实二次域Tame核的阶的3-整除性

本文证明了对某类实二次域,其类数能被3整除当且仅当其Tame核的阶能被3整除,同时给出了Browkin关于Tame核的3整除性的结果的一个不同证明.     

勾股数组一个性质的证明

在《圆和二次方程》一书中,给出了任何一组勾股数组a、b、c都可由公式a=m~2-n~2,b=2m~n,c=m~2+n~2表示(这里m、n-奇-偶,m>n,m、n均为自然数),同时指出“abc一定能被60整除”,因为它的证明“已经超出你们的知识范围,这里就不谈了”。为此,笔者给出一种浅显的证明。下面先证两个引理。引理1。任何自然数p若不能被3整除,则p~2-1能被3整除。证明:因为任何不能被3整除的自然数p均可表示勾:p=3k±1(这里k为自然数)而p~2=(3k±1)~2=9k~2±6k+1=3(3k~2±2k)+1,所以p~2-1能被3整除。引理2.任何自然数q若不能被5整除,则q~4-1能被5整除。证明:因为任何不能被5整除的自然数q可表示为q=5l±1,或q=5l±2 (这里l为自然数) 而当q=5l±1时,q-1或q+1能被5整除;当q=5l±2时,q~2=(5l±2)~2     

一道竞赛题所提及的有理数

一道竞赛题所提及的有理数４４３０００湖北三峡学院工学院金雁鸣在一次数学竞赛中有这样一道题目：设是的个位数，证明ａ（ｋ）＝０·ａａ…ａ…是有理数．首先，我们给出这个问题的证明．证明因为所以当ｓ≥１时，被１０整除．当ｓ＝０时，也被１０整除。又因为所以能被...     

关于虚二次域Tame核计算的一个注记

本文给出了一种计算数域Tame核的方法.应用到虚二次域上,证明了当 Nv>8δD6时,(？)t／:K2S'F／K2SF→k*是双射。     

关于能被两位数整除的整数的特征研究

判断一个整数能否被另一个整数整除一直是初等数论中一个饶有兴趣的问题.我们知道,能被2整除的数必是偶数,能被3或9整除的整数的特征是它的各个数字之和也必能被3或9整除,能被5整除的数的个位数一定是0或5,能被10整除的数的个位数一定是0,判断一个数能否被任意两位数整除并非易事,笔者研究发现如下规律.……     

广义Ramanujan-Nagell方程x~2-D=3~n的解数

设D是不能被3整除的正整数.本文证明了:当D>1012时,如果Pell方程U2-DV2=-1有解(U,V),则方程x2-D=3n至多有2组正整数解(x,n).     

整数的整除性证明的常用方法

在初、高中教材和一些初等数学参考书中,经常遇到一些关于整数的整除性证明的问题。本文就这一问题给出五种证明方法。一般来说,有关整数的整除性的问题,都可以用这些方法来证明。一用分解因式法证明例1 已知m是自然数,求证m~5-5m~4 4m能被120整除。证明:m~5-5m~4=m(m~2-1)(m~2-4) =m(m-2)(m-1)(m 1)(m 2) ∴m~5-5m~4 4m可化成五个连续的整数m-2,m-1,m,m 1,m 2的乘积的形式。从而知,原式能被5整除,又能被3整除。另一     

2-边连通图的线图的P3因子

如果图G有一个生成子图使得这个生成子图的每一个分支都是3个点的路，则称G有P3-因子．本文证明了对任何一个2-边连通图G，只要G的边数能被3整除，则G的线图就有P3-因子。     

争鸣

问题问题144高二的一本辅导资料上有这样一个题:在1,2,…,1000这1000个正整数中,任取一个数,它能被2或3整除的概率是解法1设A={取得的数能被2整除},B={取得的数能被3整除},则P(A B)=P(A) P(B)-P(AB)=1500000 1303030-1500000×103030≈0.667.解法2被3整除的数有3,6,9,…,999,共333个,被2整除的数有2,4,6,…,1000,共500个,被3整除又被2整除的数有6,12,18,…,996共166个,所以被2或3整除的数共333 500-166=667个.又从1,2,…,1000最1000个数中,任取一数机会是均等的,共1000种等可能结果,故P(A B)=1606070=0.667.事实上解法1的准确值为0.66…     

一个数论猜想题的否定及修正

文[1]提出问题:设M=52001 72002 92003 112004,求证:M能被8整除．同时,给出了较复杂的证明．文[2]对上述问题进行了简证,并猜想: (2n-3)m (2n-1)m 1 (2n 1)m 2 (2n 3)m 3能被2n整除(n≥2,n,m∈N)．其实,此猜想是错误的．因为972 993     

ON THE CONVERGENCE FOR NULL-ARRAYS OF THE POINT PROCESSES

The purpose of this paper is first to establish a representation of the Laplace transformfor the regular infinitely divisible point processes,and then to give a sufficient and neccesarycondition for convergence of the null-arrays toward a regular infinitely divisible pointprocess.     

DIVISIBILITY PROPERTIES OF HIGHER RANK LATTICES

We discuss a relationship between the dynamical properties of a maximal diagonalizable group A on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small: We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.     

