The book also includes exercises of varying diffi- culty and open questions to suggest avenues of research. The book is written in a nice manner, the exposition is very friendly and intriguing a delightful introductory text. Zentralblatt MATH This book explains the process of modelling real situations to obtain math- ematical problems that can be analyzed, thus solving the original problem. The presentation is in the form of case studies, which are developed much as they would be in true applications.

In many cases, an initial model is created, then modified along the way. Some cases are familiar, such as the evaluation of an annuity. Others are unique, such as the fascinating situation in which an engineer, armed only with a slide rule, had 24 hours to compute whether a valve would hold when a temporary rock plug was removed from a water tunnel. Each chapter ends with a set of exercises and some suggestions for class projects.

Some projects are extensive, as with the explorations of the predator-prey model; others are more modest. Cryptography: An Introduction V. Further, I feel that it could be very useable as a text for a first course in partial differential equations. CA L Knobel on having produced such a well-written and much-needed book! This book begins with a descrip- tion of one-dimensional waves and their visualization through computer-aided techniques.

Next, traveling waves are covered, such as solitary waves for the Klein-Gordon and KdV equations. Finally, the author gives a lucid discussion of waves arising from conservation laws, including shock and rarefaction waves. As an application, interesting models of traffic flow are used to illustrate conserva- tion laws and wave phenomena. It is suitable for independent study by undergraduate students in mathematics, engineering, and science programs.

Undergraduates will find this a very friendly and stimulating introduction to creative mathematics in higher dimensions. It takes readers from the basics of polytope theory to recent developments around secondary and state polytopes arising from point configurations. The most needed concepts are developed from scratch. Text illustrates the interaction among discrete geometry, computational algebra and combinatorics. Lectures on Generating Functions S.

Lando, Independent University of Moscow, Russia ; pp. General and Interdisciplinary A View from the Top Analysis, Combinatorics and Number Theory Alex Iosevich, University of Missouri, Columbia, MO This book brings to life the connections among different areas of math- ematics and illustrates how various subject areas flow from one another. It is designed to help readers appreciate that mathematics should not be compartmentalized into distinct subjects. The work inspires interest in research mathematics by highlighting the process in which ideas evolve.

Spain This book uses an elegant style to illustrate elementary geometrys richness as one of the most classical topics in mathematics. The topic is revisited from the higher viewpoint of university mathematics, presenting a deeper under- standing of familiar topics and an introduction to new topics that complete the picture of two-dimensional geometries. More than carefully drawn figures support the understanding of geometric concepts. Differential Geometry Curves Surfaces Manifolds, Second Edition Wolfgang Khnel, University of Stuttgart, Germany This carefully written book is an introduction to the beautiful ideas and results of differential geometry.

The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. Special topics that are explored include Frenet frames, ruled surfaces, minimal surfaces, and the Gauss-Bonnet theorem. The second part is an introduction to the geometry of general manifolds, with particular emphasis on connections and curvature.

The final two chapters are insightful exami- nations of the special cases of spaces of constant curvature and Einstein manifolds. The text is illustrated with many figures and examples. For the second edition, a number of errors were corrected and some text and a number of figures have been added. The prerequisites are under- graduate analysis and linear algebra.

McCleary offers a tight, purpose-built book, establishing the invariance of dimension, the rigorous structural distinction that differentiates lines from planes from higher-dimensional spaces. CHOICE Magazine This work departs from other texts on topology by taking a direct, focused approach grounded in solving the problem of invariance of dimension.

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Based on his Vassar College course, John McCleary addresses the basic point-set notions of topology and the fundamental group of a space, then proceeds to a proof of the Jordan Curve theorem. McClearys approach emphasizes the tools and intuitions central to topology. I am very enthusiastic about this book!

It would make an excellent text for an undergraduate course in minimal surface theory. Enough detail is included so that this book would also be suitable for an independent study. The next time I teach undergraduate differential geometry, my plan is to first teach a lead-in course using Opreas book. This provides students with easy access to soap film mathematics MAA Online Nature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations.

This book employs ingredients from each of these subjects to tell the math- ematical story of soap films. Through the Maple applications, the reader is given tools for creating the shapes that are being studied. Thus, you can see a fluid rising up an inclined plane, create minimal surfaces from complex variables data, and investigate the true shape of a balloon.

Oprea also includes descriptions of experiments and photographs that let you see real soap films on wire frames. The theory of minimal surfaces is a beautiful subject, which naturally introduces the reader to fascinating, yet accessible, topics in mathematics. Opreas presentation is rich with examples, expla- nations, and applications.

Waterloo Maple, Inc. Geometry and Billiards Serge Tabachnikov, Penn State, University Park, PA Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differ- ential geometry, classical mechanics, and geometrical optics.

The topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual or outer billiards. The book is based on an advanced undergraduate topics course but contains more material than can be realistically taught in one semester.

Although the minimum prerequisites include only the standard material usually covered in the first two years of college the entire calculus sequence, linear algebra , readers should show some mathematical maturity and strongly rely on their mathematical common sense. As a reward, they will be taken to the forefront of current research. A special feature of the book is a substantial number of digressions covering diverse topics related to billiards: evolutes and involutes of plane curves, the 4-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincar recurrence theorem, Hilberts fourth problem, Poncelet porism, and many others.

