Book Description

Liberal Arts mathematics books often cover much more material than can be addressed in a one-semester course. Karl Smith has created a solution to this problem with his new book: THE NATURE OF PROBLEM SOLVING IN GEOMETRY AND PROBABILITY. Loyal customers of Karl Smith's books laud his clear writing, coverage of historical topics, selection of topics, and emphasis on problem solving. Based on the successful NATURE OF MATHEMATICS text, this new book is designed to give you only the chapters and information you need, when you need it. Smith takes great care to provide insight into precisely what mathematics is--the nature of mathematics--what it can accomplish, and how it is pursued as a human enterprise. At the same time, Smith emphasizes Polya's problem-solving method throughout the text so students can take from the course an ability to estimate, calculate, and solve problems outside the classroom. Moreover, Smith's writing style gives students the confidence and ability to function mathematically in their everyday lives. This new text emphasizes problem solving and estimation, which, along with numerous in-text study aids, encourage students to understand the concepts as well as mastering techniques. ... Read more

Book Description

This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry. ... Read more

Reviews (1)

The beauty of geometry is captured
Originally published in 1942, this book has lost none of its power in the last half century. It is a commentary on the recent demise of geometry in many curricula that 33 years elapsed between the publication of the fifth and sixth editions. Fortunately, like so many things in the world, trends in mathematics are cyclic, and one can hope that the geometric cycle is on the rise. We in mathematics owe so much to geometry. It is generally conceded that much of the origins of mathematics is due to the simple necessity of maintaining accurate plots in settlements. The only book from the ancient history of mathematics that all mathematicians have heard of is the Elements by Euclid. It is one of the most read books of all time, arguably the only book without a religious theme still in widespread use over 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in the Euclidean classic.
There are other reasons why geometry should occupy a special place in our hearts. Most of the principles of the axiomatic method, the concept of the theorem and many of the techniques used in proofs were born and nurtured in the cradle of geometry. For many centuries, it was nearly an act of faith that all of geometry was Euclidean. That annoying fifth postulate seemed so out of place and yet it could not be made to go away. Many tried to remove it, but finally the Holmsean dictum of ,"once you have eliminated the impossible, what is left, not matter how improbable, must be true", had to be admitted. There were in fact three geometries, all of which are of equal validity. The other two, elliptic and hyperbolic, are the main topics of this wonderful book.
Coxeter is arguably the best geometer of this century but there can be no argument that he is the best explainer of geometry of this century. While fifty years is a mere spasm compared to the time since Euclid, it is certainly possible that students will be reading Coxeter far into the future with the same appreciation that we have when we read Euclid. His explanations of the non-Euclidean geometries is so clear that one cannot help but absorb the essentials. In so many ways, Euclidean geometry is but the middle way between the two other geometries. A point well made and in great detail by Coxeter.
Geometry is a jewel that was born on the banks of the Nile river and we should treasure and respect it as the seed from which so much of our basic reasoning processes sprouted. For this reason, you should buy this book and keep a copy on your shelf.

Published in Smarandache Notions Journal, reprinted with permission. ... Read more

Book Description

Shape and Shape Theory D. G. Kendall Churchill College, University of Cambridge, UK D. Barden Girton College, University of Cambridge, UK T. K. Carne King's College, University of Cambridge, UK H. Le University of Nottingham, UK The statistical theory of shape is a relatively new topic and is generating a great deal of interest and comment by statisticians, engineers and computer scientists. Mathematically, 'shape' is the geometrical information required to describe an object when location, scale and rotational effects are removed. The theory was pioneered by Professor David Kendall to solve practical problems concerning shape. This text presents an elegant account of the theory of shape that has evolved from Kendall's work. Features include:
* A comprehensive account of Kendall's shape spaces
* A variety of topological and geometric invariants of these spaces
* Emphasis on the mathematical aspects of shape analysis
* Coverage of the mathematical issues for a wide range of applications
The early chapters provide all the necessary background information, including the history and applications of shape theory. The authors then go on to analyse the topic, in brilliant detail, in a variety of different shape spaces. Kendall's own procedures for visualising distributions of shapes and shape processes are covered at length. Implications from other branches of mathematics are explored, along with more advanced applications, incorporating statistics and stochastic analysis. Applied statisticians, applied mathematicians, engineers and computer scientists working and researching in the fields of archaeology, astronomy, biology, geography and physical chemistry will find this book of great benefit. The theories presented are used today in a wide range of subjects from archaeology through to physics, and will provide fascinating reading to anyone engaged in such research. Visit our web page! http://www.wiley.com/ ... Read more

