Mathematics Study Resources – serie

The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process, the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book.In Part I, it is shown that there are 17 principally different possibilities of tesselation of the plane, the so-called 'plane crystal groups'. Complementary to this, ideas of Harald Heesch are described, who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself – following in the footsteps of Escher, so to speak – to create artistically sophisticated tesselation.In the corresponding investigations forthe complex plane in Part II, movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Möbius transformations: Kleinian groups, Schottky groups, etc. There are also interesting connections to hyperbolic geometry.Finally, in Part III, a third aspect of the subject is treated, the Penrose tesselation. This concerns results from the seventies, when easily describable and provably non-periodic parquetisations of the plane were given for the first time.

With this book, which is based on the third edition of a book first written in German about random walks, the author succeeds in a remarkably playful manner in captivating the reader with numerous surprising random phenomena and non-standard limit theorems related to simple random walks and related topics.

The aim of this book is to provide a comprehensive introduction to solving large systems of equations.In addition to direct algorithms, it presents a wide range of classical and modern solvers – from splitting methods and multigrid techniques to current Krylov subspace methods (CG, GMRES, BiCGSTAB, etc.). These methods are discussed both mathematically and in terms of their practical applications. The book also offers an in-depth treatment of preconditioning techniques to accelerate existing methods.The book covers all the necessary fundamentals, making it highly suitable for self-study. The presentation of the derived algorithms allows for straightforward implementation in any programming language. Detailed MATLAB® implementations of common Krylov methods are included in the appendix. Solutions and additional materials are available online. This book is a translation of the original German 6th edition. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book may read stylistically differently from a conventional translation.

The book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics.

This book explains compactly, without theoretical superstructure and with as little mathematical formalism as possible, the essential concepts in the encryption of messages and data worthy of protection.

This textbook is an introduction to global optimization, which treats mathematical facts stringently on the one hand, but also motivates them in great detail and illustrates them with 80 figures. The book is therefore not only aimed at mathematicians, but also at natural scientists, engineers and economists who want to understand and apply mathematically sound methods in their field.With almost two hundred pages, the book provides enough choices to use it as a basis for differently designed lectures on global optimization. The detailed treatment of the global solvability of optimization problems under application-relevant conditions sets a new accent that enriches the stock of previous textbooks on optimization. Using the theory and algorithms of smooth convex optimization, the book illustrates that the global solution of a class of optimization problems frequently encountered in practice is efficiently possible, while for the more difficult-to-handle non-convex problems itdevelops in detail the ideas of branch-and-bound methods.This book is the English translation of the 2nd edition of “Grundzüge der Globalen Optimierung” (Springer, 2021) written in German. The translation was done with the help of artificial intelligence. A subsequent revision was performed by the author to further refine the work and to ensure that the translation is appropriate concerning content and scientific correctness. It may, however, read stylistically different from a conventional translation.

Why is the squaring of the circle, why is the division of angles with compass and ruler impossible? Why are there general solution formulas for polynomial equations of degree 2, 3 and 4, but not for degree 5 or higher?This textbook deals with such classical questions in an elementary way in the context of Galois theory. It thus provides a classical introduction and at the same time deals with applications. The point of view of a constructive mathematician is consistently adopted: To prove the existence of a mathematical object, an algorithmic construction of that object is always given. Some statements are therefore formulated somewhat more cautiously than is classically customary; some proofs are more elaborately conducted, but are clearer and more comprehensible. Abstract theories and definitions are derived from concrete problems and solutions and can thus be better understood and appreciated.The material in this volume can be covered in a one-semester lecture on algebra right at the beginning of mathematics studies and is equally suitable for first-year students at the Bachelor's level and for teachers.The central statements are already summarised and concisely presented within the text, so the reader is encouraged to pause and reflect and can repeat content in a targeted manner. In addition, there is a short summary at the end of each chapter, with which the essential arguments can be comprehended step by step, as well as numerous exercises with an increasing degree of difficulty.The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.

