一類三角和數的表示式及其近似估值

<正> §1.本文的主要目的是要證明下面的結果: 定理1.設q_1,q_2為二正整數.合     

ON THE MYSOVSKICH THEOREM OF NEWTONS METHOD

In a real or oompbx Banach space X, let P be an operator with Lipsohitz continnous Frechet derivative P', and \[{X_*} \in X\] such that \[P({X_*}) = 0\] and \[{P^'}{({X_*})^{ - 1}}\] exists. It is shown that a ball with center \[{X_*}\] and best possible radius such that the theorem of Mysoyskich guarantees convergenee of Newton's method to \[{X_*}\] starting from any point \[{x_0}\] in ihe ball (theorem 3). In comparison with the corresponding results of Rall's work on Kantorovich theorem, the radius obtained is smaller than that from Kantorovich theorem. Therefore we suggest here an improved form of Mysoyskich theorem (theorem 1) . Thus, the corresponding value of the radius is augmented beyond that from Kantorovich theorem (theorem 2).     

LIMIT THEOREM FOR A CLASS OF SEQUENCE OF WEAKLY DEPENDENT RANDOM VARIABLES

In this paper, we consider a central limit theorem for the sequence of stationary m-dependent random variables, the variance of which is possibly infinite. Theorem. Let {Xn, n=l, 2,...} be a sequence of stationary m-dependent random variables with means zero. The following conditions are satisfied. (i) \[{M^2}\int_{{\text{|}}{X_1}| > M} {dP} /\int_{{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] (ii) \[\int_{\{ {X_1}| < M,|{X_i}| < M} {X_1^{}} {X_i}dP/\int_{|{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] then there are constants Bsubsub>0, such that \[\frac{1}{{{B_n}}}\sum\limits_{i = 1}^n {{X_1}} \] converges in distribution N(0,1).     

A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the same type lemma, which was proved by Bárány and Valtr.     

论局部乘积与POINCARE-ALEXANDER-LEFSCHETZ型对偶定理

<正> 在§1我们界说了局部乘积,它关联 Hausdorff 紧致空间 X 中闭子集X_0的同调以及 X_0在 X 中邻域的同调.在流形与有边流形上的 Poincaré-Alexander-Lefschetz 型对偶定理可以用这种局部乘积表示(§2).在§3,我们研讨了一类所谓摹流形状空间.局部的下调群与上调群的概念在 [3,233—263页;8]中曾不明显地使     

自正则化大偏差的一个注记

设$X_1,X_2,\cdots$为一列独立同分布的随机变量序列\bd 邵(1997)在没有任何矩条件下建立了自正则化大偏差定理, 但其上界的证明相当复杂\bd 为此, 本文给出了一个简洁的证明     

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

Classification error in multiclass discrimination from Markov data

As a model for an on-line classification setting we consider a stochastic process \((X_{-n},Y_{-n})_{n}\), the present time-point being denoted by 0, with observables \(\ldots ,X_{-n},X_{-n+1}, \ldots , X_{-1}, X_0\) from which the pattern \(Y_0\) is to be inferred. So in this classification setting, in addition to the present observation \(X_0\) a number l of preceding observations may be used for classification, thus taking a possible dependence structure into account as it occurs e.g. in an ongoing classification of handwritten characters. We treat the question how the performance of classifiers is improved by using such additional information. For our analysis, a hidden Markov model is used. Letting \(R_l\) denote the minimal risk of misclassification using l preceding observations we show that the difference \(\sup _k |R_l - R_{l+k}|\) decreases exponentially fast as l increases. This suggests that a small l might already lead to a noticeable improvement. To follow this point we look at the use of past observations for kernel classification rules. Our practical findings in simulated hidden Markov models and in the classification of handwritten characters indicate that using \(l=1\), i.e. just the last preceding observation in addition to \(X_0\), can lead to a substantial reduction of the risk of misclassification. So, in the presence of stochastic dependencies, we advocate to use \( X_{-1},X_0\) for finding the pattern \(Y_0\) instead of only \(X_0\) as one would in the independent situation.     

Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4

Siberian Mathematical Journal - Using the extension method, we study the one-parameter symmetry groups of the heat equation ?tp = Δp, where $$\Delta=X_1^2+X_2^2$$ is the sub-Laplacian...     

一类无穷阶退化抛物方程解的存在性

设X=(X_1,...,X_m)是一组无穷阶退化向量场,Δ_X=Σ_(j=1)~m X_j~*X_j,其中X_j~*是X_j的形式自伴算子.运用不动点理论得到抛物方程u_t=Δ_Xu+u log|u|解的存在性.     

σ-满正规空间的逆极限

本文证明:设X是逆系统(X_απ_β~α,A}的逆极限,|A|=λ,假设每个投射π_α:X→X_α是开且到上的。X是λ-仿紧和λ-可遮的,如果每个X_α是σ-满正规的(可遮的,σ-集体正规的),则X是σ-满正规的(可造的,σ-集体正规的)。作为这一结果的推论,我们还将证明正规σ-满正规性满足如文[1]中的通常形式的逆极限定理及遗传σ-满正规性的类似结果。     

Some properties of submatrices in a solution to the matrix equations AX=C, XB=D

Suppose that AX=C, XB=D has a common solution and partition its solution $X=\bigl[{\fontsize{7.5}{9}\selectfont \begin{array}{cc}X_{1}&X_{2}\\X_{3}&X_{4}\end{array}}\bigr]$ . In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in a solution X to matrix equations AX=C, XB=D. In addition, we investigate the uniqueness and the independence of submatrices in a solutions X to this equations.     

