Thinking in problems : how mathematicians find creative solutions / Alexander A. Roytvarf

434 Pages·2013·3.58 MB·English

by  Alexander A Roytvarf

Alexander A. Roytvarf
Thinking in Problems
How Mathematicians Find Creative Solutions
AlexanderA.Roytvarf
RishonLeZion,Israel
ISBN978-0-8176-8405-1 ISBN978-0-8176-8406-8(eBook)
DOI10.1007/978-0-8176-8406-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012950315
MathematicsSubjectClassification(2010):97H10,97I10,97K20,97K50,97K60,97K80,97M50
#SpringerScience+BusinessMedia,LLC2013
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Preface
Thisbookiswrittenforpeoplewhothinkthatmathematicsisbeautiful,forpeople
whowanttoexpandtheirmathematicalhorizonsandsharpentheirskills.Thebest
waytosharpenmathematicalskillsistousetheminsolvingproblems.
Unlikeotherproblembooks,thisbookcontainsfewroutineexercises,norisita
collection of olympiad problems. As we claim in the title, we aimed to create an
atmosphereofrealmathematicalworkforreaders.Therefore,wesoughttocollect
andexploretwokindsofproblemsunitedbyacommonmethodology.Problemsof
thefirstkindincludenicetheoreticalmaterialfromthefieldofmathematicsandare
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”[Polya1962–1965].Problemsofthesecondkindariseinreal-lifemathe-
maticalresearch.Bynecessity,thescopeofthisbookistoonarrowforamethodical
expositionofapplicationsofmathematicaltheorytoprocessesandmethodsinlife
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
discussionisonthemathematicalaspectsoftheseproblems.Havingdescribedand
explainedthetheoreticalbackgroundandmethodologyinmathematicalterms,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
opportunitytotesthimselfon solving areal-lifeprobleminwhat isessentially its
original formulation.) Thus, we seek to show the reader that the same principles
underlieworkinboth pure andapplied mathematics.Some problemsinthisbook
poseaseriouschallengeformostreaders;thosewhoarepreparedtoworkhardand
undertakethischallengewillgainthemostoutofthiswork.
v
vi Preface
Theprerequisitesforworkingwiththisbookmostlycorrespondtothegraduate
level,sothebookisaddressedprimarilytothiscategoryofreaders.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
(mainlyinlinearalgebraandanalysis)thatareoutsidethescopeofwhatisusually
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.
Professionalmathematiciansmayfindinitmaterialwhichwouldbeinterestingto
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
(adetailedspecificationoftherelatedkeynotionsandtheoremsisincludedinthe
section“UsingtheStarsonProblems”below).
Thus, we assume that the relevant definitions will be known to the reader
andthereisnoneedtoremindhimofmostofthem.However,tofacilitateworking
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,theseshortintroductionsarenoreplacementforregularuniver-
sitycourses.)
The reader is always warned if comprehending a problem requires knowledge
thatgoesbeyondwhatisdelineatedbythestars:e.g.,“Therefore,hereisanexercise
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
problemsarebasedhavealsobeenincluded.
Thisbookalsocontainsanumberofproblemsthatcouldbesuccessfullysolved
withthehelpofsometooloutsideofthestatedcurriculumlevel.Wheneverthisis
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),whilealsodiscussingthetoolandtheproblem,asrequired.Werecommend
thatifreadersdonotencounteranyunfamiliarconceptsortermswhilereadingthe
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
wemayofferthemappropriatemeanstoovercomethisdifficulty(inthiscase,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”[Polya1962–1965].
Thecompleteproblemsetconsistsof(1)problemsthatstandontheirownand
(2)problemscombinedingroupswithacommonsubject.Ingeneral,eachproblem
groupincludesmaterialmoreorlesstraditionallyrelatedtothefieldofmathemat-
ics,whichisindicatedinitstitle,butreaderswillalsonoticeanumberof“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
ofthegroupsareunitedbythesereal-lifeapplications.
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(asisdoneinrealmathematicalwork).Thus,solvingalltheproblems
inanyoftheproblemgroupswillprovideabundantpracticeinfundamentaltopics,
suchascontinuity,intermediatevalue,andimplicitfunctiontheorems,power-series
expansions (analysis), polynomials, symmetry and skew-symmetry, determinants,
eigenvalues, Jordan canonical form (algebra), all of which are indispensable for
manyreal-lifeproblems.
Groupsofrelatedproblemsareorganizedinsuchawaythattheproblemsbuild
on each other, providing students with a ladder to master increasingly difficult
problems. It is worth emphasizing that this organization also corresponds to the
sequencewhichoftenoccursinrealmathematicalwork:readersareinvitedfirstto
understand the simplest theoretical concepts, and then to examine applications of