In this book, the most important cryptographic algorithms, protocols and constructions of the last decades are explained and justified. The topics discussed range from public-key cryptography and zero-knowledge protocols to cryptography in blockchains, the Tor network and quantum.The authors succeed in presenting these complex topics in a simple and clear manner. Hardly any prior knowledge is required, as all the necessary mathematical and cryptographic background is provided in the book.The book has a modular structure so that the individual chapters can be read independently of one another. Each topic is presented at different levels: First, the concept is described as a mental model, then it is studied from a mathematical point of view, and finally the strengths and weaknesses of the respective protocol are discussed. The focus is on a high-level perspective rather than full formal proofs.The book is suitable for undergraduate students (e.g., in mathematics, computer science, electrical engineering) as well as for Internet users, tech managers, software developers and data protection/data security officers.This book is an expanded translation of the original German 9th edition Moderne Verfahren der Kryptographie by Albrecht Beutelspacher, Jörg Schwenk and Klaus-Dieter Wolfenstetter, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2022. The translation was done with the help of artificial intelligence. The authors have subsequently revised and extended the text to refine the readability of the text, and to add novel content.

In addition to elementary algebraic structures such as groups, rings and solids, Galois theory in particular is developed together with its applications to the cyclotomic fields, finite fields or the question of the resolution of polynomial equations. Special attention is paid to the natural development of the contents.

This textbook, which is based on the second edition of a book that has been previously published in German language, provides a comprehension-oriented introduction to asymptotic stochastics. It is aimed at the beginning of a master's degree course in mathematics and covers the material that can be taught in a four-hour lecture with two-hour exercises. Individual chapters are also suitable for seminars at the end of a bachelor's degree course.In addition to more basic topics such as the method of moments in connection with the convergence in distribution or the multivariate central limit theorem and the delta method, the book covers limit theorems for U-statistics, the Wiener process and Donsker's theorem, as well as the Brownian bridge, with applications to statistics. It concludes with a central limit theorem for triangular arrays of Hilbert space-valued random elements with applications to weighted L² statistics. The book is deliberately designed forself-study. It contains 138 self-questions, which are answered at the end of each chapter, as well as 194 exercises with solutions.This book is a translation of an original German edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

Starting with the Shannon-Wiener approach to mathematical information theory, allowing a mathematical "measurement" of an amount of information, the book begins by defining the terms message and information and axiomatically assigning an amount of information to a probability.

This textbook presents a direct path from real analysis of one variable to function theory.

The discussion of ways to compare mathematical fields also provides introductions to categories and sheaves, whose ever-increasing role in modern mathematics suggests a more prominent role in teaching.The book is the English translation of the second edition of “Mathematische Strukturen” (Springer, 2024) written in German.

Inspired by recent developments in dependent type theory and infinity categories, this book presents a history of ideas around the topics of truth, proof, equality and equivalence. The main focus of this textbook is on dependent type theory and its recent variant homotopy type theory.

The Musical Scale – a Triviality or a Problem?This book examines this provocative question. It soon becomes clear that combining tones into “harmonious” tonal systems represents a challenge of unexpected complexity, encompassing a remarkably large number of interconnected problems.Why does a scale have precisely 12 tones? And could there be others?Aren't 12 perfect fifths always exactly the same as 7 octaves?What exactly is "consonance" or when are intervals "pure," when "impure"?What is meant by "temperament characteristic" in relation to whole and half tones?What does "old tuning" mean – and is there a "new" one that differs from it, and if so, what exactly are the differences?and many similar ones quickly show that their answers not only require well-considered justifications but are also closely interconnected. In this examination, "mathematics" plays a key role. Starting with "simple proportions and numerical relationships", a veritable network emerges, in which both the methods of scale generation with their circle of wolf fifths and Euler grid selection procedures, as well as the models of temperament systems, can be scientifically explained. In three parts,a modern interval arithmetic of musical intervals and their theory of division, decomposition, and construction of musical intervals, driven by prime numbers,the architectural laws of musical scales with their models and patterns, their step geometries and characteristics, their semitones and commas, as well as the combinatorial diversity of all scale-internal structures,a mathematical-methodical description of the central temperament systems with a new, consistent systematics of significant historical tunings are presented and equipped with numerous examples. The music-mathematical calculations and explanatory arguments require only basic school-level knowledge, which are then developed into a suitable algebra and analysis and applied to musical themes. A small peculiarity: stories from the fairytale world of musical mythical creatures accompany the text and explain it in an amusing way.This book is a translation of the original German edition Die Tonleiter und ihre Mathematik, 4th edition, by Karlheinz Schüffler, published by Springer-Verlag GmbH, DE in 2026. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

