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We cover a few interesting classes of problems for finite state automata and then show some examples of infinite state automata and recursive regular expressions. The final problem in the book involves constructing a recursive regular expression for matching regular expressions. This book explains: * Why automata are important. * The relationship of automata to regular expressions. * The difference between deterministic and nondeterministic automata. * How to get the regular expression from an automaton. * Why two seemingly different regular expressions can belong to the same automaton. * How the regular expression for an infinite automaton is different than one for a finite one. * The relationship of a regular expression to a regular language. * What a generating function for a language tells you about the language. * How to get a generating function from a regular expression. * How the generating function of a recursive regular expression is different from that of an ordinary regular expression. * How to test divisibility properties of integers (binary and decimal based) using automata. * How to construct an automaton to search for a given pattern, or for a given pattern not occurring. * How to construct an automaton for arbitrary patterns and alphabets. * How the recursive regular expression for nested parentheses leads to the Catalan numbers. Included in this book: * Divisibility problems in binary and decimal. * Pattern search problems in binary, ternary, and quaternary alphabets. * Pattern search problems for circular strings that contain or do not contain a given pattern. * Automata, regular expressions, and generating functions for gambling games. * Automata and generating functions for finite and infinite correctly nested parentheses. * The recursive regular expression for matching regular expressions over a binary alphabet. * A further reading list.","brand":"WoB","offers":[{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":50181364416785,"sku":"GOR009848903","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51054934262033,"sku":"NIN9781887187169","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1887187162.jpg?v=1750834325"},{"product_id":"creating-rhythms-book-stefan-hollos-9781887187220","title":"Creating Rhythms","description":"\u003cp\u003eThis book is the result of a lifelong love of music and an obsession with patterns. The authors have for many years been exploring methods to find, create, describe and analyze patterns. They wrote this book to show how some of these methods can be used to generate rhythms. The methods can produce an almost endless variety of new rhythms along with popular traditional ones. For a lover of music what could be more wonderful than that?\u003c\/p\u003e \u003cp\u003eThe study of patterns at anything beyond a superficial level does require some mathematics. Fortunately the mathematics can be kept at a very elementary level. Anyone comfortable with a little algebra should have no trouble understanding and using these rhythm generation methods. Only the last chapter on stochastic rhythms requires a bit more than elementary mathematics. Any reader who faints at the sight of an equation should probably not buy the book.\u003c\/p\u003e \u003cp\u003eThe book has many example rhythms for which there are MIDI files that you can listen to on the book's website: abrazol.com\/books\/rhythm1\/. There you can also find free software for generating rhythms, doing calculations, and creating MIDI files.\u003c\/p\u003e","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50392848892177,"sku":"CIN1887187227G","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":51228520317201,"sku":"GOR014203532","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":52755957285137,"sku":"NIN9781887187220","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":52816612065553,"sku":"CIN1887187227VG","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1887187227.jpg?v=1750834326"},{"product_id":"creating-melodies-book-stefan-hollos-9781887187367","title":"Creating Melodies","description":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eCreating Melodies\u003c\/strong\u003e\u003c\/em\u003e is a companion to our book \u003cem\u003e\u003cstrong\u003eCreating Rhythms\u003c\/strong\u003e\u003c\/em\u003e. The book will show you how to explore the infinite space of melodies.\u003c\/p\u003e\u003cp\u003eLike the rhythms book, it comes with software that you can download from the book's website. With the software you can generate countless new melodies and variations on old ones. Combine those melodies with rhythms from our rhythms book and you have an endless source of inspiration for new music.\u003c\/p\u003e\u003cp\u003eYou can listen to the many example melodies in this book as MIDI and MP3 files on the book's website at http: \/\/www.abrazol.com\/books\/melody1\/\u003c\/p\u003e\u003cp\u003eWe had a lot of fun writing this book. In fact it took us so long to write it because it was hard to tear ourselves away from creating so much interesting new music and write up the results. Now you can have fun with it too \u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50392852660497,"sku":"CIN1887187367G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":52755534381329,"sku":"NIN9781887187367","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1887187367.jpg?v=1751060937"},{"product_id":"signals-from-the-subatomic-world-book-stefan-hollos-9781887187008","title":"Signals from the Subatomic World","description":null,"brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51054935802129,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":51054938456337,"sku":"NIN9781887187008","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1887187006.jpg?v=1750995881"},{"product_id":"struwwelpeter-book-stefan-hollos-9781887187435","title":"Struwwelpeter","description":"A new translation of the German children's book Der Struwwelpeter by Heinrich Hoffmann in dual language format, with German on the left facing page and English on the right. Includes the original German verse and color illustrations of 1845, and a translation of Heinrich Hoffmann's account of the book's creation. The original German book is one that virtually all German children become familiar with. The book's title is the character in the first of 10 stories comprising the book.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51054985052433,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":51054987510033,"sku":"NIN9781887187435","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/188718743X.jpg?v=1751186593"},{"product_id":"bet-smart-book-stefan-hollos-9781887187015","title":"Bet Smart","description":"In 1956, a physicist named John Kelly working at Bell Labs published a paper titled A New Interpretation of Information Rate. In the paper he draws an analogy between the outcomes of a gambling game and the transmission of symbols over a communications channel. For a positive expectation game, Kelly showed that a betting system based on a fixed fraction of the bankroll can make the bankroll grow at an exponential rate in the long run. The exponential growth rate in this case is directly analogous to the rate of information transmission through a communications channel. This book examines the Kelly system in detail. Applications of the Kelly system in both gambling and investing are considered. Python code for calculating the Kelly fractions for both a single stock investment and an investment in two stocks simultaneously is included. Included is an introductory review chapter on the probability theory needed to analyze gambling systems in general.There is also a chapter on some of the more commonly used gambling systems such as the Martingale system. 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The goal is to get the reader to the point where he or she can understand and use these filters as quickly as possible. To accomplish this we have kept the amount of mathematical background material to a minimum and have included many examples. But make no mistake, this is not a book for dummies or complete idiots. Some degree of mathematical sophistication is required. If you have never used complex numbers and do not know what Euler's identity is, then this book is not for you. If you have a basic physical science mathematics background, then you should have no problem with this book. We start with a short introduction to the minimum mathematics required to describe, use, and design recursive digital filters. This includes a description of the z-transform, filter system functions, and the frequency response. This is followed by examples of the simplest possible low pass, high pass, band pass, and band stop filters. There are examples showing how to use all these filters. A section on band stop filter banks is also included. The design portion of the book covers impulse invariance and bilinear transform design. We give a minimum theoretical description of these methods and plenty of examples. For the bilinear transform method we show how to turn analog low pass Butterworth filters into digital low pass, high pass, band pass, and band stop filters. Being able to convert analog filters to digital is useful because analog filter design is a more mature and well understood subject. The final section of the book is on analog Butterworth filters. The filter software used in this book is written by the authors, and is available free on the book's web page at http: \/\/www.abrazol.com\/books\/filter1\/ The programs are written in the C programming language, and will have to be compiled before you can use them. You do not have to know C to use the programs or understand the contents of the book. There is a C language compiler for every major operating system. 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