theseconcepts,whichallowsthemtoobserveadditionalpropertiesandtoreturnto
the theoretical analysis of the generalized concepts. (In some cases, we even
consideritnecessarythatthereaderfirstfindsacumbersomesolutionbyrelatively
simpletools,whichisfarfromthenicestpossibleone.Wearefollowingthequite
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.
Basedonourteachingexperience,webelievethatthis“inductive”approachisthe
mostproductiveforgainingmathematicalskill.Togainasmuchaspossiblefrom
thisapproach,werecommendthatreaderstrytosolvetheproblemsineachgroupin
thesameorderastheyappearinthebook.
Thepresentationwithineachproblemgroupisdividedintosubsections,someof
which include introductory, summarizing, or historical materials. To stimulate
activeperceptionofthematerial,inmanycasestheproblemsarestatedasquestions
(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
notonebutanumberofcloselyrelatedproblemscanbeproposedforsolving.For
readers’convenience,inthe“Problems”section,thefirstwordoftheeachquestion
orassignment(includingthefrequentlyusedword“Prove”)istypesetinadifferent
font(LucidaCalligraphy),andkeywordsinthematerialexpositionare typesetin
bold.Inmanycases,wegivepreferencetoassignments,as“Give...”,“Extend...”,
“Develop...”, “Find...”,“Evaluate...”, “Describe...”, butnot“Prove” because
it is more consistent with actual mathematical practice. We would also like to
emphasizethatforamathematician,theword“Why?”maybethemostimportant
questionword.Also,weusetheabbreviation“QED”(QuodEratDemonstrandum)
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 (cid:1)E (where the dimension of the vector space is
equal to 2). Any other nonstandard, or local, designations are defined in the same
placewhereused.
We have enclosed an explanation of solutions for all the problems to give the
readeranopportunitytocomparetheirsolutionswithsomeoneelse’s.Thesolutions
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.Ifhefeelsthathedidnotsucceed,heshouldlookintothe“Hint”andthentry
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),andS1.1(CompletingSolutions)correspondtoP1.1
(Problems), and so on. We sought to engage the reader in the process of problem
solvingasanactiveparticipant,sothatquestionssuchas“Why?”andofferstofill
in the necessary details are presented in these sections in the same way as in
“Problems”(readerscanseetheanswerstothesequestionsprovidedinthe“Expla-
nation”or“CompletingSolution”sections).
Althoughthetastesofreadersandtheirstylesofthinkingmaybedifferent,we
believethatitisimportantthatreaderslearntoseethesubjectfromverydifferent
angles.Manygreatdiscoverieshavebeenmadebypeoplewhohavesuchanability!
Therefore,foreachproblemthatmaybesolvedbyvariousmethods,wesoughtto
consider all the approaches that were known to us. In addition, in the “Hint,”
“Explanation,”and“CompletingSolution”sections,wediscussrelatedelementary
subjectsandothermaterials,suchastieswithmoreadvancedtheories,examplesof
applications, references, etc. The additional materials make these sections an
extremely important part of this book. Therefore, in order to get the most benefit
fromthisbook,thereaderisadvisedtolookthroughthesesections,evenifhefound
asolutiontotheproblembyhimselforhasencounteredtheproblempreviously.
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
neededifareaderthoughtthroughtheproblem,hasalreadyread“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
solvedproblemwithinwidermathematicalcontextandwouldbereadytoworkon
similarorrelatedproblemsinthefuture(becausetheproblemisrarelyagoalinand
ofitself).
Therearemanydifferencesbetweenthepresentbookandseveralwidelyknown
collectionsofnonstandardproblems.Forexample,ourworkdiffersfromaremark-
able collection [Steinhaus 1958] in that it explores more advanced topics at the
collegeundergraduateandgraduatelevel.
The main difference from another brilliant collection of problems for college
exitexams[Arnold1991,1989Russian]Isthatthisbookincludesasmallernumber
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
correspondingproblemgroupsinourbookwheretheyfitlogicallyintotheproblem
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
readertocombineideasfromdifferentbranchesofmathematicstosolveproblems.
We have achieved our goal if the reader becomes more adept in solving real
mathematicalproblemsandwewillbequitesatisfiedifthereaderdevelopsataste
forthiskindofwork.
Someoftheproblems(especiallythoserelatedtoappliedtopics)andsolutionsin
thisbookwehavesuggestedourselves.Wherethisisnotthecase,wehavetriedto
give credit to the authors of the problems and solutions. Absence of a reference
meansthatwedonotknowtheprimarysourceandalsothatthefact(ormethod)in
questionhasbecomeapartof“mathematicalfolklore.”
RishonLeZion,Israel AlexanderA.Roytvarf