Reviews (1)

Not at all a "practical" book!
I would like to point out that the back text is a bit misleading, to say the least. It makes references to engineers, computer scientists and "a variety of practical examples", but the book is completely theoretical and not at all suited for someone who actually wants to use shape theory. For those people, "Statistical shape analysis" by Dryden and Mardia is much better suited. Once you have read this, you can get some interesting theoretical background from Kendall's book. For mathematicians this is quite likely the right choice. ... Read more

Reviews (1)

Review for "Conjecture & Proof" by D.D. Schwartz
This book is primarily designed as a textbook for a "transition" college course (the kind of course at the sophomore level that helps students make the transition from calculus and calculation intensivecourses to more abstract mathematics), and it does an excellent job. Whilethe structure of the book is quite standard for a text of this kind,students find it accessible and user-friendly. To achieve the goal ofhelping students grow comfortable with abstract mathematical structures,Dr. Diane Driscoll Schwartz presents at first an introduction to logic andits use in the construction of mathematical proofs. The topics included inthe two chapters cover several standard proof techniques, such as the useof the contrapositive and of the principle of mathematical induction. Thispart is followed by the discussion of a number of mathematical topics suchas sets, functions, relations, groups, modular arithmetic, combinatoricsand cardinality. While the sets of exercises include a sufficient number ofproblems, it would be nice to see this number increased. Some of theproblems are designed for group-work and provide a good starting point forclassroom discussion and for "non standard" homework assignments.The book is accompanied by an Instructor's Guide, which includessuggestions for class preparation and hints for some of the problems. Forthose schools that provide a technology-intensive environment, the text canbe use with the computer language ISETL. The software can be downloadedfrom the Web Sites indicated in the book. ... Read more

Book Description

This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics.They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis. ... Read more

Reviews (2)

A very nice little book
This is a very nice, short introduction to the subject -- This series of blue paperbacks by CUP is excellent. Typically, all books in the series are readable introductions. Somewhat higher level than the corresponding series from Springer (the one where all exercises have full solutions).

Incidentally, the author is a very attractive woman.

Well suited as an introduction to algebraic curves
The book gives a good general overview of algebraic curves using only elementary algebra, topology, and complex analysis. There are lots of diagrams of elliptic curves in the historical introduction in the first chapter and the subject is well motivated. Hilbert's Nullstellensatz is introduced in the context of real algebraic curves as an answer to the question of when the polynomials definte the same curve. The visualization approach taken by the author in the first chapter has taken on dramatic proportions do to the computer graphics packages currently available. The author introduces complex algebraic curves in complex 2-dimensional space in the next chapter. Recognizing that such curves are not compact, he compactifies them by adding suitable points at infinity, giving complex projective curves. The algebraic properties of these curves are studied in the next chapter. He does a good job of motivating the group law on elliptic curves on the last theorem of the chapter, leaving the proof of associativity to the reader in the exercises. The topology of complex projective curves is taken up in Chapter 4. The author gives two proofs of the degree-genus formula, one geometric and the other from a holomorphic point of view. This leads to a consideration of branch points and ramified covers. The author's outline of the proofs is very detailed and therefore very helpful to one encountering the proof for the first time. The statement of the formula via the Riemann-Roch theorem in more formal treatments (and later in the book) can then be appreciated more. The subject of non-singular complex projective curves, namely Riemann surfaces, is effectively discussed in Chapter 5, with holomorphic differentials outlined in Chapter 6. The Riemann-Roch theorem makes its appearance here, and the author is careful to point out its use as an alternative characterization of the genus given earlier by topological arguments. Divisors are introduced as formal sums, but their understanding is straightforward here because the author has motivated them with a discussion of the properties of holomorphic and meromorphic functions earlier in the chapter. The proof of the Riemann-Roch theorem is very detailed and understandable. The book ends with a discussion of singular curves via resolution of singularities. Newton polygons and Puiseux expansions are used to investigate the behavior of degree d projective curves near a singular point. The geometrical constructions used here by the author are of great help in understanding the behavior of these curves. A very well-written book for students and new-comers to the area of algebraic curves. It will pave the way for more advanced reading on the subject. ... Read more

