Interesting Topics for Bachelor Theses Walter M. Bohm¨ Institute for Statistics and Mathematics Vienna University of Economics [email protected] October 4, 2018 The picture on the title page is an artwork by Jakob Bohm,¨ www.jacob-julian.com The other pictures used in this book are from Wikipedia Commons and MacTutor History of Mathematics archive maintained at The University of St. Andrews. 2 Foreword This booklet is a collection of topics which I prepared over the time for my students. The selection of topics reflects my personal interests and is therefore biased towards combinatorial mathematics, probability and statistics, opera- tions research, scheduling theory and, yes, history of mathematics. It is in the nature of things that the level of difficulty varies from topic to topic. Some are technically more demanding others somewhat easier. The only prerequisite to master these topics are courses in mathematics and statistics at an undergraduate university level. Otherwise, no special prior knowledge in mathematics is afforded. However, what is needed is serious interest in mathematics, of course. How is a Topic organized? Each topic consists of three parts: (1) An Invitation Of course, the major purpose of this invitation is to raise your interest and to draw your attention to a problem which I found very interesting, attractive and challenging. Further, in each invitation I introduce some basic terminology so that you can start reading basic literature related to the topic. (2) Where to go from here Some of my invitations are more detailed depending on the topic, so you may ask yourself: Is there anything left for me to do? Yes, there is lot of work still to be done. The second section of each topic contains questions and problems which you may study in your thesis. This list is by no means exhaustive, so there enough opportunity to unleash your creative potential and hone your skills. For some topics I explicitly indicate some issues of general interest, these are points which you should to discuss in your thesis in order to make it more or less self-contained and appealing to readers not spezialized in this topic. And sometimes there is also a section What to be avoided: here I indicate aspects and issues related to the topic which may lead too far afield or are technically too difficult (3) An annotated bibliography This is a commented list of interesting, helpful and important books and journal articles. This book has not been finished yet and probably may never be. You are free to use this material, though a proper citation is appreciated. 3 4 Contents 1 Recreational Mathematics 11 1.1 An Invitation............................. 11 1.1.1 The Challenge of Mathematical Puzzles.......... 12 1.1.2 Some Tiny Treasures From My Collection......... 14 1.2 Where to go from here........................ 18 1.3 An annotated bibliography..................... 23 1.4 References............................... 24 2 Shortest Paths in Networks 27 2.1 An Invitation............................. 27 2.1.1 The problem and its history................. 27 2.1.2 Preparing the stage - graphs, paths and cycles...... 28 2.1.3 Weighted graphs....................... 31 2.1.4 Solvability........................... 37 2.1.5 It's time to relax....................... 39 2.1.6 A sample run of the Bellman-Ford Algorithm....... 41 2.1.7 The complexity of the Bellman-Ford Algorithm...... 44 2.2 Where to go from here........................ 45 2.2.1 Issues of general interest................... 45 2.2.2 Some more suggestions.................... 46 2.2.3 To be avoided......................... 48 2.3 An Annotated Bibliography..................... 49 2.4 References............................... 50 3 The Seven Bridges of K¨onigsberg 53 3.1 An Invitation............................. 53 3.1.1 Euler's 1736 paper...................... 53 3.1.2 K¨onigsberg and a puzzle................... 54 3.1.3 Euler takes notice of the puzzle............... 54 5 Contents 3.1.4 Euler's solution........................ 56 3.1.5 What happened to the problem later?........... 60 3.1.6 An epilog: K¨onigsberg and its bridges today........ 60 3.1.7 The Chinese Postman Problem............... 61 3.2 Where to go from here........................ 65 3.2.1 Issues of general interest................... 65 3.2.2 Some more suggestions.................... 67 3.3 An Annotated Bibliography..................... 69 3.4 References............................... 70 4 The Chains of Andrei Andreevich Markov - I 73 4.1 An Invitation............................. 73 4.1.1 The Law of Large Numbers and a Theological Debate.. 73 4.1.2 Let's start with a definition................. 74 4.1.3 Example 1: Will We Have a White Christmas This Year? 77 4.1.4 Example 2: Losing Your Money - Delinquency Of Loans. 