二维输运方程离散格式的对称性

讨论了二维柱几何非定态中子输运方程离散格式的对称性问题，在几何空间和相空间连续的情况下，证明了时间离散方程的一维球对称性；而在时间和相空间离散的情况下，阐述了格式不具有一维球对称性；对时间和相空间离散情况下的几何空间间断有限元方程，得到了左右对称性。     

二维Master均值方程Backward的形式

如果二维Backward Master平均值方程的转移概率可用参数展开,我们证明了该方程可转化成一系列闭合的线性微分方程组,利用偏微分方程的特征方程组解法,在某种条件下我们求出了均值;将所得结果与线性噪音近似的结果比较,可得扩散系数Dx,Dsy,Dy;最后证明了一般情况下线性噪音近仿是不适合的.     

求二维DKdV方程精确解的方法

The exact soliton wave solution for the two-dimensional Korteweg-de Vires-Burgers equation is obtaind via introducing nonlinear transformations.This method,which is different from reference[1],here is very concise and primary.The metheod in this letter can be applied to other nonlinear equations.     

一类二维非线性Schroedinger方程的吸引子及其维数

本研究了一类二维非线性Schrodinger方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

二维Fredholm方程的Taylor展开式解法

本文讨论了一类二维Fredholm方程的一种近拟解,通过利用二元函数的Taylor展开式,积分方程转化成一个关于未知函数及其相应的偏导数的线性代数方程组.数值例子表明了该方法的有效性.     

一类二维对偶积分方程的解及其应用

二维对偶积分方程的理论与方法,在数学上尚未建立,因而完全的分析解不可能得到,从而使一些力学、物理与工程问题无法求解．利用双重展开和边界配置方法,得到了在数学和物理学上有着广泛应用的一类二维对偶积分方程的解答．把二维对偶积分方程化简成无限代数方程组,此方法的精确度取决于计算点的配置（即所谓边界配置）．通过对固体力学中某些复杂的初值-边值问题的应用说明此是方法有效的．     

求解二维Fisher-KPP方程的一类保正保界差分格式及其Richardson外推法

本文研究求解二维Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP)方程的一类保正保界差分格式.运用能量分析法证明了当网格比满足$R_{x}+R_{y}+[b\tau (p-1)]/2\leq\frac{1}{2}$时差分解具有一系列数学性质,包括保正性、保界性和单调性,且在无穷范数意义下有$O (\tau+h_{x}^{2}+h_{y}^{2})$的收敛阶.然后通过发展Richardson外推法得到收敛阶为$O (\tau^{2}+h_{x}^{4}+h_{y}^{4})$的外推解.最后数值实验表明数值结果与理论结果相吻合.值得提及的是在运用本文构造的Richardson外推法时对时空网格比没有增加更严格的条件.     

一类非线性二维奇异积分方程

本文研究复平面单位圆域内一类非线性二维奇异积分方程的可解性。文中应用泛函分析方法,在某些假设条件下,我们得到了此类非线性方程可解的几个充分条件,同时给出方程的解的表示式。     

二维RLW方程的Cauchy问题

通过椭圆积分求出了二维RLW方程椭圆余弦波解,并用先验估计方法证明了该方程Cauchy问题关于小x、y周期解的若干性质和解的唯一性、稳定性。     

二维正交变换的一些性质

本文结合解析几何和高等几何的教学,研究二维正交变换的一些性质。在一般解析几何和高等几何教程中,多是先给出正交变换的几何定义,然后求它的代数表达式。本文则利用线性变换代数表达式的特点来定义正交变换,用代数法研究其性质。     

Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.     

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.     

(D6,Z2)-等变分歧问题的识别

利用奇点理论中光滑映射芽的接触等价，研究状态变量和分歧参数均具有对称性的等变分歧问题，给出了状态变量具有D。对称性，分歧参数具有Z2对称性的等变分歧问题的两个识别条件．     

D(2)-点可区别正常边色数的一个上界

图G的D(β)-点可区别正常边染色是指G的一个正常边染色f使得对任意两点u,v∈V(G),0相似文献   

图的D(2)-点可区别一般边染色

引入了图的D(β)-点可区别一般边染色,并对β=2的情形做了讨论,得到了路,圈,星,双星,扇,轮的D(2)-点可区别一般边色数,对于2距离色数等于3及4的图的D(2)-点可区别一般边色数做了探讨,特别研究了具有稳定2距离4着色的图的D(2)-点可区别一般边染色.文中提出了一个相关猜想和一个公开问题.     

Geometric Integration of Ordinary Differential Equations on Manifolds

This article illustrates how classical integration methods for differential equations on manifolds can be modified in order to preserve certain geometric properties of the exact flow. Projection methods as well as integrators based on local coordinates are considered. The central ideas of this article have been presented at the 40th anniversary meeting of the journal BIT.This revised version was published online in October 2005 with corrections to the Cover Date.     

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

The author studies the 2D isentropic Euler equations with the ideal gas law. He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time 1 3-H¨older bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. The proof of the solutions, following the 3D construction of Buckmaster, Shkoller and Vicol (in 2023), is based on the stable 2D self-similar Burgers profile and the modulation method.     

The Level Set Method for Systems of PDEs

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996 Evans , L. C. ( 1996 ). A geometric interpretation of the heat equation with multivalued initial data . SIAM J. Math. Anal. 27 ( 4 ): 932 – 958 .[Crossref], [Web of Science ®] , [Google Scholar]) for the heat equation and by Giga and Sato (2001 Giga , Y. , Sato , M.-H. ( 2001 ). A level set approach to semicontinuous viscosity solution for Cauchy problems . Comm. Partial Differential Equations 26 ( 5–6 ): 813 – 839 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for Hamilton–Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each nonzero sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction–diffusion equations.     

W2^2（D）空间第一类算子方程近似解

０引言本文在具有再生核的Ｓｏｂｏｌｅｖ空间Ｗ＿２~２（Ｄ）中研究第一类算子方程的近似求解问题,本文利用再生核空间技巧得到的方程（１）的近似解ｕｎ（Ｍ）有如下特点：１近似解ｕｎ（Ｍ）的构成只需用到ｆ在有限个节点｛Ｗｉ｝１ｎ　Ｄ上的值;２当｛Ｍｉ｝１在Ｄ中稠密时,可以得到方程解析解ｕ（Ｍ）的表达式,并且而且误差随节点个数ｎ的增加接空间范数单调下降．数值算例表明,该方法是有效的．1Ｗ22（Ｄ）空间及其再生核记Ｄ=［ａ,ｂ］×［ｃ,ｄ］为有限区域,设是Ｄ上一组互异节点系．在［１］中,设Ｗ２２［ａ,ｂ］＝｛ｕ…