群论界最权威的青年领军人物——伽罗瓦

为纪念伽罗瓦诞辰200周年,世界上众多数学机构开展学术沙龙或研讨活动.＂伽罗瓦虽然英年早逝,但却照亮了数学界一个不为人知的隐秘天地.＂2012年4月26日,在中科院武汉物理与数学研究所举办的中法交流学术沙龙上,来自法国图卢兹大学数学学院的让·皮埃尔·米斯（Jean-Pierre Ramis）教授在介绍法国天才数学家伽罗瓦时表示.在让·皮埃尔·米斯看来,作为法国数学界的瑰宝,伽罗瓦敢于以崭新的方式去思考,     

群论界最权威的青年领军人物——伽罗瓦

<正>为纪念伽罗瓦诞辰200周年,世界上众多数学机构开展学术沙龙或研讨活动."伽罗瓦虽然英年早逝,但却照亮了数学界一个不为人知的隐秘天地."2012年4月26日,在中科院武汉物理与数学研究所举办的中法交流学术沙龙上,来自法国图卢兹大学数学学院的让·皮埃尔·米斯(Jean-Pierre Ramis)教授在介绍法国天才数学家伽罗瓦时表示.在让·皮埃尔·米斯看来,作为法国数学界的瑰宝,伽罗瓦敢于以崭新的方式去思考,其创     

伽罗瓦内积下LCD常循环码和LCD MDS码的研究

伽罗瓦内积是欧式内积和厄尔米特内积的推广,常循环码是一类结构丰富而又应用广泛的线性码,MDS码被著名学者MacWilliams和Sloane称为最富有魅力的纠错码,LCD码被广泛的应用于数据存储,通信系统,电子和密码学等.文章将研究基于伽罗瓦内积下的LCD常循环码和LCD MDS码,重点讨论了有限域上常循环码的伽罗瓦对偶码的形式及伽罗瓦LCD常循环码的充要条件,并得到了三类特殊参数的伽罗瓦LCD MDS码.     

伽罗瓦环上的一类No序列的构造

近年来,伽罗瓦环上的序列理论成为人们研究的热点问题.有限域上的No序列是一类伪随机序列,它在序列密码中占具十分重要的角色.本文利用伽罗瓦环上的置换,构造了伽罗瓦环Z_(p~e)上的一类新的No序列,并且研究了其线性复杂度.研究的结果表明此类No序列具有相当大的线性复杂度.     

伽罗瓦与奇妙的“群”

埃瓦里斯特·伽罗瓦(1811-1832)是法国数学家.他一生只活到21岁,但这位数学史上的传奇式人物,却为数学增添了全新的思想.这些思想改变了代数学发展的进程,把它引向新的天地.     

代数构造及在Ramsey理论中的应用

这篇文章在伽罗瓦域上的代数构造和关于一些特定类型图的Ramsey数之间建立了一个关系. 研究了关于伽罗瓦域上的代数构造的方程及方程组的解. 我们得到了一些关于二部图的Ramsey数的新的下界和上界.     

环上的Hopf稠密伽罗瓦扩张（英文）

给定交换整环R,以及Hopf R-代数H,且H是一个有限生成的自由R-模.设A是一个R-代数,并且A是H-余模代数.如果自然映射β:A■_(A~(coH)) A→A■RH的余核是商有限的,则称A/A~(coH)是一个Hopf稠密伽罗瓦扩张.它是域上Hopf稠密伽罗瓦扩张的推广.本文证明了R上的Hopf稠密伽罗瓦扩张隐含了一个弱化的Auslander定理.此外,假设A是几乎可交换代数,且gr(A)是一个整环.如果A/A~(coH)是Hopf稠密伽罗瓦扩张,且自然映射β是严格的,本文证明了在此情形下,H在一个包含R的代数闭域上对偶于一个有限群代数.     

