Solving Mathematical Problems

A Personal Perspective

Author: Terence Tao

Publisher: Oxford University Press on Demand

ISBN: 9780199205608

Category: Mathematics

Page: 103

View: 8172

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of 14 years and above in pure mathematics.

Solving Mathematical Problems

A Personal Perspective

Author: Terence Tao

Publisher: OUP Oxford

ISBN: 0199205612

Category: Mathematics

Page: 103

View: 4671

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of14 years and above in pure mathematics.

An Introduction to Measure Theory

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821869191

Category: Mathematics

Page: 206

View: 2290

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Compactness and Contradiction

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821894927

Category: Mathematics

Page: 256

View: 2359

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter

Analysis I

Third Edition

Author: Terence Tao

Publisher: Springer

ISBN: 9811017891

Category: Mathematics

Page: 350

View: 7993

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

Functional Equations and How to Solve Them

Author: Christopher G. Small

Publisher: Springer Science & Business Media

ISBN: 0387489010

Category: Mathematics

Page: 131

View: 2133

Many books have been written on the theory of functional equations, but very few help readers solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap. Each chapter includes a list of problems associated with the covered material. These vary in difficulty, with the easiest being accessible to any high school student who has read the chapter carefully. The most difficult will challenge students studying for the International Mathematical Olympiad or the Putnam Competition. An appendix provides a springboard for further investigation of the concepts of limits, infinite series and continuity.

Problems in Real Analysis

Advanced Calculus on the Real Axis

Author: Teodora-Liliana Radulescu,Vicentiu D. Radulescu,Titu Andreescu

Publisher: Springer Science & Business Media

ISBN: 0387773797

Category: Mathematics

Page: 452

View: 7749

Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis.

Epsilon of Room, Two

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821852809

Category: Mathematics

Page: 248

View: 7889

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such ``folklore mathematics''. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007 Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to nontechnical puzzles and expository articles. The first two years of the blog have already been published by the American Mathematical Society. The posts from the third year are being published in two volumes. This second volume contains a broad selection of mathematical expositions and self-contained technical notes in many areas of mathematics, such as logic, mathematical physics, combinatorics, number theory, statistics, theoretical computer science, and group theory. Tao has an extraordinary ability to explain deep results to his audience, which has made his blog quite popular. Some examples of this facility in the present book are the tale of two students and a multiple-choice exam being used to explain the $P = NP$ conjecture and a discussion of "no self-defeating object" arguments that starts from a schoolyard number game and ends with results in logic, game theory, and theoretical physics. The first volume consists of a second course in real analysis, together with related material from the blog, and it can be read independently.

Problem-Solving Methods in Combinatorics

An Approach to Olympiad Problems

Author: Pablo Soberón

Publisher: Springer Science & Business Media

ISBN: 3034805977

Category: Mathematics

Page: 174

View: 4092

Every year there is at least one combinatorics problem in each of the major international mathematical olympiads. These problems can only be solved with a very high level of wit and creativity. This book explains all the problem-solving techniques necessary to tackle these problems, with clear examples from recent contests. It also includes a large problem section for each topic, including hints and full solutions so that the reader can practice the material covered in the book.​ The material will be useful not only to participants in the olympiads and their coaches but also in university courses on combinatorics.

A Mathematical Mosaic

Patterns & Problem Solving

Author: Ravi Vakil

Publisher: Brendan Kelly Publishing Inc.

ISBN: 9781895997040

Category: Mathematics

Page: 254

View: 5706

Excerpt from a review in the "Mathematics Teacher." A Mathematical Mosaic is a collection of wonderful topics from nmber theory through combinatorics to game theory, presented in a fashion that seventh- and eighth- grade students can handle yet high school students will find challenging." John Cocharo, Saint Mark's School of Texas, Dallas, TX

Mathematical Bridges

Author: Titu Andreescu,Cristinel Mortici,Marian Tetiva

Publisher: Birkhäuser

ISBN: 0817646299

Category: Mathematics

Page: 309

View: 1604

Building bridges between classical results and contemporary nonstandard problems, this highly relevant work embraces important topics in analysis and algebra from a problem-solving perspective. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors and motivated mathematics students from high school juniors to college seniors will find the work a useful resource in calculus, linear and abstract algebra, analysis and differential equations. Students with an interest in mathematics competitions must have this book in their personal libraries.

