Thinking in Problems: How Mathematicians Find Creative Solutions PDF
Preview Thinking in Problems: How Mathematicians Find Creative Solutions
Alexander A. Roytvarf Thinking in Problems How Mathematicians Find Creative Solutions Alexander A. Roytvarf Rishon LeZion, Israel ISBN 978-0-8176-8405-1 ISBN 978-0-8176-8406-8 (eBook) DOI 10.1007/978-0-8176-8406-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012950315 Mathematics Subject Classification (2010): 97H10, 97I10, 97K20, 97K50, 97K60, 97K80, 97M50 # Springer Science+Business Media, LLC 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface This book is written for people who think that mathematics is beautiful, for people who want to expand their mathematical horizons and sharpen their skills. The best way to sharpen mathematical skills is to use them in solving problems. Unlike other problem books, this book contains few routine exercises, nor is it a collection of olympiad problems. As we claim in the title, we aimed to create an atmosphere of real mathematical work for readers. Therefore, we sought to collect and explore two kinds of problems united by a common methodology. Problems of the first kind include nice theoretical material from the field of mathematics and are designed to teach readers to understand the math and to help them master the mathematical techniques by working on the problems. We kept to a common approach in teaching mathematics: “For effective learning, the learner should discover by himself as large a fraction of the material to be learned as feasible under the given circumstances. This is the principle of active learning (Arbeitsprinzip). It is a very old principle: it underlies the idea of the “Socratic method” [Polya 1962–1965]. Problems of the second kind arise in real-life mathe- matical research. By necessity, the scope of this book is too narrow for a methodical exposition of applications of mathematical theory to processes and methods in life and in the work place. Such an exposition necessitates including a lengthy intro- duction to the applied aspects of the real-life problems, so the emphasis of our discussion is on the mathematical aspects of these problems. Having described and explained the theoretical background and methodology in mathematical terms, we invite the reader to go on to obtain mathematical results relating to real-life outcomes. (However, when a lengthy introduction to the applied aspect is not required, such as in the problem group “A Combinatorial Algorithm in Multiexponential Analysis,” we depart from this rule and give the reader a nice opportunity to test himself on solving a real-life problem in what is essentially its original formulation.) Thus, we seek to show the reader that the same principles underlie work in both pure and applied mathematics. Some problems in this book pose a serious challenge for most readers; those who are prepared to work hard and undertake this challenge will gain the most out of this work. v vi Preface The prerequisites for working with this book mostly correspond to the graduate level, so the book is addressed primarily to this category of readers. Undergraduate students will be able to solve a substantial number of problems in this text, including all problems that do not require the reader to wield mathematical skills (mainly in linear algebra and analysis) that are outside the scope of what is usually taught at the undergraduate university level. We also hope that this book will be useful for teachers of higher education working with students of mathematics. Professional mathematicians may find in it material which would be interesting to them (e.g., new problems and new approaches to solving some well-known problems). For the reader’s convenience, we have devised a system of stars marking the problem or problem set to indicate the required background: no stars () indicates elementary material, while one (*), two (**), or three (***) stars correspond to the recommended number of semesters of a university-level math curriculum (a detailed specification of the related key notions and theorems is included in the section “Using the Stars on Problems” below). Thus, we assume that the relevant definitions will be known to the reader and there is no need to remind him of most of them. However, to facilitate working with this book, in some cases problem statements are preceded by definitions: e.g., “Recall that a function or a map defined on a Cartesian product of vector spaces is multilinear. . .,” and “Recall that a real-valued function of one or many real variables is referred to as convex. . .,” etc. In addition, the problem groups in this book are prefaced with “zero” (“preliminary” or “introductory”) sections containing related key terms, some definitions, and a carefully selected bibliogra- phy. (Obviously, these short introductions are no replacement for regular univer- sity courses.) The reader is always warned if comprehending a problem requires knowledge that goes beyond what is delineated by the stars: e.g., “Therefore, here is an exercise for readers familiar with multivariate integration,” or “This and the next two problems are addressed toward readers familiar with normed vector spaces,” or “This problem is addressed to readers familiar with ordinary linear differential equations,” etc.; see details