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**Mathematics and technology. With the participation of Hélène Antaya and Isabelle Ascah-Coallier. Translated by Chris Hamilton.**
*(English)*
Zbl 1211.00020

Springer Undergraduate Texts in Mathematics and Technology. New York, NY: Springer (ISBN 978-0-387-69215-9/hbk; 978-1-4419-2407-0/pbk). xv, 580 p. (2008).

The presented book is devoted to the technological applications of mathematics making use of elegant mathematical concepts, and develops these concepts in the context of applications to important, practical problems. The authors highlight how mathematical modeling, together with the power of mathematical tools, has been crucial for innovation in technology. The exposition is clear, straightforward, motivated by excellent examples, and user-friendly.

Although a few of the subjects considered in the book fall outside the strict domain of technology, it becomes clear that mathematics is useful, and it plays a major role in everyday technologies. Several of the subjects treated in the text are still being actively developed, and this allows students to see, often for the first time, that the field of mathematics remains open and dynamic. An engaging quality of this book is that the authors also put the mathematical material in a historical context and not just a practical one.

The content of the book is constructed with paying particular attention to the following points:

- The applications should be recent or affect the students’ day-to-day life. Moreover, contrary to the mature mathematics typically taught in other undergraduate courses, some of the mathematics used should be modern or even still in development.

- The mathematics should be relatively elementary and if it exceeds the typical first-year undergraduate curriculum (calculus, linear algebra, probability theory), the missing pieces must be covered within the chapter. A special effort is made to make extensive use of high-school-level mathematics, particularly Euclidean geometry. Basic high-school and undergraduate mathematics form a remarkable toolkit, provided they are well understood and mastered, allowing students to readily explore their wide applications and, often for the first time, to discover their power when used together.

- The level of mathematical sophistication required should remain at a minimum: ideas are a scientist’s most precious commodity, and behind most technological successes there lies a brilliant yet sometimes elementary observation.

As a result, the mathematics used in the book covers a very wide spectrum:

- Lines and planes appear in all of their forms (regular equations, parametric equations, subspaces), often in unexpected ways (using the intersection of several planes to decode a Reed Solomon encoded message).

- A large number of subjects make use of basic geometric objects: circles, spheres, and conies. The concept of locus of points in Euclidean geometry is often repeated, for example in problems where is calculated the position of an object through triangulation (Chapter 1 on GPS, and Chapter 15 on Science Flashes).

- The different types of affine transformations in the plane or in space (in particular rotation and symmetries) appear several times: in Chapter 11 on image compression using fractals, in Chapter 2 on mosaics and friezes, and in Chapter 3 on robot motion.

- Finite groups appear as symmetry groups (Chapter 2 on mosaics and friezes) and also in the development of primality tests in cryptography (Chapter 7).

- Finite fields make an appearance in Chapter 6 on error-correcting codes, in Chapter 1 on GPS and in Chapter 8 on random-number generation.

- Chapter 7 on cryptography and Chapter 8 on random-number generation both make use of arithmetic modulo n, while Chapter 6 on error-correcting codes makes use of arithmetic modulo 2.

- Probability theory appears in several unexpected places: in Chapter 9 on Google’s Page Rank algorithm, and in the construction of large prime numbers in Chapter 7. It is also used more classically in Chapter 8 on random-number generation.

- Linear algebra is omnipresent: in Chapter 6 on Hamming and Reed Solomon codes, in Chapter 9 on the PageRank algorithm, in Chapter 3 on robot motion, in Chapter 2 on mosaics and friezes, in Chapter 1 on GPS, in Chapter 12 on the JPEG standard, etc.

The text is written for students who have a familiarity with Euclidean geometry and have mastered multivariable calculus, linear algebra, and elementary probability theory. Working-through the text requires a certain scientific maturity: it involves integrating a variety of mathematical tools in a setting different from the one in which they were originally taught. For that reason, undergraduates in their junior or senior years are the ideal audience for the course.

The text presents applications in two forms: the main chapters (all except Chapter 15) are long and detailed, while the Science Flashes (sections of Chapter 15) are short and narrow in scope. Readers will notice a certain unity in the form of the longer chapters: the first sections describe the application and the underlying mathematical problem; this is followed by an exploration of simple cases of the problem and, if necessary, a development of the required mathematics. These parts are the basic portion of the chapter. Afterward, one or more sections are exploring more-complicated examples, provide more details to the mathematical tools discussed earlier, or simply discuss the fact that mathematics alone is not always sufficient! This latter part of a chapter is the advanced portion.

Each chapter is mathematically independent, and any links among them are explicitly stated. The reader can hop from subject to subject at will. Hopefully, the reader will be equally interested in the many historical notes scattered throughout the text. The beginning of each chapter contains a brief “how-to”, describing the required basic knowledge, the relationships between the sections, and their relative difficulty. Numerous exercises at the end of every section provide practice and reinforce the material in the chapter.

