Bøger i Problem Books in Mathematics serien

Examining Sets and Relations.- Investigating Basic Properties of Functions.- Defining Distance in Sets.- Using Mathematical Induction.- Investigating Convergence of Sequences and Looking for Their Limits.- Dealing with Open, Closed and Compact Sets.- Finding Limits of Functions.- Examining Continuity and Uniform Continuity of Functions.- Finding Derivatives of Functions.- Using Derivatives to Study Certain Properties of Functions.- Dealing with Higher Derivatives and Taylor''s Formula.- Looking for Extremes and Examine Functions.- Investigating the Convergence of Series.- Finding Indefinite Integrals.- Investigating the Convergence of Sequences and Series of Functions.

This book is aimed to undergraduate STEM majors and to researchers using ordinary differential equations. It covers a wide range of STEM-oriented differential equation problems that can be solved using computational power series methods. Many examples are illustrated with figures and each chapter ends with discovery/research questions most of which are accessible to undergraduate students, and almost all of which may be extended to graduate level research. Methodologies implemented may also be useful for researchers to solve their differential equations analytically or numerically. The textbook can be used as supplementary for undergraduate coursework, graduate research, and for independent study.

This book contains a large variety of intriguing puzzles with detailed ingenious solutions generally not found elsewhere. This book is a great treasure for everybody who enjoys the beauty of fascinating world of recreational mathematics. Puzzles in the book can be browsed at random as these are not grouped in any orderly manner. Apart from puzzle enthusiasts and mathlovers, the book is considered of immense value for aspirants of Math Olympiad, CAT/MBA and job interviews of big companies like Google, Microsoft, Amazon, Apple, Facebook, Yahoo, NVidia, Oracle, Adobe, Morgan Stanley, Bloomberg, etc. The title of puzzles has been suitably framed. Repetition of similar type of puzzles has been avoided to keep the book in a concise form. However, important aspects of similar puzzles, if any, have been covered in comments section.This book is divided into four chapters (Chaps. 1-4): Chapter 1 contains 25 short riddles and brainteasers. These puzzles vary from simple buttricky to challenging ones. Chapter 2 contains detailed solutions to all the short riddles and brainteasers given in Chapter 1. Chapter 3 contains 125 creative puzzles of varying difficulty level covering arithmetic and algebraic puzzles on clock, calendar, weight, age and digital puzzles, geometric puzzles, logical reasoning puzzles, and combinatorial puzzles, match puzzles and game puzzles like "kaun banega crorepati" (who will become a Decamillionaire), new year winner, winning numbers, etc. Some famous and old puzzles like Cheryl's birthday, Bachets weight, liquid decanting, crossing bridge/river/desert, etc., have also been included so that readers can find all types of puzzles at one place. Only basic mathematics is required to solve these puzzles, but most of these puzzles are tricky and can be simplified by ingenious ideas. Chapter 4 contains detailed solutions to all the creative puzzles given in Chapter 3.

To many laymen, mathematicians appear to be problem solvers, people who do "e;hard sums"e;. Even inside the profession we dassify ourselves as either theorists or problem solvers. Mathematics is kept alive, much more than by the activities of either dass, by the appearance of a succession of unsolved problems, both from within mathematics-itself and from the in- creasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solu- tion of a problem may stifte interest in the area around it. But "e;Fermat's Last Theorem"e;, because it is not yet a theorem, has generated a great deal of "e;good"e; mathematics, whether goodness is judged by beauty, by depth or byapplicability. To pose good unsolved problems is a difficult art. The balance between triviality and hopeless unsolvability is delicate. There are many simply stated problems which experts tell us are unlikely to be solved in the next generation. But we have seen the Four Color Conjecture settled, even ifwe don't live long enough to leam the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfeet numbers. On the other hand, "e;unsolved"e; problems may not be unsolved at all, or may be much more tractable than was at first thought.

