On the Properties and Structure of Bordered Words and Generalizations

Combinatorics on words is a field of mathematics and theoretical computer science that is concerned with sequences of symbols called words, or strings. One class of words that are ubiquitous in combinatorics on words, and theoretical computer science more broadly, are the bordered words. The word w has a border u if u is a non-empty proper prefix and suffix of w. The word w is said to be bordered if it has a border. Otherwise w is said to be unbordered. This thesis is primarily concerned with variations and generalizations of bordered and unbordered words. In Chapter 1 we introduce the field of combinatorics on words and give a brief overview of the literature on borders relevant to this thesis. In Chapter 2 we give necessary definitions, and we present a more in-depth literature review on results on borders relevant to this thesis. In Chapter 3 we complete the characterization due to Harju and Nowotka of binary words with the maximum number of unbordered conjugates. We also show that for every number, up to this maximum, there exists a binary word with that number of unbordered conjugates. In Chapter 4 we give results on pairs of words that almost commute and anti-commute. Two words x and y almost commute if xy and yx differ in exactly two places, and they anti-commute if xy and yx differ in all places. We characterize and count the number of pairs of words that almost and anti-commute. We also characterize and count variations of almost-commuting words. Finally we conclude with some asymptotic results related to the number of almost-commuting pairs of words. In Chapter 5 we count the number of length-n bordered words with a unique border. We also show that the probability that a length-n word has a unique border tends to a constant. In Chapter 6 we present results on factorizations of words related to borders, called block palindromes. A block palindrome is a factorization of a word into blocks that turns into a palindrome if each identical block is replaced by a distinct character. Each block is a border of a central block. We call the number of blocks in a block palindrome the width of the block palindrome. The largest block palindrome of a word is the block palindrome of the word with the maximum width. We count all length-n words that have a width-t largest block palindrome. We also show that the expected width of a largest block palindrome tends to a constant. Finally we conclude with some results on another extremal variation of block palindromes, the smallest block palindrome. In Chapter 7 we present the main results of the thesis. Roughly speaking, a word is said to be closed if it contains a non-empty proper border that occurs exactly twice in the word. A word is said to be privileged if it is of length ≤ 1 or if it contains a non-empty proper privileged border that occurs exactly twice in the word. We give new and improved bounds on the number of length-n closed and privileged words over a k-letter alphabet. In Chapter 8 we work with a generalization of bordered words to pairs of words. The main result of this chapter is a characterization and enumeration result for this generalization of bordered words to multiple dimensions. In Chapter 9 we conclude by summarizing the results of this thesis and presenting avenues for future research

The Great War : images of reality in the French novel

War destroys the sensibility of the mind yet paradoxically it can heighten emotion and perception. Although the Great War for Civilisation destroyed the youth of an entire generation, although it irrevocably scarred the face of modern French history, its repercussions were, nevertheless, deeply lodge in the country's subsequent literary output. A link emerged between war and literature formerly unknown, as many established writers became involved in the fighting and civilians temporarily transformed into soldiers gave vent to their feelings in a secondary war of words. This thesis investigates the merging of the real with the imaginary, the threading together of historical fact and literary technique. The Introduction places emphasis upon the final weeks of peace leading up to the outbreak of hostilities and the portrayal of these events in the French novel. Each chapter then deals with different aspects of trench warfare on the Western Front. In Chapter I, I consider the innocence and naivety with which men went to war, their failure to take the situation seriously, their curiosity, their refusal to fear. The traditional heroic and patriotic spirit that has romanticized war in the past, however is completely destroyed in Chapters II and III, as a different picture is painted by those who gained first-hand experience of the horrors of the Front. In Chapter IV the telling effects of these horrors upon the wretched mortals who, day after day, month after month, endured the most inhuman existence imaginable, are assessed, and the themes of death, depersonalization and dehumanization closely examined. The absurdity of war, both in the trenches and on the "home front", is dealt with in Chapters V and VI. Chapter VII focuses attention on the utter folly of a situation where nobody wanted to fight, to be killed, and yet where no-one dared to reuse the wishes of the politicians and commanding officers who continued to accelerate the "war effort" despite rising casualties. Chapter VI adopts a different viewpoint as shared by those who remained behind. The total breakdown in communication and understanding between soldiers and civilians is clearly depicted and the wedge driven between the two, strongly emphasized. The final chapter paintes quite a different picture of war, revealing the more positive, more enjoyable, more humerous [sic] aspects of life in the trenches. The Great War was not the war to end all wars, as had been hoped, indeed, it began a new pattern of fighting more devastating, more frightening than ever before. In conclusion, I pose the question whether 1918 brought final victory or merely a temporary cease fire, an anti-climax; attention is drawn to a possible fascinating comparison between the literature of the First World War and that of the Second World War, and to the recent revival in interest in the Great War which involvement in the Second World War has inevitably brought about. Nowadays, perhaps more than at any other time during the past 40 years, is the message of the "war novel" appropriate and meaningful. It failed to prevent the outbreak of hostilities in 1939; will it succeed in maintaining world peace in the future