Book Description

Knot theory now plays a large role in modern mathematics. This unparalleled text and reference describes the main concepts of modern knot theory with full proofs accessible to both beginners and professionals alike. It presents both classical and modern knot theory, as well as the most significant results from braid theory, including the full proof of Markov's theorem, and Alexander's and Vogel's algorithms. It includes valuable information on the theory of coding knots by d- diagrams, as well as the author's own results in virtual knot theory. The material is presented at a level suitable for advanced undergraduate students, and the text is ideal for a course on knot theory. ... Read more

Reviews (1)

Best of Knots
Knot Theory by Vassily Manturov (CRC Press) The aim of the present monograph is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. Thus, in the first chapter of the second part of the book (concerning braids) we start from the very beginning and in the same chapter construct the Jones two-variable polynomial and the faithful representation of the braid groups. A large part of the present title is devoted to rapidly developing areas of modern knot theory, such as virtual knot theory and Legendrian knot theory.
In the present book, we give both the "old" theory of knots, such as the fundamental group, Alexander's polynomials, the results of Dehn, Seifert, Burau, and Artin, and the newest investigations in this field due to Conway, Matveev, Jones, Kauffman, Vassiliev, Kontsevich, Bar-Natan and Birman. We also include the most significant results from braid theory, such as the full proof of Markov's theorem, Alexander's and Vogel's algorithms, Dehornoy algorithm for braid recognition, etc. We also describe various representations of braid groups, e.g., the famous Burau representation and the newest (1999-2000) faithful Krammer-Bigelow representation. Furthermore, we give a description of braid groups in different spaces and simple newest recognition algorithms for these groups. We also describe the construction of the Jones two-variable polynomial.
In addition, we pay attention to the theory of coding of knots by d-diagrams, described in the author's papers. Also, we give an introduction to virtual knot theory, proposed recently by Louis H. Kauffman. A great part of the book is devoted to the author's results in the theory of virtual knots.
Proofs of theorems involve some constructions from other theories, which have their own interest, i.e., quandle, product integral, Hecke algebras, connection theory and the Knizhnik-Zamolodchikov equation, Hopf algebras and quantum groups, Yang-Baxter equations, LD-systems, etc.
The contents of the book are not covered by existing monographs on knot theory; the present book has been taken a much of the author's Russian lecture notes book on the subject. The latter describes the lecture course that has been delivered by the author since 1999 for undergraduate students, graduate students, and professors of the Moscow State University.
The present monograph contains many new subjects (classical and modern), which are not represented in the author's earlier Russian version of this book.
While describing the skein polynomials we have added the Przytycky-Traczyk approach and Conway algebra. We have also added the complete knot invariant, the distributive grouppoid, also known as a quandle, and its generalisation. We have rewritten the virtual knot and link theory chapter. We have added some recent author's achievements on knots, braids, and virtual braids. We also describe the Khovanov categorification of the Jones polynomial, the Jones two-variable polynomial via Hecke algebras, the Krammer-Bigelow representation, etc.
The book is divided into thematic parts. The first part describes the state of "pre-Vassiliev" knot theory. It contains the simplest invariants and tricks with knots and braids, the fundamental group, the knot quandle, known skein polynomials, Kauffman's two-variable polynomial, some pretty properties of the Jones polynomial together with the famous Kauffman-Murasugi theorem and a knot table.
The second part discusses braid theory, including Alexander's and Vogel's algorithms, Dehornoy's algorithm, Markov's theorem, Yang-Baxter equations, Burau representation and the faithful Krammer-Bigelow representation. In addition, braids in different spaces are described, and simple word recognition algorithms for these groups are presented. We would like to point out that the first chapter of the second part (Chapter 8) is central to this part. This gives a representation of the braid theory in total: from various definitions of the braid group to the milestones in modern knot and braid theory, such as the Jones polynomial constructed via Hecke algebras and the faithfulness of the Krammer-Bigelow representation.
The third part is devoted to the Vassiliev knot invariants. We give all definitions, prove that Vassiliev invariants are stronger than all polynomial invariants, study structures of the chord diagram and Feynman diagram algebras, and finally present the full proof of the invariance for Kontsevich's integral. Here we also present a sketchy introduction to Bar-Natan's theory on Lie algebra representations and knots. We also give estimates of the dimension growth for the chord diagram algebra.
In the fourth part we describe a new way for encoding knots by d-diagrams proposed by the author. This way allows us to encode topological objects (such as knot, links, and chord diagrams) by words in a finite alphabet. Some applications of d-diagrams (the author's proof of the Kauffman-Murasugi theorem, chord diagram realisability recognition, etc.) are also described.
The fifth part contains virtual knot theory together with "virtualisations" of knot invariants. We describe Kauffman's results (basic definitions, foundation of the theory, Jones and Kauffman polynomials, quandles, finite-type invariants), and the work of Vershinin (virtual braids and their representation). We also included our own results concerning new invariants of virtual knots: those coming from the "virtual quandle", matrix formulae and invariant polynomials in one and several variables, generalisation of the Jones polynomials via curves in 2-surfaces, "long virtual link" invariants, and virtual braids.
The final part gives a sketchy introduction to two theories: knots in 3-manifolds (e.g., knots in RP3 with Drobotukhina's generalisation of the Jones polynomial), the introduction to Kirby's calculus and Witten's theory, and Legendrian knots and links after Fuchs and Tabachnikov. We recommend the newest book on 3-manifolds by Matveev.
At the end of the book, a list of unsolved problems in knot and link theory and the knot table are given.
The description of the mathematical material is sufficiently closed; the mono-graph is quite accessible for undergraduate students of younger courses, thus it can be used as a course book on knots. The book can also be useful for professionals because it contains the newest and the most significant scientific developments in knot theory. Some technical details of proofs, which are not used in the sequel, are either omitted or printed in small type. ... Read more