This book provides essential foundations of mathematics, covering set theory, algebra, the theory of real and complex numbers, and topology.

This textbook is an introduction to nonlinear optimization, which treats mathematical concepts stringently on the one hand, but also motivates them in great detail and illustrates them with 42 figures.

This textbook presents the topics typically covered in a standard course on discrete structures. It is aimed at students of computer science and mathematics (teaching degree and Bachelor's/Master's) and is designed to accompany lectures, for self-study, and for exam preparation.Through explanatory introductions to definitions, numerous examples, counterexamples, diagrams, cross-references, and outlooks, the authors manage to present the wide range of topics concisely and comprehensibly.Numerous exercises facilitate the deepening of the material. Due to its compact presentation of all important discrete and algebraic structures and its extensive index, the book also serves as a reference for mathematicians, computer scientists, and natural scientists.Contents: From propositional and predicate logic to sets and combinatorics, numbers, relations and mappings, graphs, to the rich spectrum of algebraic structures, and a brief introduction to category theory. Additional chapters include rings and modules as well as matroids.This book is a translation of the second German edition. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content, so the book may read stylistically differently from a conventional translation.

The covered topics include the history of mathematical logic, formal systems, axiomatic number theory and set theory, proof theory, Gödel's incompleteness theorems, computability theory, algorithmic information theory, and model theory.

What kind of objects does mathematics investigate, and in what sense do these objects exist? Why are we justified in considering mathematical statements as part of our knowledge, and how can they be validated?A philosophy of mathematics seeks to answer such questions. In this introduction, we present the major positions in the philosophy of mathematics and formulate their core ideas into clear, accessible theses. Readers will learn which philosophers developed each position and the historical context in which they emerged. Drawing on fundamental intuitions and scientific findings, one can argue for or against these theses – such arguments form the second focus of this book.The book aims to encourage readers to reflect on the philosophy of mathematics, to develop their own stance, and to learn how to argue for it.This book is a translation of the original German 2nd edition. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book may read stylistically differently from a conventional translation.

This textbook presents the theoretical foundations of numerical mathematics in a modern, application-oriented, and comprehensive way. In addition to the standard content, it includes numerous examples and practical excursions to sustainably enhance understanding. Proofs are presented in a very detailed, step-by-step manner. The recurring core concepts of numerical mathematics like accuracy, efficiency, robustness and stability are explicitly addressed and clearly distinguished from one another. Specific example calculations for the described algorithms are carried out using provided Python codes. Numerical modeling for aspects of machine learning and neural networks are taken into consideration. Furthermore, numerical methods from linear algebra and analysis are presented separately, which significantly facilitates students’ access to numerical mathematics – based on the two main lectures within undergraduate studies in mathematics. Therefore, the book is ideally suited for students of mathematics, physics, computer science, or engineering. This book is a translation of the original German 2nd edition of “Einführung in die Numerische Mathematik” (Springer, 2024). The translation was done with the help of artificial intelligence. A subsequent revision was performed by the authors to further refine the work and to ensure that the translation is appropriate concerning content and scientific correctness.

