导　　师： 王军

学科专业： 070102

授予学位： 硕士

作　　者： ;

机构地区： 大连理工大学

摘　　要： 本文主要是研究Sturm序列因子的结构。任取Sturm序列U及其任意一个因子ω,从U中依次删去这个因子ω,我们就会得到一个新的序列,称为剩余序列。然后,根据因子的不同,对剩余序列进行分类。最后讨论剩余序列在双射作用下的变化情况:其中大多数剩余序列可以通过双射变成Sturm序列;另外的少数不能通过双射变成Sturm序列。论文的主要内容如下: 第一章简要地介绍了Sturm序列和字上的组合的发展进程,以及现在关于Sturm序列的一些研究课题。 第二章详细介绍了Sturm序列的基础知识及其等价定义。 第三章包含下述内容:首先,因为Sturm序列是一个非周期的并且其语言复杂度是最小的符号序列。所以我们对于所研究的对象进行一些合理的限制,将所研究的Sturm序列限制在标准Sturm序列及其斜率小于等于1的序列下面。其次,利用标准字的长度严格单调递增的性质,将选定的因子的长度限定在两个相邻的标准字的长度之间。再次,根据选定的因子与一些特殊因子之间的关系,分类研究其对应的剩余序列,从而得出剩余序列的因子的分类。最后,用原来的序列为Sturm序列的特点,得出如下结论:大多数类型的剩余序列可以通过双射变成Sturm序列,有一小部分的剩余序列不能通过双射变成Sturm序列。 In this thesis we mainly study the structure of the factors of Sturmian sequences. Given be a Sturmian sequence U and a factor w of U, let us delete in turn all w's in U. We then get a new sequence, which is called a remaining sequence. According to different factors, we give a classification for the remaining sequences. Finally, we discuss the remaining sequences under bijection: Most of them can be changed into a Sturmian sequence by bijection, while a few of them can't turn into a Sturmian sequence by bijection. The main content of this is summarized as follows:In the first chapter, we briefly introduce the evolution on the Sturmian sequences and combinatorics on words, as well as some topics about Sturmian sequences in recent years.In the second chapter, we introduce the preliminaries of Sturmian sequences and its other definitions, such as Balanced sequences, Cutting sequences and Rotation sequences, etc.In the last chapter, the main reseach is as follows: Firstly, since a Sturmian sequence is an aperiodic sequences with minimal language complexity, we can make some reasonable constriction for them, that is, let the Sturmian sequences be standard and with its slope less than 1. Secondly, by the exact increase of lengths of the standard words we can restrict the length of the selected factor between the lengths of two consecutive standard words. Thirdly, according to relations between the selected factor and some special factors, we study the remaining sequences, and give a classification of the factors of remaining sequences. Finally, from the properties of the original sequences, we obtain the conclusion: Most of them can be changed into a Sturmian sequence by bijection, while a few of them can't turn into a Sturmian by bijection.

关 键 词： 序列 标准字 字上的组合

领　　域： [理学] [理学]