This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.

Shortlisted for the Royal Society Science Book Prize 2019 A magisterial history of calculus (and the people behind it) from one of the world's foremost mathematicians. This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel. But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death. Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Archimedes to today's breakthroughs in chaos theory and artificial intelligence. Filled with idiosyncratic characters from Pythagoras to Fourier, Infinite Powers is a compelling human drama that reveals the legacy of calculus on nearly every aspect of modern civilisation, including science, politics, medicine, philosophy, and much besides.

The book deals with continuous piecewise linear differential systems in the plane with three pieces separated by a pair of parallel straight lines. Moreover, these differential systems are symmetric with respect to the origin of coordinates. This class of systems driven by concrete applications is of interest in engineering, in particular in control theory and the design of electric circuits. By studying these particular differential systems we will introduce the basic tools of the qualitative theory of ordinary differential equations, which allow us to describe the global dynamics of these systems including the infinity. The behavior of their solutions, their parametric stability or instability and their bifurcations are described. The book is very appropriate for a first course in the qualitative theory of differential equations or dynamical systems, mainly for engineers, mathematicians, and physicists.

This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and others. In most cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural and can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. The material is presented in a systematic and rigorous manner that allows readers to grasp the key ideas and to gain a working knowledge of the theory. Despite the fact that the work is largely self-contained, extensive bibliographic references are included, and open problems and further development directions are suggested when applicable. The monograph is targeted mainly at the mathematical research community but it is also accessible to graduate students interested in functional analysis and its applications. It could also serve as a text for an advanced course in fixed point theory of mappings acting in modular function spaces.

With applications to climate, technology, and industry, the modeling and numerical simulation of turbulent flows are rich with history and modern relevance. The complexity of the problems that arise in the study of turbulence requires tools from various scientific disciplines, including mathematics, physics, engineering and computer science.

Authored by two experts in the area with a long history of collaboration, this monograph provides a current, detailed look at several turbulence models from both the theoretical and numerical perspectives. The k-epsilon, large-eddy simulation and other models are rigorously derived and their performance is analyzed using benchmark simulations for real-world turbulent flows.

"Mathematical and Numerical Foundations of Turbulence Models and Applications" is an ideal reference for students in applied mathematics and engineering, as well as researchers in mathematical and numerical fluid dynamics. It is also a valuable resource for advanced graduate students in fluid dynamics, engineers, physical oceanographers, meteorologists and climatologists.

Discover an accessible and easy-to-use guide to calculus fundamentals In Quick Calculus: A Self-Teaching Guide, 3rd Edition, a team of expert MIT educators delivers a hands-on and practical handbook to essential calculus concepts and terms. The author explores calculus techniques and applications, showing readers how to immediately implement the concepts discussed within to help solve real-world problems. In the book, readers will find: An accessible introduction to the basics of differential and integral calculus An interactive self-teaching guide that offers frequent questions and practice problems with solutions. A format that enables them to monitor their progress and gauge their knowledge This latest edition provides new sections, rewritten introductions, and worked examples that demonstrate how to apply calculus concepts to problems in physics, health sciences, engineering, statistics, and other core sciences. Quick Calculus: A Self-Teaching Guide, 3rd Edition is an invaluable resource for students and lifelong learners hoping to strengthen their foundations in calculus.

This book presents fractional difference, integral, differential, evolution equations and inclusions, and discusses existence and asymptotic behavior of their solutions. Controllability and relaxed control results are obtained. Combining rigorous deduction with abundant examples, it is of interest to nonlinear science researchers using fractional equations as a tool, and physicists, mechanics researchers and engineers studying relevant topics. Contents Fractional Difference Equations Fractional Integral Equations Fractional Differential Equations Fractional Evolution Equations: Continued Fractional Differential Inclusions

What you ll find in this monograph is nothing less than a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but who does not have an extensive background in the subject and does not plan to make a career as a functional analyst. It develops the topological structures in connection with a number of topic areas such as measure theory, convexity, and Banach lattices, as well as covering the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature.

