关于丢番图方程x^2±2^m=y^n

本文证明了：方程ｘ２＋２ｍ＝ｙｎ，ｘ，ｙ，ｍ，ｎ∈Ｎ，ｇｃｄ（ｘ，ｙ）＝１，ｎ＞２仅有有限多组解（ｘ，ｙ，ｍ，ｎ），而且当（ｘ，ｙ，ｍ，ｎ）≠（５，３，１，３），（１１，５，２，３），（７，３，５，４）时，ｎ是适合ｎ≡７（ｍｏｄ８）以及２３≤ｎ＜８．５·１０６的奇素数，ｍａｘ（ｘ，ｙ，ｍ）＜Ｃ１；方程ｘ２－２ｍ＝ｙｎ，ｘ，ｙ，ｍ，ｎ∈Ｎ，ｇｃｄ（ｘ，ｙ）＝１，ｙ＞１；ｎ＞２仅有有限多组解（ｘ，ｙ，ｍ，ｎ），而且这些解都满足ｎ＜２·１０９炉以及ｍａｘ（ｘ，ｙ，ｍ）＜Ｃ２，这里Ｃ１，Ｃ２是可有效计算的绝对常数．     

梦游“零王国”

小毅睡得正香，忽然被一阵“零零”的声音吵醒，他翻身起床，往外一看，啊呀，外面黑乎乎的。“零零”的声音越来越响，小毅顺着声音找去，不远处出现了一座巨大的椭圆形宫殿，宫殿里灯火辉煌。小毅连忙伸头往里看，忽然，宫殿里连蹦带跳跑出一个小孩。     

乘客分流问题

某一居民小区在新建时，每天有10万居民外出，乘坐的交通工具主要有两种类型：一种是公交车，另一种是出租车，通常情况下，当天乘坐公交车的居民，在第二天出行时，仍有95％的居民乘坐公交车，剩下5％的居民将改乘出租车，而当天乘坐出租车的居民，第二天全部改乘公交车，由于小区的规模仍在不断扩大，平均每天将增加105人，     

搜索区传递2-（q，4，1）设计

对于区传递但非旗传递的可解2-（q，4，1）设计，Camina指出，当q=13，37，61，109，157，181时有具体的例子，但是否有更多的q产生具体例子有待研究。主要结果：设q是素数幂且q=13（mod24），则对于每个q〈2000，总存在区传递但非旗传递的2-（q，4，1）设计。     

排列组合中五类易混淆的问题

排列组合问题求解方法独特，灵活多变，技巧性强．但若审题不严，思考不周密，则很容易出错，有时甚至“差之毫厘，失之千里”，本文通过姊妹型例题的形式列举常见的易混淆的五类似是而非的问题，供同学们参考．     

珠韵悠扬 逸趣横生——中国南通珠算博物馆简介

南通，吞江吐海，是长江与大海孕育出的神奇土地。这里，古为崇川福地，五千年的历史舒展成百里长卷。那穿城而过、波光粼粼的濠河水，赋予南通城灵性、生机与活力。泛舟水上，“城在水中坐，人在画中游”；漫步河滨，“烟笼十里柳翠，日照万树红花”。     

一道自主招生试题的思路设计与解法赏析

这是2012年一道自主招生数学卷压轴题，此题蕴含着初等数论的气息，散发出竞赛的味道，但真正考查的仍然是中学数学的主干知识、经典内容，它要求考生抓住问题的本质，才能产生思维的顿悟．笔者在研究中发现，这道考题给人以“横看成岭侧成峰”的感觉，运用不同的知识，从不同的角度思考，得到不一样的解法，体现出不同的能力要求，它全面检测...     

关于阶为2~（α1）3~（α2）5~（α3）7~（α4）p~（α5）的单群

本文证明了以下结果：设α１，α２，α３，α４，α５均为正整数，ｐ为素数且．如果Ｇ是阶为的单群，则Ｇ同构于下列单群之一：Ａ１１，Ａ１２；Ｍ２２，Ｈｉ－Ｓ２ＭｃＬ，Ｈｅ；Ａ１（ｑ）（ｑ＝２６，５３，７４，２９，４１，７１，２５１，４４９，４８０１），Ａ２（３２），Ａ３（２２），Ａ３（７），Ａ４（２），Ａ５（２），Ｂ２（２３），Ｂ２（７２），Ｂ３（３），Ｂ４（２），Ｃ３（３），Ｄ４（３），Ｇ２（２），Ｇ２（５），２Ａ２（１９），２Ａ３（５），２Ａ３（７），２Ａ４（３），２Ａ５（２），２Ｄ４（２）．     

精彩一题，多方探求

本题为江苏省徐州市2009—2010学年度高三第三次调研考试14题（填空题的最后一题），从考后统计的结果来看，全班60多人，只有不到10人正确，可定性为难题．从试题本身而言，取材质朴，即以基本初等函数中的二次函数为载体，学生不陌生，感觉亲近，可真正求解时，却发现不是想象中的那么好操作．从命题者的角度出发，有意加大填空题的难度，以适应2010届江苏高考数学考试说明的要求．     

关于Shannon－McMillan定理的若干研究

设｛Ｘｎ，ｎ≥１｝是字母集为Ｓ＝｛１，２，…，Ｎ｝上的任意信源，其分布为ｐ｛ｘ１，…，ｘｎ｝，（ｐｋ（１），ｐｋ（２），…，ｐｋ（Ｎ），ｋ＝１，２，…，是Ｓ上的一列分布　称为｛Ｘｋ，１≤ｋ≤ｎ｝相对于乘积分布　的熵密度偏差，本文利用这个概念研究任意信源的极限性质，其中包括在没有平稳性和遍历性假设情况下对ＳｈａｎｎｏｎＭｃＭｉｌｌａｎ定理的某些讨论．     

SL(3,3~n)和SU(3,3~n)的第一Cartan不变量

确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.     

An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

The Ramsey number N(3,3,3,3; 2) is the smallest integer n such that each 4-coloring by edges of the complete graph on n vertices contains monochromatic triangles. It is well known that 51 ≤ N(3, 3, 3, 3; 2) ≤ 65. Here we prove that N(3, 3, 3, 3; 2) ≤ 64.     

3-Kings in 3-partite tournaments

We present all possible distributions of 3-kings in 3-partite tournaments with at most one transmitter.     

A 3-Local Characterization of M12 and SL3(3)

We identify the sporadic simple group M₁₂ and the simple group SL₃(3) from some part of their 3-local structure. We also present a graph theoretic analogue of our main theorem. Received: 6 October 2008, Revised: 25 October 2008     

3-LIE BIALGEBRAS

3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.     

(3, 3)-Ary Differential Rings

The main purpose of this paper is to introduce the concept of (3, 3)-ary differential rings as a generalization of differential rings. Then we study ideals, prime ideals, radical and nil ideals of them. Finally, we give a construction of hyperstructures determined by (3, 3)-ary differential rings.     

3-Lie bialgebras

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.     

Metric packing for K 3+K 3

In this paper, we consider the metric packing problem for the commodity graph of disjoint two triangles K ₃+K ₃, which is dual to the multiflow feasibility problem for the commodity graph K ₃+K ₃. We prove a strengthening of Karzanov’s conjecture concerning quarterintegral packings by certain bipartite metrics.     

DEGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE

We give a local analytic characterization that a minimal surface in the 3-sphere S3 R4 defined by an irreducible cubic polynomial is one of the Lawson’s minimal tori.This provides an alternative proof of the result by Perdomo(Characterization of order 3 algebraic immersed minimal surfaces of S3,Geom.Dedicata 129(2007),23-34).