关于МОХОНЪКО不等式的精确性

证明了如下定理：设Φ（ｚ）＝∑ｎｕ＝１∏ｔｉ＝１ｆａｕｉ０ｉ…ｆ（ｋｉ）ｉａｕｉｋｉ∑ｍｖ＝１∏ｔｉ＝１ｆｂｖｉ０ｉ…ｆ（ｋｉ）ｉｂｖｉｋｉ其中ｆｉ（ｚ），（１≤ｉ≤ｔ）是亚纯函数，ａｕｉｊ，ｂｖｉｊ为非负整数，则有Ｔ（ｒ，Φ）≤∑ｔｉ＝１｛［Ｓｉ＋ｏ（１）］ｍ（ｒ，ｆｉ）＋［△ｉ－ｕｉ＋ｏ（１）］Ｎ（ｒ，ｆｉ）＋（△ｉ－Ｓｉ）Ｎ（ｒ，ｆｉ）｝ｒＥ∈Ｌ．它是МОХОНЪКО的一个定理的精确改进．     

部分临界点轨道收敛于无穷大时的多项式动力系统

设Ｐ（ｚ）是ｄ（≥２）次多项式，Ｊ是Ｐ（ｚ）的Ｊｕｌｉａ集，σ：∑ｎ→∑ｎ是ｎ个符号的单边符号空间∑ｎ上的转移自映射．本文证明了当ｐ（ｚ）的某ｍ（１≤ｍ≤ｄ－１）个有穷临界点的轨道收敛于∞时，ｐ｜Ｊ拓扑半共轭于σ：∑（ｍ＋１）→∑（ｍ＋１），而当ｍ＝ｄ－１时，ｐ｜Ｊ拓扑共轭于σ：∑ｄ→∑ｄ。     

关于МОФОНЪКО不等式的精确性

证明了如下定理：设Φ（ｚ）＝Σ↑ｎ↓ｕ＝１∏↑ｔ↓ｉ＝１ｆｉ＾ａ＾ｕｉ０…（ｆｉ＾（ｋｉ））＾ａｉｋｉ＾ｕ／Σ↑ｍ↓ｖ＝１∏↑ｔ↓ｉ＝１ｆｉ＾ｂ＾ｖｉ０…（ｆｉ＾（ｋｉ））＾ｂｉｋｉ＾ｖ其中ｆｉ（ｚ），（１≤ｉ≤ｔ）是亚纯函数，ａｉｊ＾ｕ，ｂｉｊ＾ｖ为非负整数，则有Ｔ（ｒ，Φ）≤Σ↑ｔ↓ｉ＝１｛［Ｓｉ＋ｏ（１）］ｍ（ｒ，ｆｉ）＋［△ｉ＝ｕｉ＋ｏ（１）］Ｎ（ｒ，ｆｉ）＋（△ｉ－Ｓｉ）Ｎ↑－（ｒ，ｆｉ     

某类有界拟凸域双全纯不变量的极限

Ｅ＝Ｅ（ｍ，ｎ，ｋ）＝｛（ｚ，ω）∈Ｃ＾ｎ＋ｍ：│ｚ│＾２＋│ω│＾２ｋ〈１，ｚ∈Ｃ＾ｎ，ω∈Ｃ＾ｍ，ｋ〉０｝是Ｃ＾ｍ＋ｍ中的一类有界拟凸域。该文证明了在δＥ的强拟凸点上，当ｍ〉１时，ｌｉｍ（ｚ，ω）→δＥＪＥ（（ｚ，ω））＝π＾ｎ＋ｍ（ｎ＋ｍ＋１）＾ｎ／（ｎ＋ｍ）！．ｍ＝１时，ｌｉｎ（ｚ，ω）→δＥＪＥ（ｚ，ω））＝π＾ｎ＋１（ｎ＝）＾ｎ＋１／（ｎ＋１）！，在δＥ的弱拟凸点上，上述极限不存在。     

