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G. H. Hardy (1877-1947) ranks among the great mathematicians of the twentieth century. He did essential research in number theory and analysis, held professorships at Cambridge and Oxford, wrote important textbooks as well as the classic A Mathematician's Apology, and famously collaborated with J. E. Littlewood and Srinivasa Ramanujan. Hardy was a colorful character with remarkable expository skills. This book is a feast of G. H. Hardy's writing. There are selections of his mathematical papers, his book reviews, his tributes to departed colleagues. Some articles are serious, whereas others display a wry sense of humor. And there are recollections by those who knew Hardy, along with biographical and mathematical pieces written explicitly for this collection. Fans of Hardy should find much here to like. And for those unfamiliar with his work, The G. H. Hardy Reader provides an introduction to this extraordinary individual.

Tikhonov regularization is the most popular general-purpose method for regularization, a mathematical technique to suppress the effect of noise in data, and uses much of the machinery of Hilbert space theory. This book develops the theory of Tikhonov regularization for a certain class of linear inverse problems which are defined on Hilbert spaces. To explain why and how Tikhonov regularization works, the singular value expansion for compact operators is introduced. Tikhonov regularization with seminorms is also analyzed and for this purpose, densely defined unbounded operators are addressed and their basic properties presented. In addition, the author provides readers with a quick but thorough review of Hilbert space theory and a brief introduction to weak derivatives and Sobolev spaces. Intended as an expository work for those interested in inverse problems and Tikhonov regularization, including graduates and researchers, the author presents the theory in an engaging and straightforward style.

Several optical phenomena can occur when the sun is near the horizon, some of which can actually be seen, for instance as a dark strip in the middle of the solar disk, or as a green flash. These are a result of the way the atmosphere affects the geometry of light rays. In this book, Broer uses the Fermat principle to deduce laws for refraction and reflection. By expressing these as conservation laws, he can handle both the case of inhomogeneous layers of air and the case of continuous variation in the refraction index. Building from this elementary theory, more advanced mathematical theories are discussed, including how the Fermat Principle leads to a Riemannian metric whose geodesics are the paths of light rays, thus demonstrating the physical applicability of Riemannian geometry. Intended for teachers and professors to enlighten their courses, or for students and researchers in mathematics, science and engineering.

Functional analysis is an abstract and powerful modern theory that occupies a central role in mathematics. This book provides a quick but precise introduction to the subject, covering everything that a beginning graduate student needs to know. The subject has its roots in the theory of infinite-dimensional vector spaces which is where the book begins, with preliminaries from the theory of normed linear spaces, Hilbert spaces, operator algebras and distributions. The reader will then encounter more advanced topics such as spectral theory, convexity and fixed-point theorems. It contains plenty of examples and exercises, making it an ideal basis for advanced undergraduate and graduate level courses in a subject that has become an essential part of every analyst's toolkit.

Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. This book presents techniques for proving a variety of geometric results in three dimensions. Special attention is given to prisms, pyramids, platonic solids, cones, cylinders and spheres, as well as many new and classical results. A chapter is devoted to each of the following basic techniques for exploring space and proving theorems: enumeration, representation, dissection, plane sections, intersection, iteration, motion, projection, and folding and unfolding. The book includes a selection of Challenges for each chapter with solutions, references and a complete index. The text is aimed at secondary school and college and university teachers as an introduction to solid geometry, as a supplement in problem solving sessions, as enrichment material in a course on proofs and mathematical reasoning, or in a mathematics course for liberal arts students.

Beginning with one of the most remarkable ecological collapses of recent times, that of the passenger pigeon, this book surveys collapse processes across the entire spectrum of the natural and man-made world. The reader is taken through extreme weather events, crashing financial markets and companies, the chaotic nature of Earth's orbit, revolutionary political change, the spread of disease and many other fascinating cases. The author's key observation is the consistent occurrence of one or more of six fundamental dynamics ('six sources of collapse'), all of which can be best described and investigated using fundamental mathematical concepts which include low probability events, group dynamics, evolutionary games, instability, nonlinearity and network effects. The exposition assumes a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples. This book will also be of interest to readers in business and finance, engineering, ecology and political science.

Mistakes in mathematical reasoning can range from outlandish blunders to deep and subtle oversights that evade even the most watchful eye. This book represents the second collection of such errors to be compiled by Edward Barbeau. Like Barbeau's previous book, Mathematical Fallacies, Flaws and Flimflam, material is drawn from a variety of sources including the work of students, textbooks, the media, and even professional mathematicians. The errors presented here serve both to entertain, and to emphasise the need to subject even the most 'obvious' assertions to rigorous scrutiny, as intuition and facile reasoning can often be misleading. Each item is carefully analysed and the source of the error is exposed. All students and teachers of mathematics, from school to university level, will find this book both enlightening and entertaining.

C. Edward Sandifer has been studying Euler for decades and is one of the world's leading experts on his work. This book is the second collection of Sandifer's 'How Euler Did It' columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician in history, this volume will leave the reader marveling at Euler's clever inventiveness, which Sandifer explicates and puts in context.