Although the ?rst decades of the 20th century saw some strong debates on set theory and the foundation of mathematics, afterwards set theory has turned into a solid branch of mathematics, indeed, so solid, that it serves as the foundation of the whole building of mathematics. Later generations, honest to Hilbert's dictum, "e;No one can chase us out of the paradise that Cantor has created for us"e; proved countless deep and interesting theorems and also applied the methods of set theory to various problems in algebra, topology, in?nitary combinatorics, and real analysis. The invention of forcing produced a powerful, technically sophisticated tool for solving unsolvable problems. Still, most results of the pre-Cohen era can be digested with just the knowledge of a commonsense introduction to the topic. And it is a worthy e?ort, here we refer not just to usefulness, but, ?rst and foremost, to mathematical beauty. In this volume we o?er a collection of various problems in set theory. Most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come fromtheperiod,say,1920-1970.Manyproblemsarealsorelatedtoother?elds of mathematics such as algebra, combinatorics, topology, and real analysis. We do not concentrate on the axiomatic framework, although some - pects, such as the axiom of foundation or the role E of the axiom of choice, are elaborated.

As a student I discovered in our library a thin booklet by Frederick Mosteller entitled50 Challenging Problems in Probability. Itreferredtoas- plementary "e;regular textbook"e; by William Feller, An Introduction to Pro- bilityTheoryanditsApplications.SoItookthisonealong,too,andstartedon the ?rst of Mosteller's problems on the train riding home. From that evening, I caught on to probability. These two books were not primarily about abstract formalisms but rather about basic modeling ideas and about ways - often extremely elegant ones - to apply those notions to a surprising variety of empirical phenomena. Essentially, these books taught the reader the skill to "e;think probabilistically"e; and to apply simple probability models to real-world problems. The present book is in this tradition; it is based on the view that those cognitive skills are best acquired by solving challenging, nonstandard pro- bility problems. My own experience, both in learning and in teaching, is that challenging problems often help to develop, and to sharpen, our probabilistic intuition much better than plain-style deductions from abstract concepts.

Pell's equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. The history of this equation is long and circuitous, and involved a number of different approaches before a definitive theory was found. There were partial patterns and quite effective methods of finding solutions, but a complete theory did not emerge until the end of the eighteenth century. The topic is motivated and developed through sections of exercises which allow the student to recreate known theory and provide a focus for their algebraic practice. There are also several explorations that encourage the reader to embark on their own research. Some of these are numerical and often require the use of a calculator or computer. Others introduce relevant theory that can be followed up on elsewhere, or suggest problems that the reader may wish to pursue. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject. Edward Barbeau is Professor of Mathematics at the University of Toronto. He has published a number of books directed to students of mathematics and their teachers, including Polynomials (Springer 1989), Power Play (MAA 1997), Fallacies, Flaws and Flimflam (MAA 1999) and After Math (Wall & Emerson, Toronto 1995).

For the first two editions of the book Probability (GTM 95), each chapter included a comprehensive and diverse set of relevant exercises. While the work on the third edition was still in progress, it was decided that it would be more appropriate to publish a separate book that would comprise all of the exercises from previous editions, in addition to many newexercises.Most of the material in this book consists of exercises created by Shiryaev, collected and compiled over the course of many years while working on many interesting topics. Many of the exercises resulted from discussions that took place during special seminars for graduate and undergraduate students.  Many of the exercises included in the book contain helpful hints and other relevant information.Lastly, the author has included an appendix at the end of the book that contains a summary of the main results, notation and terminology from Probability Theory that are used throughout the present book.  This Appendix also contains additional material from Combinatorics, Potential Theory and Markov Chains, which is not covered in the book, but is nevertheless needed for many of the exercises included here.

In the pages that follow there are: A. A revised and enlarged version of Problems in analysis (PIA) . (All typographical, stylistic, and mathematical errors in PIA and known to the writer have been corrected.) B. A new section COMPLEX ANALYSIS containing problems distributed among many of the principal topics in the theory of functions of a complex variable. C. A total of 878 problems and their solutions. D. An enlarged Index/Glossary and an enlarged Symbol List. Notational and terminological conventions are to be found for the most part under Conventions at the beginnings of the chapters. Spe- cial items not included in Conventions are completely explained in the Index/Glossary. The audience to which the current book is addressed differs little from the audience for PIA. The background of the reader is assumed to include a knowledge of the basic principles and theorems in real and complex analysis as those subjects are currently viewed. The aim of the problems is to sharpen and deepen the understanding of the mechanisms that underlie modern analysis. I thank Springer-Verlag for its interest in and support of this project. State University of New York at Buffalo B. R. G. v Contents The symbol alb under Pages below indicates that the Problems for the section begin on page a and the corresponding Solutions begin on page b. Thus 3/139 on the line for Set Algebra indicates that the Problems in Set Algebra begin on page 3 and the corresponding Solutions begin on page 139.

