# Thinking in Problems: How Mathematicians Find Creative Solutions

###### Alexander A. Roytvarf
Publisher:
Birkhäuser
Publication Date:
2013
Number of Pages:
405
Format:
Hardcover
Price:
79.95
ISBN:
9780817684051
Category:
Problem Book
[Reviewed by
Annie Selden
, on
05/30/2014
]

This book has a unique structure and purpose. There are twelve groups of related problems, sort of like chapters, numbered P1 through P12. These are “organized in such a way that the problems [in each group] build on each other, providing students with a ladder to master increasingly difficult problems.” If at first you do not succeed with a given problem, there are hints, explanations, and finalizations of solutions, but no full solutions. Although some definitions are given, the author assumes that the “relevant definitions will be known to the reader”. The problems come from calculus, analysis, linear algebra, multilinear algebra, and combinatorics. The author uses stars to indicate problem difficulty level, but the author’s idea that problems with one, two, and three stars “correspond to the recommended number of semesters of a university-level math curriculum” may reflect a mathematics curriculum somewhere, but not one in the U.S. Thus, this book, if used by students going through a normal U.S. math curriculum, is mainly for highly motivated upper-level undergraduate or graduate mathematics students.

The book begins with a preface explaining the aim of the book. This is followed by a section, “Using the Stars on Problems” in which the author gives specific details about what he means by no, one, two, and three stars. For example, no star includes not only basic arithmetic but also systems of two or three equations, elementary combinatorial formulas, and basic types of mathematical argumentation — proof by contradiction and mathematical induction. One star includes basic set theory including the Cantor-Bernstein-Schroeder theorem, one variable differential calculus, systems of linear equations, topology of the real line, and multi-linear functions. Two stars includes the algebra of polynomials, Jordan canonical form, Euclidean and Hermitian finite-dimensional spaces, and power series. Three stars includes Fourier series, the Weierstrass approximation theorem, transfinite induction, and abstract algebra including Noetherian rings and Galois finite fields. There are a total of 193 problems with 6 having no stars, 58 one star, 70 two stars, and 59 three stars. The author gives references for each of these star levels, including some well-known U.S textbooks, like Apostol, “baby Rudin”, van der Waerden, Lang, and Halmos.

Next comes a section titled, “Understanding the Advanced Skill Requirements”. This is a rather long list that includes: fundamentals of differential geometry, elements of differential manifolds, metric spaces, normed linear spaces, complex analysis, elements of probability theory and mathematical statistics, and ordinary differential equations. I expect many U.S. upper-level undergraduate math majors may not have many of these skill requirements. This section is followed by acknowledgments, and only then the table of contents. The titles of the groups of problems (a.k.a., chapters) include: Jacobi Identities and Related Combinatorial Formulas, 2 × 2 Matrices That are Roots of Unity, Convexity and Related Classic Inequalities, and Least Squares and Chebyshev Systems. After these 39 pages of introductory matter, the problems begin.

To give an idea of the author’s star system of difficulty, here is a no star problem:

P11.1

Establish identities

$\sum_{i=l}^m (-1)^{i-l}\binom{m+1}{i+1}\binom{i}{l} = 1,$

for natural numbers $m\geq l$.

An example of a one star problem is:

P10.2*

Prove that for any $a>0$ and $t_0\neq 0$ there is at most one one-parameter linear group $g$ in $\mathbb{R}$ such that $g(t_0)=a$,

where a one-parameter linear group has been defined in P10.0. Could your Calculus I students do this?

Who is this book for? The author states that he “aimed to create an atmosphere of real mathematical work for readers”, and that it is “for people who think that mathematics is beautiful, for people who want to expand their mathematical horizons and sharpen their skills.” But the prerequisites for using this book are many, as indicated above — not just the classic “mathematical maturity”. So it seems to me to be a book for self-motivated U.S. graduate level mathematics students who want to hone their mathematical problem-solving skills in the areas covered and who are not overloaded with regular homework, but not one for regular undergraduate mathematics majors.

I can’t say whether undergraduate mathematics students from other countries would be more prepared to tackle the problems in this book. The author is from Israel, but no affiliation is given other than the city of Rishon LeZion, so it’s hard to speculate what undergraduate mathematics curriculum he might have been thinking of. In addition, perhaps some parts of this book could also be used in teaching some courses.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.

Section I. Problems.- 1. Jacobi Identities and Related Combinatorial Formulas.- 2. A Property of Recurrent Sequences.- 3. A Combinatorial Algorithm in Multiexponential Analysis.- 4. A Frequently Encountered Determinant.- 5. A Dynamical System with a Strange Attractor.- 6. Polar and Singular Value Decomposition Theorems.- 7. 2X2 Matrices Which Are Roots of 1.- 8. A Property of Orthogonal Matrices.- 9. Convexity and Related Classical Inequalities.- 10. One-Parameter Groups of Linear Transformations.- 11. Examples of Generating Functions in Combinatorial Theory and Analysis.- 12. Least Squares and Chebyshev Systems.- Section II. Hints.- 1. Jacobi Identities and Related Combinatorial Formulas.- 2. A Property of Recurrent Sequences.- 3. A Combinatorial Algorithm in Multiexponential Analysis.- 4. A Frequently Encountered Determinant.- 5. A Dynamical System with a Strange Attractor.- 6. Polar and Singular Value Decomposition Theorems.- 7. 2X2 Matrices Which Are Roots of 1.- 8. A Property of Orthogonal Matrices.- 9. Convexity and Related Classical Inequalities.- 10. One-Parameter Groups of Linear Transformations.- 11. Examples of Generating Functions in Combinatorial Theory and Analysis.- 12. Least Squares and Chebyshev Systems.- Section III. Explanations.-1. Jacobi Identities and Related Combinatorial Formulas.- 2. A Property of Recurrent Sequences.- 3. A Combinatorial Algorithm in Multiexponential Analysis.- 4. A Frequently Encountered Determinant.- 5. A Dynamical System with a Strange Attractor.- 6. Polar and Singular Value Decomposition Theorems.- 7. 2X2 Matrices Which Are Roots of 1.- 8. A Property of Orthogonal Matrices.- 9. Convexity and Related Classical Inequalities.- 10. One-Parameter Groups of Linear Transformations.- 11. Examples of Generating Functions in Combinatorial Theory and Analysis.- 12. Least Squares and Chebyshev Systems.- Section IV. Full Solutions.- 1. Jacobi Identities and Related Combinatorial Formulas.- 2. A Property of Recurrent Sequences.- 3. A Combinatorial Algorithm in Multiexponential Analysis.- 4. A Frequently Encountered Determinant.- 5. A Dynamical System with a Strange Attractor.- 6. Polar and Singular Value Decomposition Theorems.- 7. 2X2 Matrices Which Are Roots of 1.- 8. A Property of Orthogonal Matrices.- 9. Convexity and Related Classical Inequalities.- 10. One-Parameter Groups of Linear Transformations.- 11. Examples of Generating Functions in Combinatorial Theory and Analysis.- 12. Least Squares and Chebyshev Systems.