Book Description

In this superb topology text, the readers not only learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, but also their role in understanding molecular structures.Most results described in the text are motivated by the questions of chemists or molecular biologists, though they often go beyond answering the original question asked.No specific mathematical or chemical prerequisites are required. The text is enhanced by nearly 200 illustrations and 100 exercises. With this fascinating book, undergraduate mathematics students escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists find simple and clear but rigorous definitions of mathematical concepts they handle intuitively in their work. ... Read more

Reviews (1)

How can topology be applied to chemistry?
This question is answered in a remarkable new book by Erica Flapan, knottheorist and professor of mathematics at Pomona College.Stereochemistry,the study of the three-dimensional structure of molecules, is a recurringtheme in chemistry and related fields.Topology, the study of geometricalproperties which are invariant under continuous transformations, is asimilarly popular area for mathematicians.While it is not immediatelyobvious that the two fields have anything in common, both fields owe a debtto the other.Although she initiates her book with a historicalperspective and detailed expository on basic aspects of low dimensionaltopology (e.g., stereoisomers, chirality, nonrigid symmetries, knot andlink types, three-dimensional manifolds, and link polynomials) she doesproceed into more advanced subject matter in later chapters includingMöbius ladders, symmetries of embedded graphs, and hierarchies ofautomorphisms.Of particular interest to the molecular biologist, thefinal chapter is devoted entirely to the topology of DNA and includestopological considerations of supercoiling, toroidal winding, enzymeaction, and tangle theory. The arguments are clearly presented within theframework of interesting and relevant molecular structures, yet there isenough mathematical rigor to satisfy dyed-in-the-wool mathematicians aswell.Although there will likely be something of interest to the averageworking chemist, the supramolecular scientist, organic chemist, molecularbiologist, and biophysicist, in particular, stand to gain the most by thecontents of this book. ... Read more

Book Description

Loop groups, the simplest class of infinite dimensional Lie groups, have recently been the subject of intense study. This book gives a complete and self-contained account of what is known about them from a geometrical and analytical point of view, drawing together the many branches of mathematics from which current theory developed--algebra, geometry, analysis, combinatorics, and the mathematics of quantum field theory.The authors discuss loop groups' applications to simple particle physics and explain how the mathematics used in connection with loop groups is itself interesting and valuable, thereby making this work accessible to mathematicians in many fields. ... Read more

Book Description

The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery. ... Read more

Book Description

This unique approach to combinatorics is centered around challenging examples, fully-worked solutions, and hundreds of problems---many from Olympiads and other competitions, and many original to the authors. Each chapter highlights a particular aspect of the subject and casts combinatorial concepts in the guise of questions, illustrations, and exercises that are designed to encourage creativity, improve problem-solving techniques, and widen the reader's mathematical horizons.

