On Stability and Convergence of the Finite Difference Methods for the Nonlinear Pseudo-Parabolic System

In this paper, we deal with the finite difference method for the initial boundary value problem of the nonlinear pseudo-parabolic system $(-1)^Mu_t+A(x,t,u,u_x,\cdots,u_x 2M-1)u_x2M_t=F(x,t,u,u_x,\cdots,u_x 2M)$,$u_xk(o,t)=\psi_{0k}(t), u_xk(L,t)=\psi_{1k}(t),k=0,1,\cdots,M-1,u(x,0)=\phi (x)$ in the rectangular domain $D=[0\leq X\leq L,0\leq t\leq T]$, where $u(x,t)=(u_1(x,t),u_2(x,t),\cdots,u_m(x,t)),\phi (x),\psi_{0k}(t),\psi_{1k}(t),F(x,t,u,u_x,\cdots,u_x 2M)$ are $m$-dimensional vector functions, and $A(x,t,u,u_x,\cdots,u_x2M-1)$ is an $m\times m$ positive definite matrix. The existence and uniqueness of solution for the finite difference system are proved by fixed-point theory. Stability, convergence and error estimates are derived.     

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.     

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.     

多维局部平稳高斯过程最大值的联合渐近分布

$\{(X_1(t),\cdots,X_p(t)),0\leq t\leq T\}$为$p$维局部平稳高斯过程, 具有渐近中心化的均值$m_k(t)$和常数的方差, $M_k(T)=\sup\{X_k(t),0\leq t\leq T\},\;k=1,\cdots,p$, 当$T\rightarrow\infty$时, 本文在一定条件下获得了$M(T)=(M_1(T),\cdots,M_p(T))$的联合渐近分布.     

On the Number of Zeroes of Exponential Systems

A system $E:C^n\rightarrow C^n$ is said to be an exponential one if its terms are $ae^{im_1Z_1}.\cdots .e^{im_nZ_n}$. This paper proves that for almost every exponential system $E:C^n\rightarrow C^n$ with degree $(q_1,\cdots,q_n)$, $E$ has exactly $\Pi^n_j=1(2q_j)$ zeroes in the domain $D=\{(Z_1,\cdots,Z_n)\in C^n:Z_j=x_j+iy_j,x_j,y_j\in R,0\leq x_j<2\pi ,j=1,\cdots,n\}$, and all these zeroes can be located with the homotopy method.     

A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relations■hold as x→∞,where the random weightsθ_1,···,θ_(n )are bounded away both from 0 and from∞with no dependency assumptions,independent of the primary random variables X_1,···,X_(n )which have a certain kind of dependence structure and follow non-identically subexponential distributions.In particular,the asymptotic relations remain true whenX_1,···,X_(n )jointly follow a pairwise Sarmanov distribution.     

关于超越亚纯系数微分方程亚纯解的零点

本文研究了非齐次线性微分方程的复振荡问题，其中，D0，D1，…，D(k-1)，是超越亚纯函数．当存在某个Ds（1≤s≤k-1）比其它Dj（j≠s）有较快增长的意义下起支配作用时，得到了微分方程（I）亚纯解的零点收敛指数的精确估计式．     

On Stochastic Integrals for Strong Martingales with n-Dimensibhal Parameter

In this paper, we define PK(r)-predictable process $\Phi(\xi^1,\cdots,\xi^r)(\xi^q\in R_+^n,1\leq q \leq r \leq n)$ and its r-iterated stochastic integral with respect to a group of strong martingales $M=(M_1,\cdots,M_r)$ with n-dimensional parameter. We can express functional of strong martingale with proper properties by these stochastic integrals. Particularly, Square integrable functionals and martingale of Wiener process with 71-dimensional parameter are expressed by them so that the problem suggested by Wong, E. and Zakai, M, in[1] is solved with a method which differs from the method in[7],     

关于有限域上一类方程解的个数

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N(a1x12 + ···+anxn2 = bx1 ···xs) denote the number of solutions(x1,...,xn) of the equation a1x12 +···+ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F*q,b ∈ F*q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).     

有向循环图的支撑树数

设k≥2,1≤a_1相似文献   

有循环极大子群的素数幂阶群的作用是边传递的图(II)

假定Γ是一个有限的、单的、无向的且无孤立点的图,G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.结果为:设图Γ含有一个阶为pn(p是素数,n≥2)的自同构群,且G有一个极大子群循环,则Γ是G-边传递的,当且仅当Γ同构于下列图之一1)pmK1,pn-1-m,0≤m≤n-1;2)pmK1,pn-m,0≤m≤n;3)pmKp,pn-m-1,0≤m≤n-2;4)pn-mCpm,pm≥3,m＜n;5)2n-2K1,1;6)pn-1-mCpm,pm≥3,m≤n-1;7)2pn-mCpm,pm≥3,m≤n-1;8)2pn-mK1,pm,0≤m≤n;9)pn-mK1,2pm,0≤m≤n;10)pn-mK2,pm,0＜m≤n;11)C(2pn-m,1,pm);12)pkC(2pm-k,1,pn-m),0＜k＜m,0＜m≤n;13)(t-s,2m)C(2m 1/(t-s,2m),1,2n-1-m),其中0≤m≤n-1,2n-2(s-1)≡0(mod 2m),t≡1(mod 2),s(≠)t(mod 2m),1≤s≤2m,1≤t≤2n-1;14)∪p i=1 Ci p n-1,其中Ci p n-1=Ca1a1 [1 (i-1)pn-2]a 1 2[1 (i--1)p n-2]…a 1 (pn-1-1)[1 (i-1)p n-2]≌Cp n-1,i=1,2,…,p;15)∪2 i=1 Ci 2n-1,其中Ci 2n-1=Ca1a 1 [1 (i-1)(2n-2-1)]a1 2[1 (i-1)(2n-2-1)]…a1 (2n-1-1)[1 (i-1)(2n-2-1)]≌C2n-1,i=1,2.     