This book is intended mainly for undergraduate students in pure and applied mathematics, physics and computer science, instructors, and high school teachers. The main prerequisites are linear algebra and Euclidean geometry. A few chapters require multivariable calculus and elementary probability theory. A clear indication of the more difficult topics and relatively advanced references make it also suitable for an independent reader mastering the prerequisites. Additionally, its lack of calculus centricity as well as a clear indication of the more difficult topics and relatively advanced references makes it suitable for any curious individual with a decent command of high school math.

Although a few of the subjects considered in the book fall outside the strict domain of technology, it becomes clear that mathematics is useful, and it plays a major role in everyday technologies. Several of the subjects treated in the text are still being actively developed, and this allows students to see, often for the first time, that the field of mathematics remains open and dynamic. An engaging quality of this book is that the authors also put the mathematical material in a historical context and not just a practical one.

The content of the book is constructed with paying particular attention to the following points:

- The applications should be recent or affect the students’ day-to-day life. Moreover, contrary to the mature mathematics typically taught in other undergraduate courses, some of the mathematics used should be modern or even still in development.

- The mathematics should be relatively elementary and if it exceeds the typical first-year undergraduate curriculum (calculus, linear algebra, probability theory), the missing pieces must be covered within the chapter. A special effort is made to make extensive use of high-school-level mathematics, particularly Euclidean geometry. Basic high-school and undergraduate mathematics form a remarkable toolkit, provided they are well understood and mastered, allowing students to readily explore their wide applications and, often for the first time, to discover their power when used together.

- The level of mathematical sophistication required should remain at a minimum: ideas are a scientist’s most precious commodity, and behind most technological successes there lies a brilliant yet sometimes elementary observation.

As a result, the mathematics used in the book covers a very wide spectrum:

- Lines and planes appear in all of their forms (regular equations, parametric equations, subspaces), often in unexpected ways (using the intersection of several planes to decode a Reed Solomon encoded message).

- A large number of subjects make use of basic geometric objects: circles, spheres, and conies. The concept of locus of points in Euclidean geometry is often repeated, for example in problems where is calculated the position of an object through triangulation (Chapter 1 on GPS, and Chapter 15 on Science Flashes).

- The different types of affine transformations in the plane or in space (in particular rotation and symmetries) appear several times: in Chapter 11 on image compression using fractals, in Chapter 2 on mosaics and friezes, and in Chapter 3 on robot motion.

- Finite groups appear as symmetry groups (Chapter 2 on mosaics and friezes) and also in the development of primality tests in cryptography (Chapter 7).

- Finite fields make an appearance in Chapter 6 on error-correcting codes, in Chapter 1 on GPS and in Chapter 8 on random-number generation.

- Chapter 7 on cryptography and Chapter 8 on random-number generation both make use of arithmetic modulo n, while Chapter 6 on error-correcting codes makes use of arithmetic modulo 2.

- Probability theory appears in several unexpected places: in Chapter 9 on Google’s Page Rank algorithm, and in the construction of large prime numbers in Chapter 7. It is also used more classically in Chapter 8 on random-number generation.

- Linear algebra is omnipresent: in Chapter 6 on Hamming and Reed Solomon codes, in Chapter 9 on the PageRank algorithm, in Chapter 3 on robot motion, in Chapter 2 on mosaics and friezes, in Chapter 1 on GPS, in Chapter 12 on the JPEG standard, etc.

The text is written for students who have a familiarity with Euclidean geometry and have mastered multivariable calculus, linear algebra, and elementary probability theory. Working-through the text requires a certain scientific maturity: it involves integrating a variety of mathematical tools in a setting different from the one in which they were originally taught. For that reason, undergraduates in their junior or senior years are the ideal audience for the course.

The text presents applications in two forms: the main chapters (all except Chapter 15) are long and detailed, while the Science Flashes (sections of Chapter 15) are short and narrow in scope. Readers will notice a certain unity in the form of the longer chapters: the first sections describe the application and the underlying mathematical problem; this is followed by an exploration of simple cases of the problem and, if necessary, a development of the required mathematics. These parts are the basic portion of the chapter. Afterward, one or more sections are exploring more-complicated examples, provide more details to the mathematical tools discussed earlier, or simply discuss the fact that mathematics alone is not always sufficient! This latter part of a chapter is the advanced portion.

Each chapter is mathematically independent, and any links among them are explicitly stated. The reader can hop from subject to subject at will. Hopefully, the reader will be equally interested in the many historical notes scattered throughout the text. The beginning of each chapter contains a brief “how-to”, describing the required basic knowledge, the relationships between the sections, and their relative difficulty. Numerous exercises at the end of every section provide practice and reinforce the material in the chapter.

This book is intended mainly for undergraduate students in pure and applied mathematics, physics and computer science, instructors, and high school teachers. The main prerequisites are linear algebra and Euclidean geometry. A few chapters require multivariable calculus and elementary probability theory. A clear indication of the more difficult topics and relatively advanced references make it also suitable for an independent reader mastering the prerequisites. Additionally, its lack of calculus centricity as well as a clear indication of the more difficult topics and relatively advanced references makes it suitable for any curious individual with a decent command of high school math.

Reviewer: Tzvetan Semerdjiev (Sofia)

### MSC:

00A35 | Methodology of mathematics |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

97-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics education |

86-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geophysics |

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

00-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general |