in the section “Understanding the Advanced Skill Requirements” below. Those problems that require the reader to have a stronger mathematical background have been emphasized in this book by using a smaller font size. Paragraphs explaining the theoretical background on which these problems are based have also been included. This book also contains a number of problems that could be successfully solved with the help of some tool outside of the stated curriculum level. Whenever this is the case, the reader is warned: “If you need reference to additional definitions or tools related to this problem then look at corresponding Hint.” We introduce the tool within the “Hint,” “Explanation,” or “Completing Solution” sections (see below), while also discussing the tool and the problem, as required. We recommend that if readers do not encounter any unfamiliar concepts or terms while reading the problem’s formulation, they should attempt to solve it. Once they encounter difficulty in solving, they should try to overcome it, and only then, if necessary, Preface vii we may offer them appropriate means to overcome this difficulty (in this case, our seeds will fall into already cultivated soil). Therefore, the structure of the text follows a well-known educational method that works well for students of any background: “For efficient learning, an exploratory phase should precede the phase of verification and concept formalization and, eventually, the material learned should be merged in, and contribute to, the integral mental attitude of the learner” [Polya 1962–1965]. The complete problem set consists of (1) problems that stand on their own and (2) problems combined in groups with a common subject. In general, each problem group includes material more or less traditionally related to the field of mathemat- ics, which is indicated in its title, but readers will also notice a number of “nontra- ditional” inclusions, mainly related to applications – either in other fields of mathematics or in real-life problems (in these cases, the area is indicated). Some of the groups are united by these real-life applications. In accordance with the aim of the book, the problems are not restricted to the traditional educational categories: analysis, algebra, etc. On the contrary, the suggested solutions are obtained by combining ideas from different branches of mathematics (as is done in real mathematical work). Thus, solving all the problems in any of the problem groups will provide abundant practice in fundamental topics, such as continuity, intermediate value, and implicit function theorems, power-series expansions (analysis), polynomials, symmetry and skew-symmetry, determinants, eigenvalues, Jordan canonical form (algebra), all of which are indispensable for many real-life problems. Groups of related problems are organized in such a way that the problems build on each other, providing students with a ladder to master increasingly difficult problems. It is worth emphasizing that this organization also corresponds to the sequence which often occurs in real mathematical work: readers are invited first to understand the simplest theoretical concepts, and then to examine applications of these concepts, which allows them to observe additional properties and to return to the theoretical analysis of the generalized concepts. (In some cases, we even consider it necessary that the reader first finds a cumbersome solution by relatively simple tools, which is far from the nicest possible one. We are following the quite obvious idea that (1) the ability to deftly handle such means is useful for the researcher and should be developed, and, more importantly, (2) in this way the reader will be able to see the limits of the method’s applicability and further appreciate the more advanced subtle tools to do the same thing more efficiently. Based on our teaching experience, we believe that this “inductive” approach is the most productive for gaining mathematical skill. To gain as much as possible from this approach, we recommend that readers try to solve the problems in each group in the same order as they appear in the book. The presentation within each problem group is divided into subsections, some of which include introductory, summarizing, or historical materials. To stimulate active perception of the material, in many cases the problems are stated as questions (e.g., “Why?”, “Which exactly?”, etc.) or assignments (such as “Complete the details,” etc.) that do not interrupt the presentation; hence, within any subsection, viii Preface not one but a number of closely related problems can be proposed for solving. For readers’ convenience, in the “Problems” section, the first word of the each question or assignment (including the frequently used word “Prove”) is typeset in a different font (Lucida Calligraphy), and keywords in the material exposition are typeset in bold. In many cases, we give preference to assignments, as “Give. . .”, “Extend. . .”, “Develop. . .”, “Find. . .”, “Evaluate. . .”, “Describe. . .”, but not “Prove” because it is more consistent with actual mathematical practice. We would also like to emphasize that for a mathematician, the word “Why?” may be the most important question word. Also, we use the abbreviation “QED” (Quod Erat Demonstrandum) to denote the end of a proof, and