Manuel Kauers - Böcker

Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail.The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix.When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.

Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail.The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix.When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.

This book constitutes the proceedings of the 6th International Conference on Mathematical Software, ICMS 2018, held in South Bend, IN, USA, in July 2018.The 59 papers included in this volume were carefully reviewed and selected from numerous submissions. The program of the 2018 meeting consisted of 20 topical sessions, each of which providing an overview of the challenges, achievements and progress in a subeld of mathematical software research, development and use.

This volume contains the collected contributions of two conferences, Calcu- mus2007andMKM2007.Calculemus2007wasthe14thinaseriesofconferences dedicated to the integration of computer algebra systems (CAS) and automated deduction systems (ADS). MKM 2007 was the sixth International Conference on Mathematical Knowledge Management, an emerging interdisciplinary ?eld of research in the intersection of mathematics, computer science, library s- ence, and scienti?c publishing. Both conferences aimed to provide mechanized mathematical assistants. Although the two conferences have separate communities and separate foci, there is a signi?cant overlap in the interests in building mechanized mathem- ical assistants. For this reason it was decided to collocate the two events in 2007 for the ?rst time, at RISC in Hagenberg, Austria. The number and quality of the submissions show that this was a good decision. While the proceedings are shared, the submission process was separate. The responsibility for acc- tance/rejection rests completely with the two separate Program Committees. By this collocation we made a contribution against the fragmentation of communities which work on di?erent aspects of di?erent independent branches, traditional branches (e.g., computer algebra and theorem proving), as well as newly emerging ones (on user interfaces, knowledge management, theory exp- ration, etc.). This will also facilitate the development of integrated mechanized mathematical assistants that will be routinely used by mathematicians, c- puter scientists, and engineers in their every-day business.

Das Buch soll eine fundamentale mathematische Frage für Leserinnen und Leser ohne besondere Fachkenntnisse nachvollziehbar erörtern: Wie werden Zahlen miteinander multipliziert? Das wissen wir im Prinzip seit Jahrtausenden, und die Kinder lernen es in der Schule. Aber damit ist das Thema noch längst nicht erledigt. Für sehr große Zahlen ist das Schulverfahren nämlich unbrauchbar. Man kennt andere Verfahren, die für große Zahlen sehr viel besser funktionieren. In dem Buch wird die Funktionsweise eines solchen modernen Verfahrens allgemeinverständlich dargestellt.Dazu erklären wir zunächst, was bei den herkömmlichen Verfahren das Problem ist. Dann entwickeln wir Schritt für Schritt die Ideen, auf denen moderne Multiplikationsverfahren beruhen. Die Darstellung ist informal und richtet sich an mathematisch interessierte Laien. Auch Schülerinnen und Schüler ab Ende der Mittelstufe können den Text gut lesen, weil er keinerlei Kenntnisse der höheren Mathematik voraussetzt. Es genügt, wenn man einfache Formeln lesen und elementare Umformungen von Gleichungen nachvollziehen kann. Alles, was wir darüber hinaus benötigen, wird im Buch anschaulich erklärt. Es handelt sich dabei um mathematische Konzepte, die in der breiten Öffentlichkeit weitgehend unbekannt sind, und die auch unabhängig von der konkreten Fragestellung, wie man Zahlen multipliziert, spannend und interessant sind. So vermittelt das Buch insgesamt einen realistischen Eindruck davon, womit sich forschende Mathematikerinnen und Mathematiker beschäftigen.