Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

Allgemeine Mathematik: Mathematik und Realit?t Darmstadt, 29.9–1.10.1999

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Allgemeine Mathematik: Mathematik und Realit?t Darmstadt, 29.9–1.10.1999     

Corrigendum to the paper “Ovoidal packings of PG(3,q) for even q”

We point out that the proof of Theorem 3.3 of Bagchi and Sastry (2013) contains a serious flaw. Accordingly, this theorem needs to be modified. In consequence, we also have to retract Corollary 3.4, Corollary 3.6 and Theorem 3.8 of Bagchi and Sastry (2013).     

尺规作图助力数学探索撤回

数学课程于学生的理性思维发展、分析和解决问题能力的培养起着至关重要的作用.激发学生的数学学习兴趣和探索的欲望,要擅长发掘有意思的数学问题,将研究数学的乐趣带给学生;要培养学生分析问题、解决问题的能力,就要在平时的教学中关注分析问题、形成解决策略的过程,重视过程性教学,而不是结果导向的、单一性讲授式教学.在这个理念下,笔者设计了这样一节尺规作图的解题探究课,从两个路径出发,通过“强化”条件和“弱化”制定分级目标,辅以研究数学问题的通法帮助学生建构尺规作图题目的思考步骤,从而使学生领略尺规作图的魅力所在,完成数学思维的深度探索.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

关于Pell方程组x2-(m2-1)y2=z2-(n2-1)y2=1

设m,n,L为正整数,本文证明了:如果mε,ε∈(0,1),且m>(123789LL(1/2))⁽1/(1-ε)),或j>10.25×10¹²log⁴(2(L+1)(123789LL(1/2))⁽1/(1-ε))),Pell方程组x²-(m2-1)y²=z²-(n²-1)y²=1的正整数解满足1≤k≤δL²,这里δ∈[1/2(123787LL(1/2))⁽1/(ε-1)),1],以及■且j=k=1或k+2≤j<1/3(5-2ε)k,2|(j+k),k>3/(1-ε),并改进了文[Proc.Amer.Math.Soc.,2015,143(11):4685-4693]的结果.