乘积空间上的Z2＊Z2—指标理论

本文针对Ｚ２＊Ｚ２群在乘积空间上的自然作用定义一种新的指标。这种指标满足通常指标的一般性质，但不满足维数性质，作为个指标的应用，我们还给出了两个抽旬有的临界点定理。     

Z_2不变的C~∞函数芽的典型形式

Ｗｈｉｔｎｅｙ关于偶函数的结果给出了一个变元且在Ｚ_２群｛±１｝下不变的Ｃ~∞函数芽的典型形式：如果ｆ∈Ｅ_１且ｆ（－ｘ）＝ｆ（ｘ）,则存在ｈ∈Ｅ_１使得ｆ（ｘ）＝ｈ（ｘ~２）．该文将借助Ｍａｌｇｒａｎｇｅ预备定理和有关的计算,得出Ｒ~ｎ在原点且在群｛±Ｉ_ｎ｝下不变的Ｃ∞函数芽的典型形式．     

集合（2课时）

集合（２课时）王池富（武汉市二中４３００１０）一、教学目的１．正确理解集合概念和集合的性质，掌握集合的主要表示方法；２．能熟练使用集合语言：如集合表示法（含常见数集Ｎ，Ｚ，Ｑ，Ｒ，等）、元素与集合关系的表示法等．二、能力培养１．抽象概括能力：分析特殊...     

THE COMPUTATION OF SYMMETRY－BREAKING BIFURCATION POINTS IN Z_2×Z_2－SYMMETRIC NONLINEAR PROBLEMS

ＴＨＥＣＯＭＰＵＴＡＴＩＯＮＯＦＳＹＭＭＥＴＲＹ－ＢＲＥＡＫＩＮＧＢＩＦＵＲＣＡＴＩＯＮＰＯＩＮＴＳＩＮＺ_２×Ｚ_２－ＳＹＭＭＥＴＲＩＣＮＯＮＬＩＮＥＡＲＰＲＯＢＬＥＭＳ￥ＹＥＲＵＩＳＯＮＧＡＮＤＹＡＮＧＺＨＯＮＧＨＵＡＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａ...     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

1995年高考数学（理科）解答题另解摘编

１９９５年高考数学（理科）解答题另解摘编（２１）在复平面上，一个正方形的四个顶点按照逆时针方向依次为Ｚ１，Ｚ２，Ｚ３，Ｏ（其中Ｏ是原点），己知Ｚ２对应复数，求Ｚ１和Ｚ３对应的复数．解法１设Ｚ１，Ｚ３对应的复数分别为ｚ１，ｚ３，如图．由已知Ｚ２对应的复...     

半质环的若干交换性条件

半质环的若干交换性条件孟宪利（安徽电力职工大学，合肥２３００２２）设Ｒ表示结合环（可以设有单位元），Ｚ（Ｒ）为环Ｒ的中心，对任意ｘ，ｙ∈Ｒ，［ｘ，ｙ］＝ｘｙ－ｙｘ．郭元春［１］证明了满足（ｘｙ）２－ｘｙ２ｘ∈Ｚ（Ｒ）的半质环是交换环．魏宗宣［２］用类...     

共轭复数的一个充要条件

共轭复数的一个充要条件湖北监利县龚场中学荣延俊众所周知，Ｚ１＋ｚ２及Ｚ１·Ｚ２均为实数是Ｚ１、Ｚ２为共轭复数的必要非充分条件．本文给出两个复数为共轭复数的一个充要条件．定理设ｚ１、ｚ１６Ｃ，ｚ１＋ｚ２＝ａ，ｚ１·ｚ２＝ｂ，则复数ｚ１、ｚ２为共轭复数的...     

复数运算的几何意义

复数运算的几何意义湖北实验幼师周希冰，郑淑瑜［基本概念与结论］在运用复数运算的几何意义研究问题时，一定要深刻理解加减、乘除、乘方、开方的几何意义，才能达到运用自如的目的．１．加减运算的几何意义设Ｚ１对应于ＯＺ１，Ｚ２对应于ＯＺ２，以ＯＺ１、ＯＺ２为两...     

2-Divisible groups over Z

In this paper we construct nontrivial 2-divisible groups over Z which are isogenous to trivial groups and prove the following: THEOREM. If the height h of a 2-divisible group {G^(v)} over Z is at most 4, then {G^(v)} is isogenous to a trivial group.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 717–726, May, 1976.In conclusion, I would like to express my deep gratitude to I. R. Shafarevich for his guidance.     

A characterization of $$\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$$-linear codes

We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes.     

关于环$Z_4[u]/\langle u2-2\rangle$上的斜多元循环码

本文探索了环$R=Z_4[u]/\langle u2-2\rangle$ 上的几类斜多元循环码和多元循环码. 首先得到了环$R$上$(1,2u)$-多元循环码的生成多项式. 其次由定义的Gray映射得到了环$R$上$(1,2u)$- 多元循环码的Gray像是$Z_4$上的循环码或指数为2的逆循环码. 最后, 通过环$R$上$(1,2u)$- 多元循环码的一些例子来展示本文的主要结果.     

Z2上平延群的有理不变量

本文研究了Z2上平延群的有理不变量.利用矩阵的方法,获得了平延群的一组超越基.     

$${{\mathbb {Z}}}_2$$-double cyclic codes

A binary linear code C is a \({\mathbb {Z}}_2\)-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the \({\mathbb {Z}}_2[x]\)-module \({\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).\) We determine the structure of \({\mathbb {Z}}_2\)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of \({\mathbb {Z}}_2\)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between \({{\mathbb {Z}}}_2\)-double cyclic and other families of cyclic codes, and show some examples of distance optimal \({\mathbb {Z}}_2\)-double cyclic codes.     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.     

一类二阶常微分方程边值问题的多重解

本文利用变分原理和Z2不变群指标研究了一类二阶常微分方程奇异边值问题的多重解,得出了这类解个数的下界估计.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Some questions concerning superderivations on $${\mathbb {Z}}_{2}$$-graded rings

In this paper we pose some questions about superderivations on \({\mathbb {Z}}_{2}\)-graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special \({\mathbb {Z}}_{2}\)-gradings and we check the answer to these questions about them.