NASH EQUILIBRIUM OF TWO GAME MODELS ON THE RANDOM SOCIAL NETWORK

A graph $G$ is $k$-triangular if each of its edge is contained in at least $k$ triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1 T_2\cdots T_k$ in $G$ such that for $1\leq i\leq k-1, |E(T_i )\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j>i+1$. Two edges $e,e''\in E(G)$ are triangularly connected if there is a triangle-path $T_1,T_2,\cdots, T_k$ in $G$ such that $e\in E(T_1)$ and $e''\in E(T_k)$. Two edges $e,e''\in E(G)$ are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph $G$ is ${\mathbb Z}_3$-connected, unless it has a triangularly connected component which is not ${\mathbb Z}_3$-connected but admits a nowhere-zero 3-flow.     

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.     

图加一条边后的带宽和

令$BS(G,f)=\sum\limits_{uv\in E(G)}|f(u)-f(v)|$, 其中$f$为$V(G)\rightarrow\{1,2,\cdots,|V(G)|\}$的双射, 并称$BS(G)=\min\limits_{f}BS(G,f)$为图$G$的带宽和. 讨论顶点数为$n$的简单图$G$加上一条边$e\in\overline{E(G)}$后, 带宽和$BS(G+e)$与$BS(G)$的关系, 得其关系式$BS(G)+1\leq BS(G+e)\leq BS(G)+n-1$. 并证明此不等式中等号可取到, 即存在图$G_{1}$和$G_{2}$使得$BS(G_{1}+e)=BS(G_{1})+1$, $BS(G_{2}+e)=BS(G_{2})+n-1$.     

一类线性变换半群的格林关系

Let V be a linear space over a field F with finite dimension, L（V） the semigroup, under composition, of all linear transformations from V into itself. Suppose that V = V1 V2 ... Vm is a direct sum decomposition of V, where V1,V2,..., Vm are subspaces of V with the same dimension. A linear transformation f ∈ L（V） is said to be sum-preserving, if for each i （1 ≤ i ≤ m）, there exists some j （1 ≤ j ≤ m） such that f（Vi） Vj. It is easy to verify that all sum-preserving linear transformations form a subsemigroup of L（V） which is denoted by L （V）. In this paper, we first describe Green＇s relations on the semigroup L （V）. Then we consider the regularity of elements and give a condition for an element in L （V） to be regular. Finally, Green＇s equivalences for regular elements are also characterized.     

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…     

High dimensional reducible hyperplane sections with multigenera $\leq$ 1

Let X be an algebraic submanifold of the complex projective space $\mathbb{P}^N$ of dimension $n \geq 5$. We describe those $X \subset \mathbb{P}^N$ whose intersection with some hyperplane is a smooth simply normal crossing divisor $A_{1} + \cdots + A_{r}$ with $r \geq 2$ such that $g(A_{k}, L_{A_k}) \leq 1$ for $k=1,\ldots, r$.Received: 14 December 2001     

Partial Differential Equations of Higher Order and Symmetric Positive Systems

In this paper we study the relation between symmetric positive systems and equations of higher order. The main result is: Theorem 1. An equation of second order $L\phi =f$ can be transformed into a symmetric positive system by introducing new unknown functions $u_i=\sum\limits_{j=0}^n {\alpha_ij \varphi _j(i=0,1,\cdots,n),\varphi_0=\varphi_2,\varphi_j=\partial \varphi /\partial x_j}$ iff there exists L_1 of order 1 such that $Re(L_1 \varphi \cdot \bar {L\varphi})=\sum\limits_{i=1}^n{\frac{\partial}{\partial x_i}}+B(\varphi,\varphi)$, where P_i(\varphi,\varphi)(i=1,2,\cdots,n),B(\varphi,\varphi) are differential quadarlic forms and B(\varphi,\varphi) is positive definite. This Theorem can be extended into equations of higher order. Some examples of deducing equations of higher order into symmetric positive systems are given. Finally, we give a counter example which shows that a boundary problem of a symmetric positive system deduced from an equation of higher order is admissible, but its corresponding bounbary problem of the original equation is not well-posed.     

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本文考虑离散时间风险模型$U_n=(U_{n-1}+Y_n)(1+r_n)-X_n$,$n=1,2,\cdots$, 其中$U_0=x>0$为保险公司的初始准备金,$r_n$为在第$n$个时刻的利率, $Y_n$为到时刻$n$为止的总保费收入,$X_n$为到时刻$n$为止的所支付的全部索赔,$U_n$表示保险公司在时刻$n$的盈余. 当$Y_n$和$r_n$满足某些温和条件时,我们得到了在\, $x\to\infty$时,有限时间破产概率$\psi(x,N)=\pr\big(\min\limits_{0\leq n\leqN}U_n<0|U_0=x\big)$关于$N\geq1$的一致渐近的关系式\,$\psi(x,N)\sim\tsm_{k=1}^{N}\ol{F}_X((1+r_1)\cdots(1+r_n)x)$,其中$\ol{F}_X(x)$是$X_1$的尾分布.