On zeros of Kloosterman sums

A Kloosterman zero is a non-zero element of ${{\mathbb F}_q}$ for which the Kloosterman sum on ${{\mathbb F}_q}$ attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in ${{\mathbb F}_q}$ belongs to a proper subfield of ${{\mathbb F}_q}$ with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.     

Torsion-free abelian α-irreducible groups of finite rank

If F is a free abelian group of finite rank and α is an endomorphism or an automorphism of its divisible hull, then the α‐ hull is determined, i.e. the minimal torsion-free abelian group with this endomorphism a. Torsion-free abelian groups of finite rank are called α-irreducible if their divisible hull is α-irreducible for an automorphism a. A complete classification is given for α-irreducible groups and this result is applied to groups of rank 2.     

Characterization of a class of infinitely divisible distributions in Hilbert space

It is established that the spectral measure of an infinitely divisible distribution F in a Hilbert space H is concentrated in a sphere of finite radius if and only if the integral ∫ _H exp (α∥x∥ In (∥x∥+1))dF is finite for some numberα>0. If this integral is finite for anyα>0 then the infinitely divisible distribution F is normal (maybe, degenerate).     

关于丛属函数的几个不等式

<正> 1.引言.设(?)是单位圆中的正则函数,函数w=F(z)将|z|<1映照成黎曼面S_F.设函数(?)在单位圆中是正则的.假如w=f(z)的一切函数值都落在 S_F,上,那末说 f(z)丛属于 F(z),记此关系为 f(z)(?)F(z).我们知道 f(z)(?)F(z)的充要条件是存在|z|<1上的正则函数ω(z),适合|ω(z)|<1,ω(0)=0,和 f(z)≡F(ω(z)).     

DERIVATION ALGEBRAS OF THE MODULAR LIESUPERALGEBRAS W AND S OF CARTAN-TYPE

1NatationandPreliminariesWeknowthatthederivationalgebrasisaveryusefulsubjectintheresearchoftheLiealgebrasandLiesupealgebras.Inpapersl1]and[2]thederivationalgebrasofmodularLiealgebrasofCartan-typeareinyestigaved.Inpaper[3]thederivationalgebrasofsimpleLiesuperalgebrasoverfieldsofcharacteristiczeroaredeterndned.Inthispaperthederivational-gebrasofmodularLiesuperalgebrasWandSofCartan-typearedeterminedbythecalculatingmethod-LetFbeafieldandcharF=p>3-WesimPlydescribetheLiesuperalgbrasWandSwhic…     

Rank-1 quotient divisible groups

An Abelian group is called quotient divisible if it does not contain nonzero torsion divisible subgroups but does contain a free finite-rank subgroup such that the quotient group by it is divisible. In this paper, we will describe rank-1 quotient divisible groups with the help of cocharacteristics, and we will describe the endomorphisms of these groups as well. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 25–33, 2007.     

A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs

Hedayat and Federer (Ann. of Statist.3 (1975), 445–447) proved that Knut Vik designs do not exist for all even orders. They gave a simple algorithm for the construction of such designs for all other orders, except when the order of the design is divisible by 3. The existence of Knut Vik designs of orders divisible by 3 was left unsolved by these authors. It is shown here that Knut Vik designs do not also exist for all orders divisible by 3. An easy algorithm based on a result of Euler is provided for the construction of orthogonal Knut Vik designs for all orders not divisible by 2 or 3. Therefore, we can say that Knut Vik designs and orthogonal Knut Vik designs of order n exist if and only if n is not divisible by 2 or 3. The results are based on the concepts of a super diagonal and parallel super diagonals in an n × n array, which have been introduced and studied for the first time here. Other relevant results are also given.     

Infinitely divisible stochastic processes

Summary It often happens that a stochastic process may be approximated by a sum of a large number of independent components no one of which contributes a significant proportion of the whole. For example the depth of water in a lake with many small rivers flowing into it from distant sources, or the point process of calls entering a telephone exchange (considered as the sum of a number of point processes of calls made by individual subscribers) may approximately fulfil these hypotheses. In the present work we formulate and solve the problem of characterizing stochastic processes all of whose finite-dimensional distributions are infinitely divisible. Although some of our results could be derived from known theorems on probabilities on general algebraic structures, many could not and it seems preferable to take the vector-valued infinitely divisible laws as our starting point. We emphasize that an infinitely divisible process (in our sense) on the real line is not necessarily a decomposable process in the sense of Lévy (cf. § 4) though decomposable processes are particular cases.In § 1 a representation theorem for non-negative (and possibly infinite) stochastic processes is derived, while an analogous theorem in the real-valued case is to be found in § 2. § 3 is devoted to the statement of a central limit theorem and the investigation of some of the properties of infinitely divisible processes. The investigation is continued in § 4 by an examination of processes on the real line giving, for example, a representation theorem under weak conditions for infinitely divisible processes which are a.s. sample continuous. Finally in § 5 a study is made of infinitely divisible point processes and random measures.The author is indebted to Professor J. F. C. Kingman for advice and encouragement.