This book presents methods from the history of mathematics that have not become standard methods or even made it into classic textbooks. These alternative approaches were often popular at the time of their creation, but then lost attention compared to today's standard methods and eventually fell into oblivion. Often these methods still contain unused potential: It is worth developing them further and discovering where exactly their limits lie. The book aims to provide prospective teachers with a view beyond the usual teaching content and provide content suggestions for working with interested and talented students. For example, Archimedes developed a method for calculating the area of a parabolic segment, which in a way anticipates a piece of integral calculus. However, Archimedes' method did not develop into the standard method found in today's textbooks. Instead, you will find the methods designed by Newton and Leibniz. This book develops the Archimedean method further and shows its "residual potential": Other curves, not just parabolas, can be similarly approached and it is interesting and instructive to see how far Archimedes' method can be developed and where it ultimately reaches its limits.This book is a translation of the original German edition Seitenwege in der Mathematikgeschichte (2024). The translation was done with the help of an artificial intelligence machine translation tool. In the subsequent editing, the author and his colleague Helmer Aslaksen (Uni Oslo) carefully reviewed the translation. Still, the book may read stylistically differently from a conventional translation.

This book presents about 365 different proofs of the Pythagorean Theorem in a highly visual and accessible form, while also placing the theorem and its wide variety of proofs into both a mathematical and cultural context. Here, the theorem becomes a model for the discoveries of ancient mathematics, showing how mathematical truths build upon one another. The collection also gives students a unique insight into the methods of mathematics and its deductive structure.The collection of proofs is based on the work of Elisha Scott Loomis (1852–1940), who gathered, systematized, and published hundreds of algebraic and geometric proofs. His work became a crystallization point for the intellectual and cultural history of mathematics, exemplarily condensed in the Pythagorean theorem. After more than 80 years, the Loomis collection is now published again in a thoroughly revised and expanded edition.This book is a translation of the original German edition Der Satz des Pythagoras in 365 Beweisen by Mario Gerwig, published by Springer-Verlag GmbH Germany in 2021. It omits a chapter from the original edition that was tailored primarily to educational contexts in German-speaking countries. The translation was done with the help of artificial intelligence. The author has subsequently revised the text further in an endeavour to refine the work stylistically. Nevertheless, the book might read stylistically different from a conventional translation.From the foreword by the National Council of Teachers of Mathematics (NCTM):May this volume inspire teachers to bring the wonder of the Pythagorean Theorem to their students with renewed clarity and enthusiasm. And may it remind us all that even the most familiar theorems still hold surprises, inviting us to look again, reason again, and delight again in the power of mathematics.From the foreword by Prof. Günter M. Ziegler (President of Freie Universität Berlin, Germany):This book presents a great variety of proofs for a very specific, interesting and fascinating statement – from a historical source, expertly and critically commented. There is much to learn from this book: Learn about the variety of proofs, be inspired by them and take pleasure in them.

This book is a guide for self-study of mathematics as a scientific discipline. It addresses a diverse audience: young people interested in gaining a deeper understanding of the material presented in school, teachers who run math clubs and are looking for material that connects to the school curriculum, university members curious about an alternative perspective on mathematics as a field of study, professionals from other disciplines who have a basic mathematical background and wish to deepen their understanding of mathematics.This book assumes an intuitive grasp of natural numbers and solid mastery of the calculation techniques taught in lower secondary school. Starting from elementary problems in enumerative combinatorics, it first offers a systematic description of the natural numbers. Considerations on solving equations then lead, step by step, to the introduction of real numbers. This enables the development of powerful concepts for modeling distances and other quantities such as areas and volumes. The study of systems of equations further introduces the concepts of “vector space” and “linearization,” which allow computations through linear approximation.In this way, this book conveys the fundamental content of the first three to four semesters of a mathematics degree program. It deliberately avoids the usual division into separate subjects such as “Linear Algebra,” “Analysis,” or “Probability.” Instead, the problems arise from questions about mathematical modeling or from interesting mathematical ideas.This book is a translation of the original German edition, enriched with exercises and solutions. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that this book may read stylistically differently from a conventional translation.