typeset it in Arial Black. Mathematical and other symbols that we use in the book are traditional and widely used; perhaps the only exception (for the English-speaking reader) is that we prefer to denote the identity matrix (and the identity operator) by “E”, and the symbol “I” is reserved for a square root of E (where the dimension of the vector space is equal to 2). Any other nonstandard, or local, designations are defined in the same place where used. We have enclosed an explanation of solutions for all the problems to give the reader an opportunity to compare their solutions with someone else’s. The solutions to most of the problems are discussed in stages: first a hint, then a more in-depth explanation, and, finally, everything else that needs to be done to complete the solution. First of all, the reader should try to deal with the problem on his own. If he feels that he did not succeed, he should look into the “Hint” and then try to complete the solution of the problem. If that is not enough, the “Explanation” should be used, etc. The sections “Hint,” “Explanation,” and “Completing Solu- tion” are numbered similarly to the “Problems” sections: e.g., subsections H1.1 (Hints), E1.1 (Explanations), and S1.1 (Completing Solutions) correspond to P1.1 (Problems), and so on. We sought to engage the reader in the process of problem solving as an active participant, so that questions such as “Why?” and offers to fill in the necessary details are presented in these sections in the same way as in “Problems” (readers can see the answers to these questions provided in the “Expla- nation” or “Completing Solution” sections). Although the tastes of readers and their styles of thinking may be different, we believe that it is important that readers learn to see the subject from very different angles. Many great discoveries have been made by people who have such an ability! Therefore, for each problem that may be solved by various methods, we sought to consider all the approaches that were known to us. In addition, in the “Hint,” “Explanation,” and “Completing Solution” sections, we discuss related elementary subjects and other materials, such as ties with more advanced theories, examples of applications, references, etc. The additional materials make these sections an extremely important part of this book. Therefore, in order to get the most benefit from this book, the reader is advised to look through these sections, even if he found a solution to the problem by himself or has encountered the problem previously. As should be clear from the above, we assume that the reader will be actively working on problems from the book “with pencil and paper.” In this regard, we would like to make clear that the “Completing Solution” section of the book Preface ix does not contain the presentations of solutions “from ‘A’ to ‘Z’” (which are not needed if a reader thought through the problem, has already read “Hint” – “Expla- nation,” and therefore wants only the remaining instructions, so that his mind has developed a complete solution). Mainly, this section contains details such as proofs of the lemmas that were formulated and used without proof in “Hint” – “Explanation” sections, and further discussion (sometimes including related problem formulations). The solution may be qualified as complete, only if (1) all gaps in the proof are filled and (2) the reader has a clear view of the place of the solved problem within wider mathematical context and would be ready to work on similar or related problems in the future (because the problem is rarely a goal in and of itself). There are many differences between the present book and several widely known collections of nonstandard problems. For example, our work differs from a remark- able collection [Steinhaus 1958] in that it explores more advanced topics at the college undergraduate and graduate level. The main difference from another brilliant collection of problems for college exit exams [Arnold 1991, 1989 Russian] Is that this book includes a smaller number of topics and larger number of problems per topic, allowing detailed and gradual topic development. A few problems from Arnold collection were included in the corresponding problem groups in our book where they fit logically into the problem sequence. Finally, the present book differs from the famous Polya et al. [1964] by the relatively small number of topics that are explored deeply and in its orientation toward readers with a relatively limited experience in mathematics – namely, undergraduate and graduate students. Unlike Polya, we focus on stimulating the reader to combine ideas from different branches of mathematics to solve problems. We have achieved our goal if the reader becomes more adept in solving real mathematical problems and we will be quite satisfied if the reader develops a taste for this kind of work. Some of the problems (especially those related to applied topics) and solutions in this book we have suggested ourselves. Where this is not the case, we have tried to give credit to the authors of the problems and solutions. Absence of a reference means that we do not know the primary source and also that the fact (or method) in question has become a part of “mathematical folklore.” Rishon LeZion, Israel Alexander A. Roytvarf
The list of books you might like