Learning and Teaching Real World Problem Solving in School Mathematics

A Multiple-Perspective Framework for Crossing the Boundary

Author: Murad Jurdak

Publisher: Springer

ISBN: 3319082043

Category: Education

Page: 199

View: 5857

The ultimate aim of this book is to identify the conceptual tools and the instructional modalities which enable students and teachers to cross the boundary between school mathematics and real world problem solving. The book identifies, examines, and integrates seven conceptual tools, of which five are constructs (activity theory, narrative, modeling, critical mathematics education, ethnomathematics) and two are contexts (STEM and the workplace). The author develops two closely linked multiple-perspective frameworks: one for learning real world problem solving in school mathematics, which sets the foundations of learning real world problem solving in school mathematics; and one for teaching real world problem solving in school mathematics, which explores the modalities of teaching real world problem solving in school mathematics. “The book is composed as, on the one hand, a high-level theoretical scholarly work on real world problem solving in school mathematics, and, on the other hand, a set of twelve narratives which, put together, constitute a thought-provoking and moving personal and professional autobiography.” - Mogens Niss “These narratives combine aspects of Murad’s personal trajectory as an individual with those points in his professional career at which he became aware of perspectives on and approaches to mathematics education that were both significant in and of themselves, and instrumental for the specific scholarly endeavor presented in the book.” - Mogens Niss

Techniques of Problem Solving

Author: S.G. Krantz

Publisher: Universities Press

ISBN: 9788173711169


Page: 465

View: 5021

The purpose of this book is to teach the basic principles of problem solving, including both mathematical and non-mathematical problems. Talking a direct and practical approach to the subject matter, Krantz's book stands apart from others like it in that it incorporates exercises throughout the text. Additional problems are included for readers to takle at the end of each chapter. There are more than 350 problems in all. A Solutions Manual to most end-of-chapter exercises is available.

How to Solve Mathematical Problems

Author: Wayne A. Wickelgren

Publisher: Courier Corporation

ISBN: 0486152685

Category: Science

Page: 288

View: 1789

Seven problem-solving techniques include inference, classification of action sequences, subgoals, contradiction, working backward, relations between problems, and mathematical representation. Also, problems from mathematics, science, and engineering with complete solutions.

Problem-Solving Strategies

Author: Arthur Engel

Publisher: Springer Science & Business Media

ISBN: 0387226419

Category: Mathematics

Page: 403

View: 7490

A unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week", thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market.

Putnam and Beyond

Author: Razvan Gelca,Titu Andreescu

Publisher: Springer Science & Business Media

ISBN: 038768445X

Category: Mathematics

Page: 798

View: 4210

Putnam and Beyond takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis in one and several variables, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research.

The Stanford Mathematics Problem Book

With Hints and Solutions

Author: George Polya,Jeremy Kilpatrick

Publisher: Courier Corporation

ISBN: 048631832X

Category: Mathematics

Page: 80

View: 9128

Based on Stanford University's well-known competitive exam, this excellent mathematics workbook offers students at both high school and college levels a complete set of problems, hints, and solutions. 1974 edition.

The Art and Craft of Problem Solving

Author: Paul Zeitz

Publisher: Wiley Global Education

ISBN: 1118916662

Category: Mathematics

Page: 380

View: 6007

Appealing to everyone from college-level majors to independent learners, The Art and Craft of Problem Solving, 3rd Edition introduces a problem-solving approach to mathematics, as opposed to the traditional exercises approach. The goal of The Art and Craft of Problem Solving is to develop strong problem solving skills, which it achieves by encouraging students to do math rather than just study it. Paul Zeitz draws upon his experience as a coach for the international mathematics Olympiad to give students an enhanced sense of mathematics and the ability to investigate and solve problems.

Additive Combinatorics

Author: Terence Tao,Van H. Vu

Publisher: Cambridge University Press

ISBN: 1139458345

Category: Mathematics

Page: N.A

View: 7441

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.

Mathematical Constants

Author: Steven R. Finch

Publisher: Cambridge University Press

ISBN: 9780521818056

Category: Mathematics

Page: 602

View: 3025

Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. This book is helpful both to readers seeking information about a specific constant, and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This work represents an outstanding scholarly attempt to bring together all significant mathematical constants in one place.