Many phenomena of interest for applications are represented by differential equations which are defined in a domain whose boundary is a priori unknown, and is accordingly named a "free boundary." A further quantitative condition is then provided in order to exclude indeterminacy. Free boundary problems thus encompass a broad spectrum which is represented in this state-of-the-art volume by a variety of contributions of researchers in mathematics and applied fields like physics, biology and material sciences. Special emphasis has been reserved for mathematical modelling and for the formulation of new problems.

The present book is a valuable continuation of the large cycle of the author's investigations on harmonic analysis in the complex domain. For certain sets of segments in the complex domain, the elegant and explicit apparatus of the biorthogonal Fourier type systems (basis systems in the Rieszian sense), is constructed by purely analytic methods of classical function theory. This is done using the remarkable asymptotic properties of the Mittag-Leffler type entire functions and new interpolation theorems for the Banach spaces of entire functions. It is especially noteworthy that at the same time the constructed basis systems are eigenfunctions for quite non-ordinary boundary value problems in the complex domain for differential equations of fractional order. Such boundary value problems, the solutions of which are carried through to a logical conclusion, i.e. up to the theorems of expansions into eigenfunctions, are considered in the book for the first time.

The purpose of the corona workshop was to consider the corona problem in both one and several complex variables, both in the context of function theory and harmonic analysis as well as the context of operator theory and functional analysis. It was held in June 2012 at the Fields Institute in Toronto, and attended by about fifty mathematicians. This volume validates and commemorates the workshop, and records some of the ideas that were developed within. The corona problem dates back to 1941. It has exerted a powerful influence over mathematical analysis for nearly 75 years. There is material to help bring people up to speed in the latest ideas of the subject, as well as historical material to provide background. Particularly noteworthy is a history of the corona problem, authored by the five organizers, that provides a unique glimpse at how the problem and its many different solutions have developed. There has never been a meeting of this kind, and there has never been a volume of this kind. Mathematicians-both veterans and newcomers-will benefit from reading this book. This volume makes a unique contribution to the analysis literature and will be a valuable part of the canon for many years to come.

This is an introduction to classical and quantum mechanics on two-point homogenous Riemannian spaces, empahsizing spaces with constant curvature. Chapters 1-4 provide basic notations for studying two-body dynamics. Chapter 5 deals with the problem of finding explicitly invariant expressions for the two-body quantum Hamiltonian. Chapter 6 addresses one-body problems in a central potential. Chapter 7 investigates the classical counterpart of the quantum system introduced in Chapter 5. Chapter 8 discusses applications in the quantum realm.

This volume presents a collection of selected papers by the prominent Brazilian mathematician Djairo G. de Figueiredo, who has made significant contributions in the area of Differential Equations and Analysis. His work has been highly influential as a challenge and inspiration to young mathematicians as well as in development of the general area of analysis in his home country of Brazil.

In addition to a large body of research covering a variety of areas including geometry of Banach spaces, monotone operators, nonlinear elliptic problems and variational methods applied to differential equations, de Figueiredo is known for his many monographs and books. Among others, this book offers a sample of the work of Djairo, as he is commonly addressed, advancing the study of superlinear elliptic problems (both scalar and system cases), including questions on critical Sobolev exponents and maximum principles for non-cooperative elliptic systems in Hamiltonian form.

The present monograph is the successor of Direct methods in the calculus of variations which was published in the Applied Mathematical Sciences series and is currently out of print. Although the core and the structure of the present book is similar to the old one, it is much more than a revised version. Fifteen years have passed since the publication of the Direct methods in the calculus of variations book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, which is reflected in the change of the title. Indeed a new subject, quasiconvex analysis has now been developed. The present monograph, which is essentially a reference book on the subject of quasiconvex analysis, can be used, as was the earlier book for an advanced course on the calculus of variations.