关于代数体函数的Nevanlinna第二基本定理

设（Ａ，（ｚ））＾ｎｉ＝０为复平面上的整函数且没肥公共零点，令ｋ＋１是（Ａｉ（ｚ））＾ｎｉ＝０在复数域上的极大线性无关数，设Ｗ＝Ｗ（ｚ）由下列不可约方程Ａｎ（ｚ）Ｗ＾ｎ＋Ａｎ－１（ｚ）Ｗ＾ｎ－１＋．．．＋Ａ１（ｚ）ｗ＋Ａ０（ｚ）＝０所定义，我们称Ｗ＝Ｗ（Ｚ）为ｎ值ｋ型代数体函数（１≤Ｋ≤ｎ）。     

星形函数族的一个子族的极值点与支撑点

设Ｆ（｛ｎ｝）＝｛ｆ（ｚ）：ｆ（ｚ）在｜ｚ｜＜１内解析，ｆ（ｚ）＝ｚ－∞ｎ＝１ａｎｚｎ，ａｎ≥０，＋∞ｎ＝２ｎａｎ≤１｝，则Ｆ（｛ｎ｝）是星形函数族的一个子族．许多学者研究了这个函数族．设Ｍ＝｛ｆ（ｚ）：ｆ（ｚ）在｜ｚ｜＜１内解析，ｆ（ｚ）＝ｚ－∞ｎ＝１ａｎｚｎ，ａｎ≥ａｎ＋１≥０，＋∞ｎ＝２ｎａｎ≤１｝．在本文中我们找出了函数族Ｍ的极值点与支撑点．     

GL(n,Z)中的局部有限子群的一点注记

证明了：若Ｇ是一般线性群ＧＬ（ｎ,Ｚ）中的局部有限子群,则Ｇ含有一个２~ｍ阶的初等阿贝尔２－子群,且 Ｇ同构于 ＧＬ（ｎ,Ｚ_ｐ）的一个子群,其中户为任意奇素数．当 ｎ＝１,２,３,４时,Ｇ的阶分别是 ２,３· ２~ｋ（ｋ＝ｍｉｎ（４,ｍ＋１）,０≤ｍ≤４）,３·２~ｋ（ｋ＝ｍｉｎ｛５,ｍ＋１｝,０≤ｍ≤５）,３~２·５·２~ｋ（ｋ＝ｍｉｎ｛９,ｍ＋６｝,０≤ｍ≤９）的一个因子,而当ｎ≥５时,Ｇ的阶是（ｐ~ｉ－１）的一个因子,其中ｐ为任意素数．     

一种级数的维数

令其中Ｓ∈（１，２），对ｎ∈ｚ，ｉ）若ｂｎ＋１／ｂｎ单凋趋于无穷．则图形Ｇ（ｆ，Ｉ）的Ｈａｕｓｄｏｒｆｆ维数ｉｉ）若ｂｎ＝ｂｎ（ｂ＞１），则Ｓ≥ｄｉｍＧ（ｆ，Ｉ）≥Ｓ－Ｃｂ１－ｓ其中Ｃ是一个常数．     

关于Thue方程的解数

设ｍ是正整数，ｆ（Ｘ，Ｙ）＝ａ０Ｘｎ＋ａ１Ｘ（ｎ－１）Ｙ＋．．．＋ａｎＹｎ∈Ｚ［Ｘ，Ｙ］是Ｑ上不可约化的叫ｎ（ｎ≥３）次齐次多项式。本文证明了：当ｇｃｄ（ｍ，ａ０）＝１，ｎ≥４００且ｍ≥１０（３５）时，方程｜ｆ（ｘ，ｙ）｜＝ｍ，ｘ，ｙ∈ｚ，ｇｃｄ（ｘ，ｙ）＝１，至多有６ｎｖ（ｍ）组解（ｘ，ｙ），其中ｖ（ｍ）是同余式Ｆ（ｚ）＝ｆ（ｚ，１）≡０（ｍｏｄｍ）的解数。特别是当ｇｃｄ（ｍ，ＤＦ）＝１时，该方程至多有６ｎ（ω（ｍ）＋１）组解（ｘ，ｙ），其中ＤＦ是多项式Ｆ的判别式，ω（ｍ）是ｍ的不同素因数的个数．     

