Recent book reviews

Five recent book reviews

Problem – Solving and Selected Topics in Euclidean Geometry. In the Spirit of the Mathematical Olympiads

Author(s): 
Sotirios E. Louridas and Michael Th. Rassias
Publisher: 
Springer
Year: 
2013
ISBN: 
978 1-4614–7272–8
Short description: 

The book is a wonderful presentation of the essential concepts, ideas and results of Euclidean Geometry useful in solving olympiad problems of various level of difficulties. The theoretical part is excellently illustrated by challenging olympiad problems. The complete solutions to these problems are carefully presented, most of them together with several interesting comments and remarks.

MSC main category: 
51 Geometry
MSC category: 
51-01
Review: 

Geometry is one of the most important and active fields in Mathematics with a substantial and large variety of applications in several disciplines, and with a very high impact in all levels of mathematical education.

This book deals with the essential results in plane Euclidean Geometry that are useful in solving difficult olympiad problems. The reader will become acquainted with well - known theorems such as Menelaus theorem, Ceva theorem, Ptolemy theorem, Stewart theorem, Euler nine point circle and the Euler line, etc., in the context of some complex geometric problems.

This book provides a very synthetic presentation of concepts and ideas in Euclidean Geometry, most of them without proof since its main goal is to illustrate by nonstandard problems how these ideas can be used. The book clearly demonstrates how instrumental it is to use various tools for the formulation of basic geometrical questions in order to find the simplest and the most intuitive arguments to solve a variety of problems. The book under review fully fits this purpose. In several situations and from different points of view the book presents the power of some natural geometric ideas. Most of the material is really suitable for advanced high-school classes and the book itself could offer a great service of attracting bright students to Mathematics.

The textbook is organized into six chapters. The first four chapters present some theoretical results including suggestive examples on the following aspects : Euclid's Elements, logic, methods of proof, fundamentals on geometric transformations and some important theoretical results in solving problems. Chapter 5 contains carefully selected Olympiad - caliber problems and it is organized into three sections : geometric problems with basic theory, geometric problems with more advanced theory, geometric inequalities.

The book concludes with a useful and relevant bibliography containing 99 references. It also contains an index of symbols and a subject index.

I would like to conclude this review with the statement of appreciation of the Fields Medalist Michael H. Freedman who wrote the foreword of the book :
`"........Young people need such texts, grounded in our shared intellectual history and challenging them to excel and create a continuity with the past. Geometry has seemed destined to give way in our modern computerized world to algebra. As with Michael Th. Rassias' previous homonymous book on number theory, it is a pleasure to see the mental discipline of the ancient Greeks so well represented to a youthful audience ".

All in all the text is a highly recommendable choice for any olympiad training program, and fills some gaps in the existing literature in Euclidean Geometry. The book is a very useful source of models and ideas for students, teachers, heads of national teams and authors of problems, as well as for people who are interested in mathematics and solving difficult problems.

Reviewer: 
Mihaly Bencze
Affiliation: 
PhD, University of Craiova

Topological Library. Volume III. Spectral Sequences in Topology

Author(s): 
Editors: S.P. Novikov & I.A. Taimanov
Publisher: 
World Scientific
Year: 
2012
ISBN: 
9789814401302
Short description: 

This volume belongs to a three-volume edition devoted to reprint classical papers on algebraic and differential topology published in the 1950s-1960s. This is a collection of carefully chosen classical papers who have had a big impact in the development of algebraic topology during the XX century.

MSC main category: 
55 Algebraic topology
Review: 

This volume belongs to a three-volume edition devoted to reprint classical papers on algebraic and differential topology published in the 1950s-1960s. Some of these papers may be difficult to find, and several of them have been translated (by V.P. Golubyatnikov) into English for this volume.

The first volume was dedicated to Cobordism theory, the second one to Smooth structures on manifolds. The present volume is devoted, as the title says, to Spectral sequences in topology. The papers appearing in the volume are the following:

J.P. Serre. Singular homology of fiber spaces, Ann. Math. 1951.

J.P. Serre. Homotopy groups and classes of abelian groups, Ann. Math. 58 (1953) 268-294.

J.P. Serre. Cohomology modulo 2 of Eilenberg-MacLane complexes, Comment. Math. Helv. 27 (1953) 198-232.

A. Borel. On cohomology of principal fiber bundles and homogeneous spaces of compact Lie groups, Ann. Math. 57 (1953) 115-207.

A. Borel, Cohomology mod 2 of some homogeneous spaces, Comment. Math. Helv. 27 (1953) 165-197.

J. Milnor. The Steenrod algebra and its dual, Ann. Math. 67 (1958) 150-171.

J.F. Adams. On the structure and applications of the Steenrond algebra, Comm. Math. Helv. 32 (1958) 180-214.

M.F. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces, Proc. Symp. Purre Math. vol. III, AMS (1961) 7-38.

S.P. Novikov. The methods of algebraic topology from viewpoint of cobordism theory, Izv. Akad. Nauk SSSR, Ser. Mat. Tom 31 (1967) No. 4.

This is a collection of carefully chosen classical papers who have had a big impact in the development of algebraic topology during the XX century. The papers that were published in other languages have been translated into English. All the papers are accompanied by useful comments about further developments or related publications.

This is a nice volume that should not be missing in any Mathematics Library.

