在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.     

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   

限制零的m进制数集之Hausdorff测度

对于任意整数表示mkz的分数部分.给出了数集Fm的Hausdorff测度是Fm的HausdorfF维数.     

Series Representation of Daubechies' Wavelets

1,Iotroduction.InthispaPerwe8tudytherepresentationofDaubechies'wavelets.DaubechiesI1]constructedaf4milyofcompartlysupportedregularscallngfUnctionsrk.(x)andtheassoci4tedregularwpeletsop.(x)(N32):where4.eL'(R)definedbythep0lyn0mia:withZq.(k)=1'q.(k)ER,k=0,1,')N-1.Itisknownthat[1]f0reachN32,k=Osuppgh.=[0,2N-l],suppop.=[-(N-1),N]andthewaveletop.generatesbyitsdilatiOnsandtranslati0nsan0rth0rn0rmalbasis{m.(2ix-k)}i,k6Z0fL'(R).Thefunctionsrk.andop.havebeenprovedtobeveryusefulinnumericalanal…     

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.     

一类复线性微分差分方程亚纯解的唯一性

本文主要研究一类复线性微分差分方程超越亚纯解的唯一性.特别地,假设$f(z)$为复线性微分差分方程: $W_{1}(z)f''(z+1)+W_{2}(z)f(z)=W_{3}(z)$的一个有穷级超越亚纯解,其中$W_{1}(z)$, $W_{2}(z)$, $W_{3}(z)$为增长级小于1的非零亚纯函数并且满足$W_{1}(z)+W_{2}(z)\not\equiv 0$.若$f(z)$与亚纯函数$g(z)$, $CM$分担0,1,$\infty$,则$f(z)\equiv g(z)$或$f(z)+g(z)\equiv f(z)g(z)$或$f^{2}(z)(g(z)-1)^2+g^{2}(z)(f(z)-1)^2=g(z)f(z)(g(z)f(z)-1)$或存在一个多项式$\varphi(z)=az+b_{0}$使得$f(z)=\frac{1-e^{\varphi(z)}}{e^{\varphi(z)}(e^{a_{0}-b_{0}}-1)}$与$g(z)=\frac{1-e^{\varphi(z)}}{1-e^{b_{0}-a_{0}}}$,其中$a(\neq 0)$, $a_{0}$ $b_{0}$均为常数且$a_{0}\neq b_{0}$.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is $y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. We find that a Newton-Kantorovich iteration always converges from an $m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x), \xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to discretize $x$. The Lane-Emden equation has branch point singularities at the endpoint $x=1$ whenever $m$ is not an integer; we show that the Chebyshev coefficients are $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a Chebyshev truncation of $N=100$ always gives at least ten decimal places of accuracy — much more accuracy when $m$ is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table.     

The evaluation and application of some modified moments

Recurrence formulas for the calculation of the modified moments $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta T_n (x)dx} $$ and $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta \ln \left( {\frac{{1 + x}}{2}} \right)T_n (x)dx} $$ are presented. Some applications of these modified moments are discussed, such as the numerical calculation of integrals of functions having branch points, the computation of Chebyshev series coefficients and the construction of Gaussian quadrature formulas for integrals with logarithmic singularity.     

单位圆上解析系数的高阶复微分方程

The growth of solutions of the following differential equation ■ is studied, where A_j(z) is analytic in the unit disc D = {z : |z| 1} for j = 0, 1,..., k-1. Some precise estimates of [p, q]-order of solutions of the equation are obtained by using a notion of new[p, q]-type on coefficients.     

非齐次线性微分方程解的增长性

研究了非齐次线性微分方程f^{(k)}+A_{k-1}(z)f^{(k-1)}+...+A_{s}(z)f^{(s)}+...+A_{0}(z)f=F(z) 解的增长性,其中A_{j}(j=0,1,\cdots,k-1)及F是整函数. 在A_{s}比其他系数有较快增 长的情况下,得到了上述非齐次微分方程在一定条件下的超越整函数解的超级的精确估计.     

图P2r,2m的优美标号

设u,v是两个固定顶点,用b条内部互不相交且长度皆为a的道路连接u,v所得图用P_(a,b)表示．K．M．Kathiresan证明：P_(2r,2m-1)(r,m皆为任意正整数)是优美的,且猜想,除了(a,b)=(2r 1,4s 2)外,所有的P_(a,b)都是优美的．杨元生已证明P_(2r 1,2m-1)是优美的,并且证明了,当r=1,2,3,4,5,6,7时,P_(2r,2m)也是优美的．作者证明：r为任意奇数时,P_(2r,2m)也是优美的．     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

Higher Order Turán Inequalities

The celebrated Turán inequalities , where denotes the Legendre polynomial of degree , are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities , which hold for the Maclaurin coefficients of the real entire function in the Laguerre-Pólya class, .

     

一类具梯度项和 源的抛物方程有界弱解的存在性

我们考虑了一类原型为$$\begin{cases}u_t-\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma|\nabla u|^2-\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega,\end{cases}$$的一类抛物方程. 其中, 函数$|\overrightarrow{b}(x,t)|^2,|\overrightarrow{F}(x,t)|^2,f(x,t)$位于空间$L^r{(0,T;L^q(\Omega))}$, $\gamma$是一个正常数. 在源项和梯度的系数项在空间$L^r{(0,T;L^q(\Omega))}$具有合适的可积条件下, 本文的目的在于证明先验的$L^\infty$估计以及方程存在有界解. 主要的方法包括通过正则化建立扰动问题, 用非线性的检验函数实现Stampacchia迭代技术以及极限过程中的紧性论断.     

系数的级相同的高阶微分方程解的增长性

In this paper,we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations.It is proved that under some conditions for entire functions F,A_(ji) and polynomials P_j(z),Q_j(z)(j=0,1,…,k-1;i=1,2)with degree n≥1,the equation f~(k)+(A_(k-1,1)(z)e~(p_(k-1)(z))+A_(k-1,2)(z)e~(Q_(k-1(z)))/~f~(k-1)+…+(A_(0,1)(z)e~(P_o(z))+A_(0,2)(z)e~(Q_0(z)))f=F,where k≥2,satisfies the properties:When F ≡0,all the non-zero solutions are of infinite order;when F=0,there exists at most one exceptional solution fo with finite order,and all other solutions satisfy λ(f)=λ(f)=σ(f)=∞.     

On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.