What do mathematicians think of themselves, and what do others think of them? These musings, the theme of a special session at the San Jose MathFest of 2007, resonate with a large and diverse group of mathematicians and students, for these are the questions that govern the way that mathematicians live. A large and diverse group of mathematicians and mathematical people were assembled to offer their views on these matters. The contributions represent a vast array of perspectives on the psychology of the mathematician. It is hoped that readers will find the thoughts assembled here stimulating and cause for further rumination.

Founded in 1961 and still thriving today, the recreational magazine Pythagoras has given generations of high school students in the Netherlands a perspective on many branches of mathematics that are not taught in schools. This book is a selection of the best and most inspiring articles from Pythagoras magazine. It contains a mix of problems - some easy, some challenging, always original - as well as playful introductions to subjects in algebra, geometry, topology, number theory, and more. Concepts like sudoku and the magic square are given a whole new dimension. Articles on Escher's 'impossible objects', a recurring subject over the years, are part of a special section on mathematics and art. Smart high school students and anyone else with a sharp and inquisitive mind will find in this book a treasure trove rich enough to keep one's mind engaged for many weeks and months.

The set of real numbers is one of the fundamental concepts of mathematics. This book surveys alternative number systems: systems that generalise the real numbers yet stay close to the properties that make the reals central to mathematics. There are many alternative number systems, such as multidimensional numbers (complex numbers, quarternions), infinitely small and infinitely large numbers (hyperreal numbers) and numbers that represent positions in games (surreal numbers). Each system has a well-developed theory with applications in other areas of mathematics and science. They all feature in active areas of research and each has unique features that are explored in this book. Alternative number systems reveal the central role of the real numbers and motivate some exciting and eccentric areas of mathematics. What Numbers Are Real? will be an illuminating read for anyone with an interest in numbers, but specifically for advanced undergraduates, graduate students and teachers of university-level mathematics.

This volume is a republication and expansion of the much-loved Wohascum County Problem Book, published in 1993. The original 130 problems have been retained and supplemented by an additional 78 problems. The puzzles contained within, which are accessible but never routine, have been specially selected for their mathematical appeal, and detailed solutions are provided. The reader will encounter puzzles involving calculus, algebra, discrete mathematics, geometry and number theory, and the volume includes an appendix identifying the prerequisite knowledge for each problem. A second appendix organises the problems by subject matter so that readers can focus their attention on particular types of problems if they wish. This collection will provide enjoyment for seasoned problem solvers and for those who wish to hone their skills.

G. H. Hardy (1877-1947) ranks among the great mathematicians of the twentieth century. He did essential research in number theory and analysis, held professorships at Cambridge and Oxford, wrote important textbooks as well as the classic A Mathematician's Apology, and famously collaborated with J. E. Littlewood and Srinivasa Ramanujan. Hardy was a colorful character with remarkable expository skills. This book is a feast of G. H. Hardy's writing. There are selections of his mathematical papers, his book reviews, his tributes to departed colleagues. Some articles are serious, whereas others display a wry sense of humor. And there are recollections by those who knew Hardy, along with biographical and mathematical pieces written explicitly for this collection. Fans of Hardy should find much here to like. And for those unfamiliar with his work, The G. H. Hardy Reader provides an introduction to this extraordinary individual.

Celebrating a century of geometry and geometry teaching, this book will give the reader an enjoyable insight into all things geometrical. There are a wealth of popular articles including sections on Pythagoras, the golden ratio and recreational geometry. Historical items, drawn principally from the Mathematical Gazette, are authored by mathematicians such as G. H. Hardy, Rouse Ball, Thomas Heath and Bertrand Russell as well as some more recent expositors. Thirty 'Desert Island Theorems' from distinguished mathematicians and educationalists give light to some surprising and beautiful results. Contributors include H. S. M. Coxeter, Michael Atiyah, Tom Apostol, Solomon Golomb, Keith Devlin, Nobel Laureate Leon Lederman, Carlo Séquin, Simon Singh, Christopher Zeeman and Pulitzer Prizewinner Douglas Hofstadter. The book also features the wonderful Eyeball Theorems of Peruvian geometer and web designer, Antonio Gutierrez. For anyone with an interest in mathematics and mathematics education this book will be an enjoyable and rewarding read.

Celebrating a century of geometry and geometry teaching, this book will give the reader an enjoyable insight into all things geometrical. There are a wealth of popular articles including sections on Pythagoras, the golden ratio and recreational geometry. Historical items, drawn principally from the Mathematical Gazette, are authored by mathematicians such as G. H. Hardy, Rouse Ball, Thomas Heath and Bertrand Russell as well as some more recent expositors. Thirty 'Desert Island Theorems' from distinguished mathematicians and educationalists give light to some surprising and beautiful results. Contributors include H. S. M. Coxeter, Michael Atiyah, Tom Apostol, Solomon Golomb, Keith Devlin, Nobel Laureate Leon Lederman, Carlo Séquin, Simon Singh, Christopher Zeeman and Pulitzer Prizewinner Douglas Hofstadter. The book also features the wonderful Eyeball Theorems of Peruvian geometer and web designer, Antonio Gutierrez. For anyone with an interest in mathematics and mathematics education this book will be an enjoyable and rewarding read.