Topics encompass permutations and combinations, binomial coefficients and their applications, recursion, bijections, inclusions and exclusions, and generating functions. The work is replete with a broad range of useful methods and results, such as Sperner's Theorem, Catalan paths, integer partitions and Young's diagrams, and Lucas' and Kummer's Theorems on divisibility. Strong emphasis is placed on connections between combinatorial and graph-theoretic reasoning and on links between algebra and geometry.

The authors' previous text, 102 Combinatorial Problems, makes a fine companion volume to the present work, which is ideal for Olympiad participants and coaches, advanced high school students, undergraduates, and college instructors. The book's unusual problems and examples will stimulate seasoned mathematicians as well. A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but to mathematical ingenuity, rigor, and the joy of solving puzzles. ... Read more

Book Description

Curves and Surfaces for Geometric Design offers both a theoretically unifying understanding of polynomial curves and surfaces and an effective approach to implementation that you can bring to bear on your own work-whether you're a graduate student, scientist, or practitioner.

Inside, the focus is on "blossoming"-the process of converting a polynomial to its polar form-as a natural, purely geometric explanation of the behavior of curves and surfaces.This insight is important for far more than its theoretical elegance, for the author proceeds to demonstrate the value of blossoming as a practical algorithmic tool for generating and manipulating curves and surfaces that meet many different criteria.You'll learn to use this and related techniques drawn from affine geometry for computing and adjusting control points, deriving the continuity conditions for splines, creating subdivision surfaces, and more.

The product of groundbreaking research by a noteworthy computer scientist and mathematician, this book is destined to emerge as a classic work on this complex subject.It will be an essential acquisition for readers in many different areas, including computer graphics and animation, robotics, virtual reality, geometric modeling and design, medical imaging, computer vision, and motion planning.

Reviews (3)

Best text on geometric design
This is a great book, definitely the best among the various books on geometric design and CAGD (other good ones include Farin, Mortsenson, Piegl and Tiller, Hoscheck and Lasser). It is not as encyclopedic as the sources listed above, but it a lot more coherent and a lot clearer, because it follows the unifying concept of blossoming. As a result, one gets multiple complementary views of polynomial curves and surfaces: algebraic, geometric, combinatorial, and algorithmic. For example, we can see where the Bernstein polynomials come from, instead of mysteriously being dropped from the sky. The systematic use of blossoms (polar forms) is particularly elegant in the presentation of surfaces, where it clarifies greatly the differences between rectangular and triangular patches. The discussion of subdivision versions of the de Casteljau algorithm is very thorough and unique. Gallier's book is also the only book to discuss subdivision surfaces in some detail (Doo-Sabin, Catmull-Clark, and Loop). In particular, an analysis of the convergence of Loop's scheme is given. For this, the author gives a remarkable crash course on the discrete Fourier transform. However, this chapter is too dense and should have been split. Also, much more pictures are needed. It seems that the author was in a rush. The appendix on vector spaces is gorgeous, and the one on differentials is also excellent. This book is highly recommended to mathematically inclined readers interested in geometric modeling and computer graphics. Too bad that applications to medicine such as organ modeling, or to computer animation, are not presented. Nevertheless, Mathematica code is provided for most of the algorithms. A web site would be helpful.

A good mathematical review for practicing graphics engineers
This book is a good review of the concepts of geometry for Modeling. The presentation is original. The mathematical treatment is sound. This a "required reading" for those in Computer Graphics research and did not have a good course in geometry. Those who have had a good course in geometry will appreciate the original style of presentation. This book fills a long felt gap in the treatment of geometry from the perspective of Computer Graphics. The book assumes minimal background in mathematics, and is almost self-contained.

There are fewer graphics programmers who have an adequate understanding of the underlying mathematical concepts. This book can partially help the graphics programmers to cross over to that select group. Problems at the end of each chapter enhance the value of the book. The material is updated with latest developments in the field such as subdivision surfaces.

People interested in Computer Graphics, Geometric Modeling, Computer Vision, and Robotics will benefit from studying this book.