84 4.2 Where to go from here........................ 88 4.2.1 Make up your mind - absorbing or regular chains?.... 88 4.2.2 Google's PageRank Algorithm............... 89 4.2.3 Credit Ratings........................ 90 4.2.4 Generating Random Text, maybe Bullshit......... 91 4.2.5 Other Applications...................... 92 4.3 An Annotated Bibliography..................... 93 4.4 A note on software.......................... 93 4.5 References............................... 94 5 The Chains of Andrei Andreevich Markov - II 97 5.1 An Inivitation............................. 97 5.2 An Annotated Bibliography..................... 97 5.3 References............................... 98 6 Benford's Law 99 6.1 An Invitation............................. 99 6.1.1 Simon Newcomb and the First Digit Law......... 99 6.1.2 The significand function................... 102 6 Contents 6.1.3 Benford's Law and the uniform distribution........ 103 6.1.4 The general digit law..................... 104 6.1.5 Testing the Hypothesis.................... 106 6.1.6 Remarkable Properties of Benford's Law.......... 109 6.2 Where to go from here........................ 113 6.2.1 Statistical Forensics..................... 113 6.2.2 Experimental Statistics................... 115 6.3 An Annotated Bibliography..................... 117 6.4 References............................... 118 7 The Invention of the Logarithm 121 7.1 An Invitation............................. 121 7.1.1 A personal remembrance................... 121 7.1.2 Tycho Brahe - the man with the silver nose........ 122 7.1.3 Prostaphaeresis........................ 123 7.1.4 John Napier and Henry Briggs............... 125 7.2 Where to go from here........................ 129 7.2.1 Historical Issues....................... 129 7.2.2 Technical Issues........................ 132 7.3 An Annotated Bibliography..................... 136 7.4 References............................... 137 8 Exercise Number One 139 8.1 An Invitation............................. 139 8.1.1 Exercise number one..................... 139 8.1.2 Partitions of integers..................... 139 8.1.3 Partitions with restricted parts............... 141 8.1.4 Generating functions..................... 142 8.2 Where to go from here........................ 143 8.2.1 Issues of general interest................... 143 8.2.2 Some more suggestions.................... 143 8.3 An Annotated Bibliography..................... 143 8.4 References............................... 144 9 The Ubiquitious Binomialcoefficient 145 7 Contents 9.1 An Invitation............................. 145 9.1.1 The classical binomialtheorem............... 145 9.1.2 Pascal's triangle....................... 146 9.1.3 Newton's binomial theorem................. 148 9.1.4 Binomial sums........................ 150 9.2 Where to go from here........................ 151 9.3 An Annotated Bibliography..................... 152 9.4 References............................... 152 10 Prime Time for a Prime Number 153 10.1 An Invitation............................. 153 10.1.1 A new world record..................... 153 10.1.2 Why primes are interesting................. 153 10.1.3 Primes and RSA-encryption................. 154 10.1.4 Really big numbers...................... 156 10.1.5 Mersenne numbers...................... 156 10.1.6 Primality testing....................... 157 10.1.7 Generating prime numbers................. 160 10.1.8 Factoring of integers..................... 161 10.2 Where to go from here........................ 162 10.2.1 Computational issues.................... 162 10.2.2 Issues of general interest................... 163 10.2.3 Some more suggestions.................... 164 10.2.4 What to be avoided..................... 164 10.3 An Annotated Bibliography..................... 164 10.4 References............................... 165 11 Elementary Methods of Cryptology 167 11.1 An Invitation............................. 167 11.1.1 Some basic terms....................... 168 11.1.2 Caesar's Cipher........................ 169 11.1.3 Frequency analysis...................... 172 11.1.4 Monoalphabetic substitution................ 174 11.1.5 Combinatorial Optimization................. 177 11.1.6 The Vigen`ereCipher, le chiffre ind´echiffrable....... 179 8 Contents 11.1.7 Transposition Ciphers.................... 184 11.1.8 Perfect Secrecy........................ 185 11.2 Where to go from here........................ 186 11.2.1 Issues of general interest................... 187 11.2.2 Some more suggestions.................... 187 11.2.3 What to be avoided..................... 188 11.3 An Annotated Bibliography..................... 189 11.4 References............................... 190 12 Parrondo's Paradox 191 12.1 An Invitation............................. 191 12.1.1 Favorable