AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.     

关于杨全会和陈永高提出的一个问题

对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.     

关于S\''{a}rk\"{o}zy和S\''{o}s问题的一个注记

令$k,\ell \geq 2$是正整数.令$A$是无限非负整数的集合.对$n\in \mathbb{N}$, 令$r_{1,k,\ldots,k^{\ell-1}}(A, n)$表示方程$n=a_0+ka_1+\cdots +k^{\ell-1}a_{\ell-1}$, $a_0, \ldots, a_{\ell-1}\in A$解的个数. 在本文中, 我们证明了对所有$n\geq 0$, $r_{1,k,\ldots,k^{\ell-1}}(A, n)=1$当且仅当$A$是$k^\ell$进制展开中数位小于$k$的所有非负整数的集合. 这个结果部分回答了S\''{a}rk\"{o}zy and S\''{o}s关于多维线性型表示的一个问题.     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

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.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.     

Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions

Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of $f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order $\lambda$ iff $\lambda_0\leq\lambda<1$ where $\lambda_0=0.654222...$ is the unique solution $\lambda\in(\frac{1}{2},1)$ of the equation $\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes the unit disk in the complex plane $\C$. This result is the best possible with respect to $\lambda_0$. In particular, it shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all $n\in\N,\ x\in[-1,1]$, and $0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000).     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

特殊集上素变数方程的解

令n=e_0+e_12+…+e_k2~k,其中e_j=0,1(j=0,…,k)表示自然数n的二进制展开式,N_0表示二进制展开式中项数为偶数的自然数的集合.分别给出了这个特殊集上素变数方程p_1+p_2+p_3~k=N和p_1+p_2~2+p_3~2+p_4~2=N解的个数的渐近公式.     

一个素变量丢番图问题

设k和r是满足k≥3及r≥Ψ(k)+1的正整数,这里当3≤k≤4时,Ψ(k)=2~(k-1);而当k≥5时,Ψ(k)=1/2k(k+1).假定δ和ε是给定的足够小的正数,λ_1,λ_2,…,λ_(r+1)是不全同号且两两之比不全为有理数的非零实数.对于任意实数η与0σ2~(1-2k)/r-1,证明了:存在一个正数序列X→+∞,使得不等式|λ_1p_1~k+λ_2p_2~k+···+λ_rp_r~k+λ_(r+1)p_(r+1)+η|(max(1≤j≤r+1)p_j)~(-σ)有》■X~(■-(2~(1-2k))/(r-1)+ε组素数解(p_1,p_2,…,p_(r+1)),这里(δX)~(1/k)≤p_j≤X~(1/k)(1≤j≤r)及δX≤p_(r+1)≤X.这改进了之前的结果.     

限制零点个数的亚纯函数的正规定则

设k,n(≥k+1)是两个正整数,a(≠0),b是两个有穷复数,F为区域D内的一族亚纯函数.如果对于任意的f∈F,f的零点重级大于等于k+1,并且在D内满足f+a[L(f)]~n-b至多有n-k-1个判别的零点,那么F在D内正规·这里L(f)=f~((k))(z)+a_1f~((k-1))(z)+…+a_(k-1)f'(z)+a_kf(z),其中a_1(z),a_2(z),…,a_k(z)是区域D上的全纯函数.     

On the Number of Integral Ideals in Two Different Quadratic Number Fields

Let K be an algebraic number field of finite degree over the rational field Q,and a K(n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of a K(n),Σn≤x a K(n)~l, l = 1, 2, 3, ···.This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum Σn≤x aK_1(n~j)~laK_2(n~j)~l, j = 1, 2, l = 2, 3, ···,where K_1 and K_2 are two different quadratic fields.     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

拟线性次椭圆方程组在Morrey空间上的部分正则性

证明了拟线性次椭圆方程组-X_α~*(a_(ij)~(αβ)(x,u)X_βu~j)=-X_α~*f_i~α+g_i,i=1,2,…,N,x∈Ω的弱解广义梯度Xu在Morrey空间L_x~(p,λ)(Ω,R~(mN))(p2)上的部分正则性,其中光滑实向量场族X=(X_1,X_2,…,X_m)满足H(o|¨)rmander有限秩条件,X_α~*是X_α的共轭;而且主项系数a_(ij)~(αβ)(x,u)关于x一致VMO(Vanishing Mean Oscillation的缩写,消失平均震荡)间断,且关于u为一致连续.     

一类连分数的有理逼近

设f(n)是非负函数,k,b,s_i,t_i(i=1,2,…)是正常数,研究形如[a_0,a_1,a_2…]=[■]_m~∞=0和[■]_n~∞=1的连分数有理逼近的下界．     

The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.