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The theory of slice regular functions over quaternions is the central subject of the present volume. This recent theory has expanded rapidly, producing a variety of new results that have caught the attention of the international research community. At the same time, the theory has already developed sturdy foundations. The richness of the theory of the holomorphic functions of one complex variable and its wide variety of applications are a strong motivation for the study of its analogs in higher dimensions. In this respect, the four-dimensional case is particularly interesting due to its relevance in physics and its algebraic properties, as the quaternion forms the only associative real division algebra with a finite dimension n>2. Among other interesting function theories introduced in the quaternionic setting, that of (slice) regular functions shows particularly appealing features. For instance, this class of functions naturally includes polynomials and power series. The zero set of a slice regular function has an interesting structure, strictly linked to a multiplicative operation, and it allows the study of singularities. Integral representation formulas enrich the theory and they are a fundamental tool for one of the applications, the construction of a noncommutative functional calculus. The volume presents a state-of-the-art survey of the theory and a brief overview of its generalizations and applications. It is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.

Algebra. --- Mathematics. --- Polynomials. --- Functions, Quaternion --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Functional analysis. --- Functions of complex variables. --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Complex variables --- Functional calculus --- Math --- Sequences (Mathematics). --- Functions of a Complex Variable. --- Sequences, Series, Summability. --- Functional Analysis. --- Elliptic functions --- Functions of real variables --- Calculus of variations --- Functional equations --- Integral equations --- Science --- Algebra

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Nestled between number theory, combinatorics, algebra, and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e. sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.  .

Mathematics. --- Additive combinatorics --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Algebra. --- Ordered algebraic structures. --- Sequences (Mathematics). --- Number theory. --- Number Theory. --- Sequences, Series, Summability. --- Order, Lattices, Ordered Algebraic Structures. --- Isoperimetric inequalities. --- Math --- Science --- Geometry, Plane --- Inequalities (Mathematics) --- Mathematical analysis --- Mathematical sequences --- Numerical sequences --- Number study --- Numbers, Theory of --- Algebraic structures, Ordered --- Structures, Ordered algebraic

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The purpose of these lecture notes is to develop a theory of asymptotic expansions for functions involving two variables, while at the same time using functions involving one variable and functions of the quotient of these two variables. Such composite asymptotic expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed ordinary differential equations near turning points. CAsEs imply inner and outer expansions near turning points. Thus our approach is closely related to the method of matched asymptotic expansions. CAsEs offer two unique advantages, however. First, they provide uniform expansions near a turning point and away from it. Second, a Gevrey version of CAsEs is available and detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The first application concerns canard solutions near a multiple turning point. The second application concerns so-called non-smooth or angular canard solutions. Finally an Ackerberg-O’Malley resonance problem is solved.

Asymptotic expansions --- Differential equations --- Integral equations --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Mathematical Theory --- Operations Research --- Asymptotic theory --- Asymptotic expansions. --- Asymptotic theory. --- Asymptotic theory in integral equations --- 517.91 Differential equations --- Asymptotic developments --- Mathematics. --- Approximation theory. --- Differential equations. --- Sequences (Mathematics). --- Approximations and Expansions. --- Ordinary Differential Equations. --- Sequences, Series, Summability. --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Differential Equations. --- Mathematical sequences --- Numerical sequences --- Algebra --- Math --- Science --- Theory of approximation --- Functional analysis --- Polynomials --- Chebyshev systems

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This is the first comprehensive monograph on the mathematical theory of the solitaire game “The Tower of Hanoi” which was invented in the 19th century by the French number theorist Édouard Lucas. The book comprises a survey of the historical development from the game’s predecessors up to recent research in mathematics and applications in computer science and psychology. Apart from long-standing myths it contains a thorough, largely self-contained presentation of the essential mathematical facts with complete proofs, including also unpublished material. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms and their correctness proofs form an essential part of the book. In view of the most important practical applications of the Tower of Hanoi and its variants, namely in physics, network theory, and cognitive (neuro)psychology, other related structures and puzzles like, e.g., the “Tower of London”, are addressed. Numerous captivating integer sequences arise along the way, but also many open questions impose themselves. Central among these is the famed Frame-Stewart conjecture. Despite many attempts to decide it and large-scale numerical experiments supporting its truth, it remains unsettled after more than 70 years and thus demonstrates the timeliness of the topic. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike.

Mathematical recreations. --- Mathematics -- Humor. --- Mathematics. --- Puzzles. --- Mathematical recreations --- Mathematics --- Physical Sciences & Mathematics --- Elementary Mathematics & Arithmetic --- Mathematical Theory --- History --- History. --- Mathematical puzzles --- Number games --- Recreational mathematics --- Recreations, Mathematical --- Algorithms. --- Sequences (Mathematics). --- Game theory. --- Combinatorics. --- Mathematics, general. --- History of Mathematical Sciences. --- Sequences, Series, Summability. --- Game Theory, Economics, Social and Behav. Sciences. --- Algorithm Analysis and Problem Complexity. --- Puzzles --- Scientific recreations --- Games in mathematics education --- Magic squares --- Magic tricks in mathematics education --- Computer software. --- Mathematical sequences --- Numerical sequences --- Algebra --- Math --- Science --- Software, Computer --- Computer systems --- Combinatorics --- Mathematical analysis --- Algorism --- Arithmetic --- Games, Theory of --- Theory of games --- Mathematical models --- Annals --- Auxiliary sciences of history --- Foundations

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The book begins at an undergraduate student level, assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, the Lebesgue integral, vector calculus and differential equations. After having created a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis as understood by mathematicians today.

Mathematical analysis. --- Mathematics. --- Matrix theory. --- Algebra. --- Functions of complex variables. --- Measure theory. --- Differential equations. --- Functions of real variables. --- Sequences (Mathematics). --- Real Functions. --- Linear and Multilinear Algebras, Matrix Theory. --- Measure and Integration. --- Functions of a Complex Variable. --- Ordinary Differential Equations. --- Sequences, Series, Summability. --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Real variables --- Functions of complex variables --- 517.91 Differential equations --- Differential equations --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Complex variables --- Elliptic functions --- Functions of real variables --- Mathematical analysis --- Math --- Science --- Differential Equations.