非参数回归函数最近邻估计的强相合性

设(X,Y),(X₁,Y₁),…,(X_n,Y_n)为取值R^d×R的独立同分布随机向量,E|Y|<∞。设m_n(x)为m(x)=E(Y|X=x)的最近邻估计。本文在E|Y|^r<∞(对某个r>1)或E{exp(t|Y|^λ)}<∞(对某个λ>0及t>0)的条件下,建立了m_n(x)的强相合性。其它附加条件均与(X,Y)的分布无关。     

On the Pointwisise L_p Convergence Rates of Nearest Neighbor Estimate of Nonparametric Regression Function

Let (X,y) be a R~d×R~1-valued radom vector with E(?)=E(Y|X=x) be the regression funvion of Y with respect to X.Su (X_i, Y_i),i=1, …,n, are iid samples drawn ftc, from (X,Y).(?)desired to estimate m(x) based on these samples,everoye discussed in (?) the pointwise convergence of the nearest neigthoor estimate m_n(X) (see (5) of the present paper). Ih this article we further study the rate of this eonyer-gence.It is shown that if there exists p≥2 sach that E 、|Y|<∞,then for suitabte choice of the weighte C_m (see(4) of the present paper).     

一类具不变性质的变系数偏微分方程的特解

In this paper, a few properties of general patial differential operator P beinginvariant with respect to the form F(|x|_p^p+|y|_q^q-|z|_r^r) are studied, where x∈Rⁿ,y∈ R^m, z∈R^l.and explicit formulas are given for certain solution of the equation Pu=Aδ with P being a differential operator with power function coefficientswhich preserves the form (|x|_p^p+|y|_q^q-|z|_r^r)^1/v for arbitrary even integers p, q, r,and odd integers v.     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

概率密度的均匀核估计与近邻估计的均方误差

In this paper we consider the Mean Square Error (MSE) of two uaual estimates of density function f(x) at a point x: The uniform kernel estimate f_n(x) and the NN estimate fn(x). we- show that when f is differentiable for sufficiently high order at x. these MSE can be expanded in a form E(f_n(x)-f(x))²=A₁(x)n^-4/5 +A₂(x)n^-1+A₃(x)n^-6/5+…;E(f_n(x)-f(x))²=B₁(x)n^-4/5 +B₂(x)n^-1+B₃(x)n^-6/5+… And if we suitably choose the parameters in f_n and f_n to make A₁(x) and B₁(x)to assume its minimunm value, then we also, have A₂(x) =B₂(x) but A₃(X) differs form B₃(X). This result shows that while the two estimates are not identical with respect to MSE. each one can be superior to the other in various special cases.     

NN-模式识别中条件误差概率收敛性的充要条件

Let (θ₁,X₁),…, (θ_n,X_n), (θ, X) be iid random vectors ,where θ∈{0,1},X∈R^d Denote by θ′_n the nearest neighbour discriminator of θ based on the training samples (θ₁,X₁),…, (θ_n,X_n) and the observed X; put(?). This paper gives a sufficient and necessary condition for (?) as n→∞, namely (P(θ=0, X=x)-P(θ=1, X=x))²·P(θ=0, X=x)·P(θ=1, X=x)=0 for every x∈R^d.This generalizes a previous result of the authors [5] and improves a result of Wagner, T.J. [2].     

关于线性模型中误差方差估计中心极限定理余项的非一致性估计

in linear models, the error varianceσ2 of random errors is usually estimat-ted by the residual sum of squares (divided by a su ita ble degree of free-dom), based on the first n observation, (denote it by σ_n²). it is well known t that under certain conditions, the distribution of this estimate, when standardised, converges to the standard normal distribution. In this paper, it is shown that |G_n(x)-Ф(x)|=O(n^-δ/2(1+|x|)^-(2+δ)). when the errors are indepedent (maynot be identically distributed) and their 4 + 2δ order moments exist, where G_n(x) is the distribution of (σ_n²-σ²/(varσ_n²)^1/2,Ф(x)=1/(2π)^1/2∫_-∞^xe^-r2/2dt.     

关于环模的n-类比

In this paper, we define a kind of the n-versions of R-module, it is an Abelian n-group under the action of an m-ring, where m-1|n-1 or n-1|m-1, and is called a left R(m)-n-module. The category of left R(m)-n-modules is denoted by R_m -M_m¹;Theorem 1 An Abelian n-group M is a left R(n)-n-module with R =Z(n), where Z(n) = {1+s(n-1) :s∈Z}.Theorem 2 R_m - M_n¹ is an n-preadditive category.     

Local Well-posedness of the Derivative Schr¨odinger Equation in Higher Dimension for Any Large Data

In this paper, the authors consider the local well-posedness for the derivative Schr¨odinger equation in higher dimension u_t ? i△u + |u|²(?→γ · ?u) + u²(?→λ · ?u) = 0, (x, t) ∈ Rⁿ × R,?→γ ,?→λ ∈ Rⁿ; n ≥ 2.It is shown that the Cauchy problem of the derivative Schr¨odinger equation in higher dimension is locally well-posed in H^s(Rⁿ) (s > n/2) for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space H^n/2.     

极大高阶交换子的端点估计

该文主要确立了当b∈BMO 时, 极大高阶奇异积分算子交换子T_{b, m}* 满足如下不等式 |{y∈Rⁿ:T_{b, m}*f(y)>λ}|≤C||b||^m_BMO∫_Rⁿ|f(y)|/λ (1+log⁺|f(y)|/λ)^mdy 且T_{b, m}* 在L^p(Rⁿ)(1 < p <∞上有界.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 395–400, May–June, 1976.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.