Table Of ContentAlexander 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
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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