The revised and enlarged third edition of this successful book presents a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied and updated applications. In an effort to make the book more useful for a diverse readership, updated modern examples of applications are chosen from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation.

"Nonlinear Partial Differential Equations for Scientists and Engineers, Third Edition, "improves onanalready highlycomplete and accessible resource for graduate students and professionals in mathematics, physics, science, and engineering. It may be used to great effect as a course textbook, research reference, or self-study guide."

This book collects lectures given by the plenary speakers at the 10th International ISAAC Congress, held in Macau, China in 2015. The contributions, authored by eminent specialists, present some of the most exciting recent developments in mathematical analysis, probability theory, and related applications. Topics include: partial differential equations in mathematical physics, Fourier analysis, probability and Brownian motion, numerical analysis, and reproducing kernels. The volume also presents a lecture on the visual exploration of complex functions using the domain coloring technique. Thanks to the accessible style used, readers only need a basic command of calculus.

The volume covers wide-ranging topics from Theory: structure of finite fields, normal bases, polynomials, function fields, APN functions. Computation: algorithms and complexity, polynomial factorization, decomposition and irreducibility testing, sequences and functions. Applications: algebraic coding theory, cryptography, algebraic geometry over finite fields, finite incidence geometry, designs, combinatorics, quantum information science.

Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis and in particular the theory of operator semigroups.

This book discusses basic topics in the spectral theory of dynamical systems. It also includes two advanced theorems, one by H. Helson and W. Parry, and another by B. Host. Moreover, Ornstein's family of mixing rank-one automorphisms is given with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are also examined. Baire category theorems of ergodic theory, scattered in literature, are discussed in a unified way in the book. Riesz products are introduced and applied to describe the spectral types and eigenvalues of rank-one automorphisms. Lastly, the second edition includes a new chapter "Calculus of Generalized Riesz Products", which discusses the recent work connecting generalized Riesz products, Hardy classes, Banach's problem of simple Lebesgue spectrum in ergodic theory and flat polynomials.

The Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication. The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural. These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.

"This book addresses mathematical problems motivated by various applications in physics, engineering, chemistry and biology. It gathers the lecture notes from the mini-course presented by Jean-Christophe Mourrat on the construction of the various stochastic "basic" terms involved in the formulation of the dynamic OE4 theory in three space dimensions, as well as selected contributions presented at the fourth meeting on Particle Systems and PDEs, which was held at the University of Minho's Centre of Mathematics in December 2015. The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, offering them a forum to present their recent results and discuss their topics of expertise. The meeting was also intended to present to a vast and varied public, including young researchers, the area of interacting particle systems, its underlying motivation, and its relation to partial differential equations. The book will be of great interest to probabilists, analysts, and all mathematicians whose work focuses on topics in mathematical physics, stochastic processes and differential equations in general, as well as physicists working in statistical mechanics and kinetic theory."

Nonlinear Optimization is an intriguing area of study where mathematical theory, algorithms and applications converge to calculate the optimal values of continuous functions. Within this subject, Global Optimization aims at finding global optima for difficult problems in which many local optima might exist.

This book provides a compelling introduction to global and non-linear optimization providing interdisciplinary readers with a strong background to continue their studies into these and other related fields. The book offers insight in relevant concepts such as "region of attraction" and "Branch-and-Bound" by elaborating small numerical examples and exercises for the reader to follow.

This book presents a rational scheme of analysis for the periodic and quasi-periodic solution of a broad class of problems within technical and celestial mechanics. It develops steps for the determination of sufficiently general averaged equations of motion, which have a clear physical interpretation and are valid for a broad class of weak-interaction problems in mechanics. The criteria of stability regarding stationary solutions of these equations are derived explicitly and correspond to the extremum of a special "potential" function. Much consideration is given to applications in vibrational technology, electrical engineering and quantum mechanics, and a number of results are presented that are immediately useful in engineering practice. The book is intended for mechanical engineers, physicists, as well as applied mathematicians specializing in the field of ordinary differential equations.