关于极大强单调算子的不精确邻近点算法的收敛性分析

该文研究集值映象方程0∈T(z)的解的迭代逼近，其中T是极大强单调算子．设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列，满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了，{x^k}收敛到T的一个根当且仅当 lim inf_{k→+∞} d(x^k,Z)=0，其中Z是方程0∈T(z)的解集     

一类三角方程解集的简化

在解三角方程时,有一类三角方程的解集可以简化.首先看卜面的例子: 例:解方程:cos3x一cosx=0 解:原方程可化为:sinx sinZx=0.由sinx=0得解集为:{川x=k二,k(公.又由sinZx二0得解集为:{,},一夸,k〔团. 若说原方程的解集为:{x}x二k;r、k(才U lxlx=午,*〔才是欠佳的。由于*〔z.…*可表示为:k=Zn或k二Zn一l,(其中厅〔刀. 于是集合:{二}二=夸,乏〔公=伙}x二。二,。、团u{‘r}二=架井二,。。团即{二I、一*;r,*〔刁c十二4二一夸,*(刁.…{xI二一*,,k〔公u{二}二一夸,*〔刀={二I二一夸,*。二}.故原方程的解集为:{二}二一夸,k〔才 一般地.对形如sin…     

对流扩散方程的解

关注如下的对流扩散方程 $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}} $$ 的初边值问题. 若 $p>1+\frac{1}{m}$, 通过考虑正则化问题的解 $u_{k}$, 利用 Moser 迭代技巧, 得到了$u_{k}$ 的 $L^{\infty}$ 模与 梯度 $\nabla u_{k}$ 的 $L^{p}$ 模的局部有界性. 利用紧致性定理, 得到了对流扩散方程本身解的存在性. 若 $p<1+\frac{1}{m},\ p>2$ 或者 $p=1+\frac{1}{m}$, 利用类似的方法可以得到解的存在性. 证明了解的唯一性, 同时讨论了正性和熄灭性等解的性质.     

对数列通项最值问题的一个解法的思考

问题 已知数列{an}，求通项an的最大值或最小值及相应的n的值．     

一类线性循环数列的通项公式

定义:若数列{a_n}满足循环方程 a_n=C_1a_(n-1) C_2a_(n-2) ¨ C_ka_(n-k)其中n=k_1,k 2,…;C_k0,就称数列{a_n}是一个k阶线性循环数列。方程     

一个包含Smarandache原函数的方程

设p为素数,n为任意正整数,我们定义Smarandache原函数S_p(n)为最小正整数k,使得p~n|k!,即S_p(n)=min{k∈N:p~n|k!}．本文利用初等方法研究了方程S_p(1)+S_p(2)+…+S_p(n)=S_p((n(n+1))/2)的可解性,并给出了该方程的所有正整数解．     

一个包含Euler函数及k阶Smarandache ceil函数的方程及其正整数解

设k≥2为给定的整数．对任意正整数n,k阶Smarandache ceil函数Sk（n）定义为Sk（n）=min{x：x∈N,n｜x^k}．本文的主要目的是利用初等方法研究函数方程Sk（n）=Ф（n）的可解性,并给出该方程的所有正整数解,其中Ф（n）为Euler函数．     

正余弦级数和的几个公式及其应用

数学竞赛中很多三角函数的求和都涉及到数列{sinαk}、{cosαk}、{（-1）^k＋1sinαk}及{（-1）^k＋1cosαk}的求和题，其中数列{αk}为等差数列。这类问题的解法具有一定的灵活性。     

某类$q$差分方程亚纯解的性质

对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.     

有关正项等比数列的不等式

本文研究由正项等比数列若干项构成的不等式．为了简便起见，以下约定{an}是正项等比数列，公比为q（q〉0，且q≠1），m,n，k均为正整数，且优≠k．     

r-log-concavity of partition functions

The Ramanujan Journal - Let $$\hat{\mathscr {L}}$$ be the operator given by $$\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}$$ . A sequence $$\{ a_n \}_{n \ge 0}$$...     

在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

关于广义{\Phi}-增生算子的最速下降法的收敛定理

设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$[0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$.     

Oscillatory and asymptotic behavior of solutions of higher order damped nonlinear difference equations

The asymptotic and oscillatory behavior of solutions of mth order damped nonlinear difference equation of the form where m is even, is studied. Examples are included to illustrate the results.     

Banach空间中有限族渐进伪压缩映象的迭代程序

Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ [1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in [0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.     

Palindromes in linear recurrence sequences

We prove that for any base $b\ge 2$ and for any linear homogeneous recurrence sequence $\{a_n\}_{n\ge 1}$ satisfying certain conditions, there exits a positive constant $c>0$ such that $\# \{n\le x:\ a_n \;\text{ is} \text{ palindromic} \text{ in} \text{ base}\; b\} \ll x^{1-c}$ .     

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.     

Central limit theorem and infinitely divisible probabilities associated with partial differential operators

We provide

Oscillation of certain difference equations

Some new criteria for the oscillation of difference equations of the form

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .