPPINGS ARE STARLIKE ON A CLASS OF REINHARDT DOMAINS

This paper studies the Reinhardt domains B defined as(?)The Schwartz lemma for B is established.Using it the authors give a necessary andsu fficient condition that a local biholomorphic mapping from B to C~n is starlike.It isreduced to the Suffridge's theorem in the case p_1=p_2=…=p_n>1.     

SINGULAR PERTURBATIONS FOR QUASILINEAR HYPERBOLIC EQUATIONS

This paper deals with the following mixed problem for Quasilinear hyperbolic equationsThe M order uniformly valid asymptotic solutions are obtained and there errors areestimated.     

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.     

A NECESSARY AND SUFFICIENT CONDITION FOR CONVERGENCE OF ERROR PROBABILITY ESTIMATES IN K-NN DISCRIMINATION

Let(X,θ)be R~d×{1,…,s}valued random vector,(X_j,θ_j),j=1,…,n,be its observedvalues, be the K-nearest neighbor estimate of θ_j,R~((K)) be the limit of error probabilityand be the error probability estimate.In this paper it is shown thatA_ε>0, constants α>0,c<∞ such thatif add only if there is no unregular atom of(X,θ)defined below and the various conver-gences R_(nk)→R~(k) are equivalent.     

On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation

In this paper, we study the behavior of the Fourier--Haar coefficients $a_{m_1 , \ldots ,m_n } \left( f \right)$ of functions $f$ Lebesgue integrable on the $n$ -dimensional cube $D_n = \left[ {0,1} \right]^n $ and having a bounded Vitali variation $V_{D_n } f$ on it. It is proved that $$\sum\limits_{m_1 = 2}^\infty \cdots \sum\limits_{m_n = 2}^\infty {\left| {a_{m_1 , \ldots ,m_n } \left( f \right)} \right|} \leqslant \left( {\frac{{2 + \sqrt 2 }}{3}} \right)^n {\text{ }}.{\text{ }}V_{D_n } f$$ and shown that this estimate holds for some function of bounded finite nonzero Vitali variation.     

The Stability of Systems of Neutral Type with Small Time Lag

In this paper, we have obtained the equivalence theorems of stability between the system of differential equations $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t)} (i = 1,2, \cdots ,n)\]$ and the system of differential-difference equations of neutral type $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t - {\Delta _{ij}})} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t - {\Delta _{ij}})} (i = 1,2, \cdots ,n)\]$ where a_ij, b_ij, c_ij are given constants, and \Delta_ij are non-negative real constants.     

The Existence Theorem for Solutions of General Boundary Value Problem of Quasi-Linear Second Order Elliptic Differential Equations

In this paper, we provide the existence theorem for solutions of general boundary value problem of quasi-linear second order elliptic differential equations in the following form: $\[\sum\limits_{i,j = 1}^n {({a_{ij}}(x,u)\frac{{\partial u}}{{\partial {x_j}}}) + a(x,u,{u_{{x_k}}}),{\rm{ }}in} {\rm{ }}\Omega \]$, $\[\alpha (x,u)\frac{{\partial u}}{{\partial \gamma }} + \beta (x,u) = 0,{\rm{ on }}\partial \Omega \]$, where \alpha(x, u) \geq 0,\alpha_u(x, u) \leq 0 and \gamma is some direction, defining on $\[\partial \Omega \]$.     

On best approximation of functions of two variables

$\mathop {\lim \sup }\limits_{r \to \infty } \frac{{E_{n_i ,m_i } (f)_L }}{{[E_{n_i ,\infty } (f)_L + E_{\infty ,m_i } (f)_L ]ln\{ 2 + min(n_i ,m_i )\} }}\underset{\raise0.3em\hbox{$\mathop {\lim \sup }\limits_{r \to \infty } \frac{{E_{n_i ,m_i } (f)_L }}{{[E_{n_i ,\infty } (f)_L + E_{\infty ,m_i } (f)_L ]ln\{ 2 + min(n_i ,m_i )\} }}\underset{\raise0.3em\hbox{     

SOME GENERALIZATION OF GRONWALL-BIHARI INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

The interest of this paper lies in the estimates of solutions of the three kinds of Gronwail-Bihari integral inequalities:(Ⅰ) y(x)≤f(x) sum from i=1 to n(g_i(x)integral from n=0 to x(h_i(d)y(s)ds)),(Ⅱ) y(x)≤f(x) g(x)φ(integral from n=0 to x(h(s)w(y(s))ds))(Ⅲ) y(x)≤f(x) sum from i=1 to n(g_i(x)integral from n=0 to a(h_i(s)y(s)ds g_(n 1)φ(integral from n=0 to x(h_(n 1)(s)w(y(t))ds)).The results include some modifications and generalizations of the results of D. Willett, U. D. Dhongade and Zhang Binggen. Furthermore, applying the conclusion on the above inequalities to a Volterra integral equation and a differential equation, the authors obtain some new better results.     

On Estimate of Complete Trigonometric Suns

In this paper, the following theorem is proved: Let f(x)=a_kx^k+\cdots+a_1x+a_0 be a polynomial with integral coefficients such that (\alpha_1,\cdots,\alpha_k,q)=1, where q is a positive integer. Then, for k\geq 3, $[|\sum\limits_{x = 1}^q {{e^{2\pi if(x)/q}}} | \le {e^{2k}}{q^{1 - 1/k}}\]$     

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 Existence of Solutions for First Order Quasilinear Symmetrizable Systenis of Partial Differential Equations

In this paper we study the first order quasilinear symmetrizable system of partial differential equations $\sum\limits_{i = 1}^n {{a_i}(x,u)\frac{{\partial u}}{{\partial {x_i}}} + \lambda u = f(x,u)}$ (1) where a_i(x,y) are k*k matrices.     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.