A fascinating collection of numerical curiosities and wonders. The Professor in Owen O Shea s book is the imaginary American Richard Stein. As Owen O Shea and the Professor travel through Ireland, O Shea notes the Professor s collection of amazing magic numbers in fascinating detail. His mathematical curiosities are wide ranging, concerning the 1915 sinking of the Lusitania to coincidences about Apollo 11 to the first moon walk to new numerical curiosities. The new curiosities, among many others, center on Presidents Lincoln and Kennedy, the USA and Ireland, the two World Wars, the King James Version of the Bible, and James Joyce.The Magic Numbers of the Professor reveals astonishing details about the year 1776; the year of American Independence. It contains discussions on prime numbers, gives some wonderful number patterns, and reveals many other eye-opening properties of numbers. It asks, for instance, if you know in how many different ways a US dollar can be changed. The Professor gives the answer to this and other currency questions. The number of the Beast 666, is discussed as well, as are many new equations involving that famous number all appearing here for the first time. And for those fascinated by games and gambling, a number of curious proposition bets involving dice, darts, and playing cards, and various mathematical puzzles are scattered throughout this singularly entertaining book.

This book picks up the history of mathematics from where Sherlock Holmes in Babylon left it. The forty articles of Who Gave You the Epsilon? continue the story of the development of mathematics into the nineteenth and twentieth centuries. The articles have all been published in the Mathematical Association of America journals and are in many cases written by distinguished mathematicians such as G. H. Hardy and B. van der Waerden. The articles are arranged thematically to show the development of analysis, geometry, algebra and number theory through this period of time. Each chapter is preceded by a foreword, giving the historical background and setting and the scene, and is followed by an afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book is ideal for anyone wanting to explore the history of mathematics.

This book has a dual objective: first to introduce the reader to some of the most important and widespread environmental issues of the day; and second to illustrate the vital role played by mathematical models in investigating these issues. The subjects covered are ground water contamination, air pollution, and hazardous materials emergencies. These issues are presented in their full real-world context and are used to develop important classical mathematical themes. The emphasis throughout is on fundamental principles and concepts, not on achieving technical mastery of state-of-the-art models.

This book is a collection of materials gathered by the author while teaching real analysis over a period of years. It is intended for use as a supplement to a traditional analysis textbook, or to provide material for seminars or independent study in analysis and its historical development. The book includes historical and biographical information, a wide range of problem types, selected readings on a variety of topics, and many references for additional study. Since all these materials are collected into a single book, teachers and students can easily choose items most suitable for their purpose. Teachers may use the book as a supplement to their courses, while students may read much of the book on their own. No other book has been written specifically as a supplement for a real analysis course.

Mistakes in mathematical reasoning can range from outlandish blunders to deep and subtle oversights that evade even the most watchful eye. This book represents the second collection of such errors to be compiled by Edward Barbeau. Like Barbeau's previous book, Mathematical Fallacies, Flaws and Flimflam, material is drawn from a variety of sources including the work of students, textbooks, the media, and even professional mathematicians. The errors presented here serve both to entertain, and to emphasize the need to subject even the most "obvious" assertions to rigorous scrutiny, as intuition and facile reasoning can often be misleading. Each item is carefully analysed and the source of the error is exposed. All students and teachers of mathematics, from school to university level, will find this book both enlightening and entertaining.

This collection of refereed papers celebrates the contributions, achievements, and progress of female mathematicians, mostly in the 20th and 21st centuries. Emerging from the themed paper session “The Contributions of Women to Mathematics: 100 Years and Counting” at MAA's 2015 MathFest, this volume contains a diverse mix of current scholarship and exposition on women and mathematics, including biographies, histories, and cultural discussions. The multiplicity of authors also ensures a wide variety of perspectives. In inspiring and informative chapters, the authors featured in this volume reflect on the accomplishments of women in mathematics, showcasing the changes in mathematical culture that resulted as more women obtained tenure-track and tenured academic positions, received prestigious awards and honors, served in leadership roles in professional societies, and became more visibly active in the mathematical community. Readers will find discussions of mathematical excellence at Girton College, Cambridge, in the late 19th and early 20th centuries; of perseverance by Polish women in mathematics during and after World War II and by Black women in mathematics in the United States from the 1880s onward; and of the impact of outreach programs ranging from EDGE's promotion of graduate education to the Daughters of Hypatia dance performances. The volume also provides informative biographies of a variety of women from mathematics and statistics, many of them well-known and others less well-known, including Charlotte Angas Scott, Emmy Noether, Mina Rees, Gertrude Cox, Euphemia Lofton Haynes, Norma Hernandez, Deborah Tepper Haimo, and Teri Perl. These essays provide compelling reading for a wide audience, including mathematicians, historians of science, teachers of mathematics, and students at the high school, college, and graduate levels. Anyone interested in attracting more girls and women as students, faculty, and/or employees will also find this volume engaging and enlightening.