U-统计量的精致渐近性

设{X_n．n≥1}是一非退化的i．i．d．随机变量序列,U_n是以二维Borel可测对称函数h(x,y)为核函数的U-统计量．记U_n=2/(n(n-1))Σ_≤i≤j≤nh(X_i,X_j)．本文分别在核函数h(x,y)只有4/3阶矩或4/3+δ,0＜δ≤1的情况下,对非常广泛的一类权函数(x)与边界函数b(x)得到了如下关于U-统计量U_n的精致渐近性:不仅使得已有的结果成为我们的特况,还大大降低了其中的矩条件．     

Equations and rational points of the modular curves $$X_0^+(p)$$

Let p be an odd prime number and let \(X_0^+(p)\) be the quotient of the classical modular curve \(X_0(p)\) by the action of the Atkin–Lehner operator \(w_p\). In this paper, we show how to compute explicit equations for the canonical model of \(X_0^+(p)\). Then we show how to compute the modular parametrization, when it exists, from \(X_0^+(p)\) to an isogeny factor E of dimension 1 of its Jacobian \(J_0^+(p)\). Finally, we show how to use this map to determine the rational points on \(X_0^+(p)\) up to a large fixed height.     

Irreducibility criteria for compositions of multivariate polynomials

We provide irreducibility criteria for compositions of multivariate polynomials over a field K, of the form \({f(X_{1},\ldots ,X_{r-1},g(X_{1},\ldots ,X_{r}))}\), with \({f,g\in K[X_{1},\ldots ,X_{r}]}\), for the case that \({f}\) as a polynomial in X_r is irreducible over \({K(X_{1},\ldots ,X_{r-1})}\) and has leading coefficient divisible by a power of an irreducible polynomial \({p(X_{1},\ldots ,X_{r-1})}\) of sufficiently large degree with respect to \({X_{r-1}}\).     

非平稳高斯序列超过数点过程与部分和的联合渐近分布

设$\{X_{i}\}^{\infty}_{i=1}$是标准化非平稳高斯序列, $N_{n}$为$X_{1},X_{2},\cdots,X_{n}$对水平$\mu_{n}(x)$的超过数形成的点过程, $r_{ij}=\ep X_{i}X_{j}$, $S_{n}=\tsm_{i=1}^{n}X_{i}$. 在$r_{ij}$满足一定条件时, 本文得到了$N_{n}$与$S_{n}$的渐近独立性.     

$l^{0}(\{X_{i}\})$型赋$F$-范空间的共轭空间的表示定理

作者在《数学学报》(2016, {\bf 59}(4))上的一篇文章中, 给出了几个$l^{0}$型赋$F$-范空间的共轭空间的表示定理. 对于赋范空间序列$\{X_{i}\}$,本文研究$l^{0}(\{X_{i}\})$型赋$F$- 范空间的共轭空间的表示问题,得到代数表示连等式$\left(l^{0}(\{X_{i}\})\right)^{*}\stackrel{A}{=}\left(c^{0}_{00}(\{X_{i}\})\right)^{*}\stackrel{A}{=}c_{00}(\{X^{*}_{i}\})$,$$\left(l^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c_{0}^{0}(X)\right)^{*}\stackrel{\mathrm{A}}{=}\left(c^{0}_{00}(X)\right)^{*}\stackrel{\mathrm{A}}{=}c_{00}(X^{*}),$$以及序列弱星拓扑下的拓扑表示定理$\left(c^{0}_{00}(\{X^{*}_{i}\}),sw^{*}\right)=c^{0}_{00}(\{X^{*}_{i}\})$. 对于内积空间序列与通常拓扑下的数域空间序列,文章最后给出了基本表示定理的具体表现形式.     

截断和随机乘积的渐近性质

对一列独立同分布平方可积的随机变量序列{Xn,n≥1},当随机变量的分布具有中尾分布时,讨论了其截断和Tn(a)的随机乘积的渐近正态性质,其中Tn(a)=Sn-Sn(a),n=1,2,…,Sn(a)=n∑ j=1 XjI{Mn-a＜Xj≤Mn},a为某一大于零的常数'Mn=max 1≤k≤n{Xk}.     

Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk

Queueing Systems - We consider a discrete-time d-dimensional process $$\{{\varvec{X}}_n\}=\{(X_{1,n},X_{2,n},\ldots ,X_{d,n})\}$$ on $${\mathbb {Z}}^d$$ with a background process $$\{J_n\}$$ on a...     

Narrow operators and the Daugavet property for ultraproducts

We show that if T is a narrow operator (for the definition see below) on or , then the restrictions to X₁ and X₂ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.