1954年苏联科学院在数学研究方面的成就

1954年数学家们继续研究着现代数学、理论物理和力学方面的很多最重要的问题。在解析数论方面,得出了维诺格拉多夫院士的熟知方法的一种新的、更简单的变体,使得对三维区域中整点数目的渐近公式中的馀项得出显著的改善。某些特殊的狄黎克雷L-函数的零点界限得到精密化。在逆伽罗瓦问题方面得到有趣的结果:在正面的意义上,解决了关于系数取自已知代数数域并具有预给的可解伽罗瓦韦的方程的存在问题。制定了场的量子理     

一位辛勤的开拓者——纪念孙本旺教授

从1982年秋被确诊为肠癌,孙本旺教授在病床上仍顽强拼搏了将近两年。只要一息尚存,时刻关心湖南省数学学会的活动和研究教育的情况;还身兼中国人民政治协商会议全国委员会委员、中国人民解放军国防科技大学副校长等重要职务,常从医院返校视事;并出版了《伽罗瓦理论》一书,将稿酬捐赠给母校南开大学姜立夫基金会,1983年     

基于格值逻辑的模糊概念格

从逻辑的角度,将非经典逻辑之一的格值逻辑引入概念格,建立了格值模糊形式背景,通过格结构来刻画对象与属性之间的模糊关系,证明了由蕴涵算子诱导的算子对是伽罗瓦连接,并讨论了相关的一些性质,进而给出了格值模糊概念格的构造算法.格值模糊概念格的建立为模糊性与不可比较性信息的处理提供了可靠的数学工具.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of Möbius transformations

By introducing the conception “relativistic differential Galois group” for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of Möbius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of Möbius transformations at the first author’s article published in this journal in 1996 are refreshed in this paper.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of Mbius transformations

By introducing the conception "relativistic differential Galois group" for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of Mobius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of Mobius transformations at the first author's article published in this journal in 1996 are refreshed in this paper.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of M(o)bius transformations

By introducing the conception "relativistic differential Galois group" for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of M(o)bius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of M(o)bius transformations at the first author's article published in this journal in 1996 are refreshed in this paper.     

Évariste Galois: Principles and Applications

Évariste Galois formulated his famous theory in 1831 in the first part of his Mémoire sur les conditions de résolubilité des équations par radicaux. It is titled Principes. Even though the theory is completely understood today, it is hard to follow Galois's original. The style is brief, almost aphoristic and the approach quite different from today's. The aim of this paper is to make Galois's Principes readable for contemporary mathematicians (Sections 1 and 2) and to give a survey of Galois's Applications concerning equations of prime degree, primitive equations, and the modular equation in the theory of elliptic functions (Section 3). Remarks show the relationship to the work of Lagrange and Gauss. © 2002 Elsevier Science (USA).Évariste Galois formulierte seine berühmte Theorie 1831 im ersten Teil seines Mémoire sur les conditions de résolubilité des équations par radicaux. Er trägt den Titel Principes. Auch wenn die Theorie heute vollkommen verstanden ist, ist es bisweilen schwer, Galois' ursprünglichen, äußerst knappen Formulierungen zu folgen. Ziel des Artikels ist es, Galois' Principes für heutige Mathematiker lesbar zu machen (Abschnitte 1 und 2) und einen Überblick über Galois' Applications zu geben, betreffend Gleichungen von Primzahlgrad, primitive Gleichungen und die Modulargleichungen aus der Theorie der elliptischen Funktionen (Abschnitt 3). Bemerkungen verweisen jeweils auf die nahe Beziehung zum Werk von Lagrange und Gauss. © 2002 Elsevier Science (USA).MSC subject classification: 12E10.     

Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of Livšic (Soobshch Akad Nauk Gruzin SSSR 91(2):281–284, 1978). Our work may be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm–Liouville differentiable equations on the half axis (0,∞) with singularity at 0. We also develop a rich and interesting theory of vessels with deep connections to the notion of the τ function, arising in non linear differential equations (LDE), and to the Galois differential theory for LDEs.     

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two-parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t.     

Construction inconditionnelle de groupes de Galois motiviquesAn unconditional construction of motivic Galois groups

We attach to any “classical” Weil cohomology theory over a field a motivic Galois group, defined up to an inner automorphism. We also study the specialisation of numerical motives and the behaviour of motivic Galois group by specialisation. To cite this article: Y. André, B. Kahn, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 989–994.     

Des extensions plus petites que leurs groupes de Galois

In this article we construct some Galois extensions L∕K with finite Galois groups and such that |Gal(L∕K)|>[L:K]. Using an analog of the Noether method, we explain how to obtain, with a fixed center, such a Galois curiosity with a Galois group as large as we want.

Résumé : Dans cet article, nous construisons une extension galoisienne L∕K à groupe de Galois fini et telle que |Gal(L∕K)|>[L:K]. En utilisant un analogue non commutatif de la méthode de Noether, nous expliquons ensuite comment, à centre fixé, l’on peut construire une telle curiosité galoisienne avec un groupe aussi gros que l’on veut.

Mots clés : Corps gauches; théorie de Galois.