仿射差序列的研究

Abstract

對於一個正整數的遞增序列 < s1, s2,...,sn > ，如果存在一種分割 將 [1,2n] = {1, 2,..., 2} 分割成兩個兩個一組的集合 {ai. bi} ，使得它對於所有 i = 1, 2,...,n 滿足 |kai-hbi| = si , 則我們稱它為一個 (k;h)-n 維的仿射差序列 (RAD sequence)。收集所有的 < s1, s2,...,sn > 中的元素所成的集合是一個 (k;h)-RAD 集 (重集)。我們可以發現一個(1, 1)- RAD 集在特殊情況下會滿足 s1 < s2 <...<sn 和Σ_{i=1}^n si = n^2。這樣的集合我們稱之為完全極值索克蘭集。 在這個論文裡，我們將研究分成兩個部分。第一部分是針對完全極值索克蘭集的一些特徵下去觀察。我們主要提供一個方法去建構這種集合。然後在第二個部分我們討論 (k,h) =(2,1) 和 (3,2) 的狀況。我們得到一些新結果特別是我們證明了 < 1, 2,...,n> 是一個 (2,1)-n維 RAD sequence 對於所有 n 2，同時，在 (3; 2) 的情況也有一些部分的研究成果。

An increasing sequence of positive integers <s1,s2,...,sn > is called a (k;h)-realizable affine difference sequence (RAD sequence) of order n if there exists a partition of [1,2n] = {1,2,...,2n} into 2-subsets {ai,bi} such that |kai-hbi| = si for i = 1,2,...,n.In such partition, {s1, s2,...,sn} will be called a (k;h)-RAD set .Note that a (1,1)-RAD set is also known as aperfect extremal Skolem set if s1 < s2 <...<sn andΣ_{i=1}^n si = n^2. In this thesis, our study focuses on two-topics. The first one is concerned about the characterization of a perfect extremal Skolem set. Mainly, we provide an idea to construct such sets. Then, in the second topic, we consider the case (k,h) =(2, 1) and(3, 2). Several new results are obtained, especially we prove that < 1, 2, ...,n> is a (2,1)- RAD sequence of order n for all n>=2 and partial partial results on the (3, 2)-case and obtained.