This volume is copublished with the Mathematics Advanced Studies Seminars. A concise treatment of differential and algebraic topology.

American Mathematical Monthly. This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc.

The author notes, The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs. He concludes, As a rule, only those proofs or sketches of proofs that are interesting per se and have important generalizations are presented. Almgren, Jr. There is also a fair number of good exercises.

Roman Smirnov, Dalhousie University The main notions of set theory cardinals, ordinals, transfinite induction are fundamental to all mathematicians, not only to those who specialize in math- ematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own dedicated treatment.

The text introduces all main subjects of naive nonaxiomatic set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals.

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With over problems, the book is a complete and accessible introduction to the subject. Also available in Geometry and topology Computable Functions A. This slender volume is extremely well-written and contains a wealth of material. It is a lucid and accessible introduction to a rich and fascinating area of mathematics, written by the worlds leading expert. For anyone with a knowledge of calculus wanting to learn about the mathematical work of Ramanujan, this book is the best place to start.

Berndt, the foremost authority on Indias greatest mathematician, presents the first introduction to Ramanujans work in number theory. The text addresses the important subjects of theta functions and q-series and discusses a number of topics in number theory that are intertwined with these disciplines. The author includes detailed proofs of theorems from some of Ramanujans most famous writings. This short book presents a nice enjoyable introduction to Diophantine analysis, which invites the motivated reader to rediscover by himself or herself many of the fundamental results of the subject, with hints given in an appendix for the more difficult results.

Mathematical Reviews Welcome to diophantine analysisan area of number theory in which we attempt to discover hidden treasures and truths within the jungle of numbers by exploring rational numbers. Diophantine analysis comprises two different but interconnected domainsdiophantine approximation and diophantine equations. Through an engaging style, readers participate in a journey through these areas of number theory. Each mathematical theme is presented in a self-contained manner and is motivated by very basic notions.

The reader becomes an active participant in the explorations, as each module includes a sequence of numbered questions to be answered and statements to be verified. Many hints and remarks are provided to be freely used and enjoyed. Each module then closes with a Big Picture Question that invites the reader to step back from all the technical details and take a panoramic view of how the ideas at hand fit into the larger mathematical landscape.

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This book enlists the reader to build intuition, develop ideas and prove results in a very user-friendly and enjoyable environment. Edwards, New York University, NY This text demonstrates the shift in the focus of studies in number theory toward arithmetical computations. Students can use the text to master the fundamental ideas of basic number theory through the application of algorithms to computational examples. Exercises, many with answers given, appear at the end of nearly every chapter. Svetlana Katok, Pennsylvania State University, University Park, PA This book examines p -adic numbers from the point of view of number theory, topology and analysis, unlike other texts that look at only one of these three areas.

Several topics from real analysis and elementary topology that are not usually covered in undergraduate courses are included. A large number of exercises are provided, along with hints and solutions for most. The authors have chosen to highlight some of the most important points of the area, and the exposition and the translation are excellent. Reading this book is equivalent to ascending a major summit. MAA Monthly. One notable new direction this century in the study of primes has been the influx of ideas from probability.

The goal of this book is to provide insights into the prime numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness. The book opens with some classic topics of number theory. It ends with a discussion of some of the outstanding conjectures in number theory.

In between are an excellent chapter on the stochastic properties of primes and a walk through an elementary proof of the Prime Number Theorem. This book is suitable for anyone who has had a little number theory and some advanced calculus involving estimates. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians.

Fristedt, N. Jain, and N. Krylov, University of Minnesota, Minneapolis, MN This book maintains a rigorous treatment of the theory of filtering and prediction for random processes without burdening the reader with unnec- essary technical machinery. The goal is to present an introduction to the modern theory of filtering and prediction on the examples of Markov chains in discrete and continuous time and on stationary sequences. Numerous exercises appear in each chapter. It is helpful for the probability community to have access to this book, which contains a unified and elementary presentation of limit theorems Beyond the introductory ideas, there are many wonderful results that are unfamiliar to the layman, but which are well within our grasp to understand and appreciate.

Some of the most remarkable results in probability are those that are related to limit theoremsstatements about what happens when the trial is repeated many times. The most famous of these is the Law of Large Numbers, which mathematicians, engineers, economists, and many others use every day. In this book, Lesigne has made these limit theorems accessible by stating everything in terms of a game of tossing of a coin: heads or tails. In this way, the analysis becomes much clearer, helping establish the readers intuition about probability. Moreover, very little generality is lost, as many situ- ations can be modelled from combinations of coin tosses.

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## Invariant Theory

Carousel Previous Carousel Next. Neusel established Emmy Noether High School Mathematics Days in May [7] which continues to be celebrated with workshops and mathematical competitions. Neusel began her career at Texas Tech University in as an associate professor and was promoted to full professor in From Wikipedia, the free encyclopedia. Mara Dicle Neusel. Stuttgart , Germany. Retrieved 12 March Invariant Theory of Finite Groups.

Mathematical Surveys and Monographs. Providence, R. Retrieved 10 March Invariant Theory. Student Mathematical Library. Inverse Invariant Theory and Steenrod Operations. Memoirs of the American Mathematical Society.