The author, the founder of the Greek Statistical Institute, has based this book on the two volumes of his Greek edition which has been used by over ten thousand students during the past fifteen years. It can serve as a companion text for an introductory or intermediate level probability course. Those will benefit most who have a good grasp of calculus, yet, many others, with less formal mathematical background can also benefit from the large variety of solved problems ranging from classical combinatorial problems to limit theorems and the law of iterated logarithms. It contains 329 problems with solutions as well as an addendum of over 160 exercises and certain complements of theory and problems.

Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, ¿The Critic as Artist,¿ 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying.

This book covers topics in the theory and practice of functional equations. Special emphasis is given to methods for solving functional equations that appear in mathematics contests, such as the Putnam competition and the International Mathematical Olympiad. This book will be of particular interest to university students studying for the Putnam competition, and to high school students working to improve their skills on mathematics competitions at the national and international level. Mathematics educators who train students for these competitions will find a wealth of material for training on functional equations problems. The book also provides a number of brief biographical sketches of some of the mathematicians who pioneered the theory of functional equations. The work of Oresme, Cauchy, Babbage, and others, is explained within the context of the mathematical problems of interest at the time.Christopher Small is a Professor in the Department of Statistics and Actuarial Science at the University of Waterloo. He has served as the co-coach on the Canadian team at the IMO (1997, 1998, 2000, 2001, and 2004), as well as the Waterloo Putnam team for the William Lowell Putnam Competition (1986-2004). His previous books include Numerical Methods for Nonlinear Estimating Equations (Oxford 2003), The Statistical Theory of Shape (Springer 1996), Hilbert Space Methods in Probability and Statistical Inference (Wiley 1994). From the reviews:Functional Equations and How to Solve Them fills a need and is a valuable contribution to the literature of problem solving. - Henry Ricardo, MAA ReviewsThe main purpose and merits of the book...are the many solved, unsolved, partially solved problems and hints about several particular functional equations.- Janos Aczel, Zentralblatt

The idea of writing this book came roughly at the time of publication of my graduate text Lectures on Modules and Rings, Springer GTM Vol. 189, 1999. Since that time, teaching obligations and intermittent intervention of other projects caused prolonged delays in the work on this volume. Only a lucky break in my schedule in 2006 enabled me to put the finishing touches on the completion of this long overdue book. This book is intended to serve a dual purpose. First, it is designed as a "problem book" for Lectures. As such, it contains the statements and full solutions of the many exercises that appeared in Lectures. Second, this book is also offered as a reference and repository for general information in the theory of modules and rings that may be hard to find in the standard textbooks in the field. As a companion volume to Lectures, this work covers the same math­ ematical material as its parent work; namely, the part of ring theory that makes substantial use of the notion of modules. The two books thus share the same table of contents, with the first half treating projective, injective, and flat modules, homological and uniform dimensions, and the second half dealing with noncommutative localizations and Goldie's theorems, maximal rings of quotients, Frobenius and quasi-Frobenius rings, conclud­ ing with Morita's theory of category equivalences and dualities.

Mathematics is kept alive by the appearance of new unsolved problems, problems posed from within mathematics itself, and also from the increasing number of disciplines where mathematics is applied. This book provides a steady supply of easily understood, if not easily solved, problems which can be considered in varying depths by mathematicians at all levels of mathematical maturity.For this new edition, the author has included new problems on symmetric and asymmetric primes, sums of higher powers, Diophantine m-tuples, and Conway's RATS and palindromes. The author has also included a useful new feature at the end of several of the sections: lists of references to OEIS, Neil Sloane's Online Encyclopedia of Integer Sequences. About the First Edition:"...many talented young mathematicians will write their first papers starting out from problems found in this book."- András Sárközi, MathSciNet

The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 "explorations" invite the reader to investigate research problems and related topics.