数的横加与类余数

将一个数的每一数位上的数相加——横加,这是一个很有趣的数学问题。例如:一个自然数能被3整除的充要条件是:该数的横加值为3的倍数。 本文将证明:若a是一个m进制的有限数,经有限次以m进制的数的加法横加后的值是一个不超过m-1的“一位数”,(并非是十进制的一位数)且类似于以m-1除a的最小正余数——类余数。最后,作为应用给出一种检验乘法运算的简便方法,这种方法只需要“一位”数的连加和一位数的乘法即可。

On Infinite Disjoint Congruence Covering Systems

In this paper we show that for any α∈(0,l] there exists an infinite disjoint covering system {a1(modni)}i=1∞ such that

On the Favard Class of Cosine Operator Function

In this paper we give some basic properties of the Favard class of cosine operator function, and concern with the question when for a given cosine fuction we have Fav(A) = D(A).

A density theorem and its application

We compute the density of primes represented by a special quadratic form in a fixed square residue class. Using this result and a new method introduced by Thaine we prove the fact that for a prime p > 3congruent to 3 modulo 4, the component e(p+1)/2of the p-Sylow subgroup of the ideal class group of Q(ζp) is trivial.

Variables separated equations:Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification

Davenport's Problem asks:What can we expect of two polynomials,over Z,with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport,Lewis and Schinzel.By bounding the degrees,but expanding the maps and variables in Davenport's Problem,Galois stratification enhanced the separated variable theme,solving an Ax and Kochen problem from their Artin Conjecture work.Denef and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement.By restricting the variables,but leaving the degrees unbounded,we found the striking distinction between Davenport's problem over Q,solved by applying the Branch Cycle Lemma,and its generalization over any number field,solved by using the simple group classification.This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups.Guralnick and Thompson led its solution in stages.We look at two developments since the solution of Davenport's problem.Stemming from MacCluer's 1967 thesis,identifying a general class of problems,including Davenport's,as monodromy precise.R(iemann)E(xistence)T(heorem)'s role as a converse to problems generalizing Davenport's,and Schinzel's (on reducibility).We use these to consider:Going beyond the simple group classification to handle imprimitive groups,and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

From projective representations to quasi-quantum groups

This is a contribution to the project of quiver approaches to quasi-quantum groups.We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations.This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras,or equivalently cofree pointed coalgebras,and helps to provide a projective representation-theoretic description of the gauge equivalence of graded pointed Majid algebras.We apply this machinery to construct some concrete examples and obtain a classification of finite-dimensional graded pointed Majid algebras with the set of group-likes equal to the cyclic group of order 2.