Reviewer: 
Vicente Muñoz
Affiliation: 
UCM

Jim Totten's Problems of the Week

Author(s): 
John Grant McLoughlin, Joseph Khoury, Bruce Shawyer (eds.)
Publisher: 
World Scientific
Year: 
2013
ISBN: 
978-981-4513-30-2 (hbk)
Price (tentative): 
GBP 45 (hbk)
Short description: 

This is a collection of problems and solutions of miscellaneous mathematical problems at an advanced undergraduate level. They are selected from the notes of Jim Totten (1947-2008) after he passed away unexpectedly. Jim Totten was problem editor and later editor in chief of Crux Mathematicorum of the Canadian Mathematical Society.

URL for publisher, author, or book: 
www.worldscientific.com/worldscibooks/10.1142/8836
MSC main category: 
00 General
MSC category: 
00A07
Review: 

Jim Totten was problem editor and later editor in chief of the journal Crux Mathematicorum of the Canadian Mathematical Society. In 1986 he collected 80 of his problems under the title Problems of the Week as volume 7 of the ATOM (A Taste of Mathematics) series published by the CMS. The origin of the title is that when he started teaching in 1976, he posted a weekly problem to challenge his students. The response to these was so positive that it tempted him to continue the idea for about 30 more years.

The present book contains 406 problems and solutions that were collected from Totten's notes that he left after his unexpected death in 2008. Although, to solve the problems, the required mathematics are at an undergraduate level, covering many different topics such as logic, geometry, functions, number theory, statistics, etc., the problems are often quite challenging. Even for professional mathematicians they are not at all trivial. They are often formulated as brain teasing puzzles with an underlying recreational flavour. Whatever the formulation is, the solution always requires some sound mathematical reasoning. The witty solutions are included and often require more than just the application of standard class room recipes. Not that they are terribly complicated, once you know the trick, but it may take a pencil and possibly several pages of trial an error to arrive at the key that will set you on the rails to find the solution. Since Fermat's margin note, we know that a problem with a simple formulation can take a highly advanced body of mathematics to solve a seemingly simple problem. Not in the case of these problems. All what is needed stays well within the package that an undergraduate student should be able to deal with. Just a clear and agile mind that is sensitive for this type of puzzles suffices.

The kind of problems can best be illustrated by giving two examples that I selected just because they are short to formulate. Here is one from number theory: "How old is the captain, how many children has he, and how long is his boat, given the product 32118 of the three desired integers? The length of his boat is given in feet (several feet), the captain has both sons and daughters, he has more years than children, but he is not yet one hundred years old." And here is one from plane geometry: "Three circles or radius r each pass through the centers of the other two. What is the area of their common intersection?".

Solving all of the problems will provide many hours and days of puzzling pass-time. There is however the temptation to read the answer prematurely because the proposed solution immediately follows the problem formulation, enticing the reader to read it before he or she even tried to work it out him- or herself. If the purpose is that the reader should indeed find the solutions, then it might have been a better idea to give all the problems in the first part and all the solutions in a second part. However, if the reader is looking for problems to give to others and at the same time estimate the level of difficulty, then the present format is of course the better one. It is technically also the simpler solution because many of the geometric problems require drawings, which should then be repeated for problem and answer to avoid an annoying flipping back and forth to match the text with the figure.

There are of course many other popular puzzler books on the market that also have a lot of brain teasers and logical recreations, but these are usually requiring less mathematics and less computation. Martin Gardner's mathematical puzzles come close, but there the emphasis is often on the recreational aspect, rather than on the mathematical side. In Totten's problems the scales tip much more to the mathematical side.

Reviewer: 
A. Bultheel
Affiliation: 
KU Leuven

Figurate Numbers

Author(s): 
Elena Deza, Michael Marie Deza
Publisher: 
World Scientific
Year: 
2012
ISBN: 
13 978-9814355-48-3
Short description: 

The book provides a systematic and thorough presentation of the
theory of figurate numbers.

MSC main category: 
00 General
Reviewer: 
Capi Corrales Rodrigáñez
Affiliation: 
Mathematics, Universidad Complutense de Madrid

Stable Perturbations of Operators and related topics

Author(s): 
Yifeng Xue
Publisher: 
Word Scientific
Year: 
2012
ISBN: 
13 978-981-4383-59-2
Short description: 

The book studies different analysis perturbation of operators , including Moore-Penrose inverse, and Drazin inverse of operaors .Several miscellaneous applications are also included.

MSC main category: 
46 Functional analysis
Review: 

This book concerns with the theory of stable perturbations of operators in
Banach spaces . Thus the perturbation analysis for generalized inverses, the
Moore-Penrose inverse and Drazin inverse of operators under stable pertur-
bation. After a preliminary chapter including several basic results in Func-
tional Analysis ( Hilbert spaces, operators and C + algebras ) the book studies
carefully the relationships among the densely-defined operators with closed
range and the reduced minimum modulus of densely defined operators . The
Moore-Penrose inverse and its stable perturbations in Hilbert spaces and in
C ∗ -algebras is also presented . Several results of the K-theory in C + algebras
are also included. The last chapters of the book contain some miscellaneous
applications and related topics to the perturbation theory, like the approxi-
mate polar decomposition in C ∗ -algebras, and some applications of Moore-
Penrose inverses in frame theory.The list of references in the Bibliography is
quite extensive.

Reviewer: 
Francisco L. Hernandez
Affiliation: 
Universidad complutense de Madrid