关于广义Thue－Mahler方程的解数

设ａ，ｂ是非零整数，ｐ１，…，ｐｒ是不同的素数，Ｐ＝｛±｜ｍ１，…，ｍｒ是非负整数｝．设Ｋ是ｎ（ｎ≥３）次代数数域，α１，…，αｍ∈ｋ（１＜ｍ＜ｎ），△（α１，…，αｍ）是α１，…，αｍ的判别式，ｆ（ｘ１，…，ｘｍ）＝αＮｋ／Ｑ（α１ｘ１＋…＋αｍｘｍ）∈ｚ［ｘ１，…，ｘｍ］．本文证明了：当ｆ（ｘ１，…，ｘｍ）非退化且Ｐｉ△（α１，…，αｍ）（ｉ＝１，…，ｒ）时，方程ｆ（ｘ１，…，ｘｍ）＝ｂｙ，ｘ１，…，ｘｍ∈ｚ，ｇｃｄ（ｘ１，…，ｘｍ）＝１，ｙ∈Ｐ至多有（４Ｓｄ２）（Ｓｄ）组解（ｘ１，…，ｘｍ，ｙ），其中ｄ＝ｎ！，Ｓ＝ｒ＋ω是ｂ的不同素因数的个数，ｈＡ是Ｋ的类数．     

THE FUNDAMENTAL GROUP OFTHE AUTOMORPHISM GROUP OFA NONCOMMUTATIVE TORUSTHE FUNDAMENTAL GROUP OFTHE AUTOMORPHISM GROUP OFA NONCOMMUTATIVE TORU

1. IntroductionGiven a locally compact abelian group G and a multiplier p on G, one can associate tothem the twisted group C*-algebra C*(G, p), which is the universal object for unitary prepresentations of G. C* (Zm, p) is said to be a noncommutative torns of rank m and denotedby A.. The multiplier p determines a subgroup S. of G, called its symmetry group, andthe multiplier p is called totally skew if the symmetry group S. is trivial. And A. is calledcompletely irrational if p is totally…     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ）× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

THE SEMICENTRE OF A GROUP ALGEBRA. II

The semicentre of an arbitrary group algebra K[G] is studied. In particular necessary and sufficient conditions are shown such that the semicentre is equal to a fixed subring of K[Δ(G)]. Also the semicentre is always equal to a sum of centres of certain subgroup algebras.     

关于n—可解群与n—幂零群的两个结果

本文证明了下面主要结果：设Ｇ是ｎ－可解群，π是一些素数之集，若对任意ｐ∈∩π（Ｇ），（ｐ，ｎ（１－ｎ））＝１，则Ｇ的π－Ｈａｌｌ子群的个数ｒ＝ｋ１ｋ２．．．ｋｔ，每ｋｉ≡１（ｍｏｄｐ），某Ｐ∈π，且每ｋｉ整除Ｇ的一个主因子。     

GENERALIZATION OF THE NOTION OF FUNDAMENTAL GROUP

The classical definition of fundamental group for a topological space is based on the pathwise connectedness. A space with less path will not be able to be described effectively by its fundamental group. The author introduces a definition of generalized fundamental group for a given topological space by means of its own connectedness. For a well-behaved space, e.g., a locally pathwise and semilocally simply connected compact metric space, the generalized fundamental group coincides with the classical one.     

NEW OBSTRUCTIONS FOR THE SURJECTIVITY OF THE JOHNSON HOMOMORPHISM OF THE AUTOMORPHISM GROUP OF A FREE GROUP

In this paper we construct new obstructions for the surjectivityof the Johnson homomorphism of the automorphism group of a freegroup. We also determine the structure of the cokernel of theJohnson homomorphism for degrees 2 and 3.