A brillian geometry book
I found this book an exellent introduction to advanced geometry concepts used in computer graphics, vision, robotics, geometric modeling and many other related areas. Gallier has struck a perfect balance between formal mathematical rigour and intuition and readability which the book lends easily with its many beautiful illustrations and examples. The concept of "blossoming" is a rarely-seen but extremely elegant way of presenting the curves and surfaces. This book is a must for anyone who loves the elegance of geometry. ... Read more

Book Description

This book is the first systematic and comprehensive study ofthe theory of continuous selections of multivalued mappings. Thisinteresting branch of modern topology was introduced by E.A. Michaelin the 1950s and has since witnessed an intensive development withvarious applications outside topology, e.g. in geometry of Banachspaces, manifolds theory, convex sets, fixed points theory,differential inclusions, optimal control, approximation theory, andmathematical economics. The work can be used in different ways: the first part is anexposition of the basic theory, with details. The second part is acomprehensive survey of the main results. Lastly, the third partcollects various kinds of applications of the theory. Audience: This volume will be of interest to graduate studentsand research mathematicians whose work involves general topology,convex sets and related geometric topics, functional analysis, globalanalysis, analysis on manifolds, manifolds and cell complexes, andmathematical economics. ... Read more

Book Description

This volume is the second edition of the highly successful Fractals Everywhere.The Focus of this text is how fractal geometry can be used to model real objects in the physical world.

This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.

Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software.The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry

* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.
* Problems and tools emphasizing fractal applciations.
* An all-new answer key to problems in the text, with solutions and hints. ... Read more

Reviews (9)

Sometimes annoying but instructive
Although instructive, this book is sometimes annoying to read. The author seems to be playing his cards very close to the vest and not telling us everything.

For instance, there is little or no instruction on how to implement the IFS attractors presented as a panacea for data compression. This seems to be proprietary to his company. It also seems that hands-on manipulation is crucial to the images' production, contrary to the author's claims.

If you can understand the mathematics you may find the book useful, as I did when writing my book Fractals in MUsic.

A bad book for 7 th graders like me
this is a bad and very confusing book for a young student in, say... 7th grade, like me. The language is incomprehensible and there are no visual aids.

Opinión general
HUmmm!! parece interesante este librito. Pero la verdad busco uno donde encuentre aplicaciones a la ingeniería.
Estos libros de teoría suelen ponerse aburridos al no tener sufuciente información sobre aplicaciones.
De todos modos apenas lo tenga en las manos y lo mire doy una opinión más seria de este.

Good book for applications of fractal geometry, but....
This is a good book on applications of fractals; the chapters on modeling natural objects with iterated function systems (IFS) and fractal interpolation are especially useful. Many standard topics are included, for example, fractal dimension, Julia and Mandelbrot sets, chaos and the shift dynamical system. Some of the illustrations are captivating.

However, the book is not well organized, and the writing is extremely wordy to the point of being irritating. Some paragraphs read as if they belonged to a "Dummies" handbook. Also, I have to agree with one reviewer that the treatment of fractal dimension is poor. For one thing, it does not fully develop the intuition behind the concept-- much less the math. This same remark holds for the chapter on chaotic dynamics.

In summary, the book is fine for applications, but supplement your reading with a more substantial text.

Good computer graphics book. Bad mathematics book.
This book was written by a regarded expert in the fields of digital image processing and data compression, and illustrates well how some "abstract" mathematical concepts can be applied successfully to such purposes.

However, it is evident that it was written in a rush, and the results can be seen. I have found a lot of typographic mistakes, errors in the exercises, and even errors in some of its mathematical proofs. Also, the author pays almost no attention to the fundamental concept of fractal geometry: the fractal dimension.

I read this book because I needed a strong background in fractal geometry to write my Bachelor's thesis, but got dissapointed because of its mathematical defficiencies, and eventually decided to move to better sources on the subject.

Please check my other reviews in my member page (just click on my name above). ... Read more

Book Description

This textbook demonstrates the excitement and beauty of geometry. The approach is that of Klein in his Erlangen program: a geometry is a space together with a set of transformations of that space. The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case they carefully explain key results and discuss the relationship among geometries. This richly illustrated and clearly written text includes full solutions to over 200 problems and is suitable both for undergraduate courses on geometry and as a resource for self study. ... Read more

Reviews (3)

Good and enjoyable for a wide range of readers
A quarter century ago I noticed that some of the graduate physics students in my university were carrying around copies of Scientific American. Armed with that clue, I dug out every article on the newly discovered fundamental particles. Within the space of a week of fairly easy reading I was able to acquire a good sense of what this subject was all about. These articles explained the basic stuff our professors assumed we must know (but most of us surely didn't).

Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects.

Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas.

The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").

One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels.

The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed.

Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed.

This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on.

The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter.

The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance.

Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.

A lovely Introduction to all kinds of Plane Geometries
This book gives a beautiful overview of geometry of 2 dimensions. All of the book is about many plane geometries I have heard of, but didn't really know. This book changed that.

The first chapter treats some basics about conics. The second chapter is on affine geometry. The third and fourth chapters are about projective geometry. In the fifth chapter you will be led through Inversive geometry which functions as a base for the sixth and seventh chapter. The sixth chapter has as itst title Non-Euclidean geometry, but it is in fact the Hyperbolic geometry of Boljay in a formulation of Henry Poincaré. The seventh chapter is about Spherical Geometry. In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. It is clearly demonstrated that this is less trivial than you would expect.

I learned two things from this book. The first is, that you can, in principle, prove every theorem of geometry by just using Euclidean geometry. But if you do this, the amount of work it takes can be very huge indeed. It is a far better strategy to try to determine what geometry is best suited for the problem at hand, and solve it within that geometry.

Since the book gives a very clear picture not only of the particular geometries, but also to how the geometries relate to each other, you have, as an extra bonus, insight in the level of abstraction and the scope of your theorem.

The second thing I learned is how you can use geometry to make concepts as simple as 'triangle' precise. What I mean is this: a right angle triangle is not the same as an equilateral triangle. But both are the same in the sense that they are both triangles. The question is this: how can two 'things' be the same and at the same time not 'the same'? The book gives an answer to this 'question about the meaning of abstractions'. It gives the following solution. Take a triangle, ANY triangle. Consider the group of all affine transformations A (which consists of an uncountably infinite set of transformations.) If you subject this one triangle Tr to every affine transformation in this group A, you will have created a set consisting of exactly ALL triangles. In other words, the abstract idea of 'triangle' consists of ONE triangle Tr together with the set of ALL affine transformations. You can denote this as the pair (Tr, A). In the same way you can express the abstract idea of ellipse by the pair (El, A), and the abstract idea of parabola by the pair (Par, A). And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.

The book presupposes some group theory and some knowledge of linear algebra. Furthermore you have to know a little calculus. I have very little knowledge of group theory, and I have just about enough knowledge and skill about linear algebra to know the difference between an orthogonal and unitary matrix, and to know what eigenvectors are. I have studied the first 5 chapters of CALCULUS from Tom M. Apostol, which does not go too deep into linear algebra. This proved to be enough.

I have only one point of critique. Virtually all problems in the book are of the 'plug in type', even those at the end of every chapter (from which, by the way, you cannot find the solutions at the end of the book, while the solutions of those in the text can be found in an appendix). If you have understood the text, you have no difficulties whatsoever to solve them. The problems are not challenging enough to give you a real skill in all of these geometries, although they do become more challenging in later chapters. They are only intended to help you to understand the basic principles of all of these geometries, no more, no less. So if you want to have a tool to help you in obtaining a greater skill in, say, the special theory of relativity by studying hyperbolic geometry, this is not a suitable book. That is why I have given it 4 stars, and not the full 5 stars.

I also have a piece of advise. Although the problems are, from a conceptual point of view, not challenging, a mistake is easily made. Therefore it is best to solve the problems by making use of a mathematical program like Maple or Mathematica. If you then have made a mistake, you can backtrack exactly where you have made it, and let the program take care of all of the tedious calculations. This has also stimulated to try to calculate some outcomes by following a different approach, and then to compare the results.

I have enjoyed studying this book immensely.

Best on affine transformations used in computer graphics
I'm trying to understand transformations used in computer graphics, for example world transformation used in Windows GDI API. And I found this book to be the best description on the topic, that is affine transformations ... Read more

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Last of A Dying Breed
This is a classic textbook which introduces students to the fundamentals of analytic geometry. In order to understand calculus, one must have a thorough background in analytic geometry. Unfortunately, the current trend in math publishing is to forgo this great subject and present students with an motley mix of topics in a course called PreCalculus. Trigonometry and analytic geometry end up getting the short shrift, with greater emphasis being placed on algebraic manipulation. This is a tragic loss for students on a number of levels. They need analytic geometry in order to succeed in even first semester calculus, and yet for many analytic geometry is relegated to the second or third semester of a calculus class in college, or in one chapter (or less) in a precalculus book! Also, the analytic geometry allows the high school student a glimpse of mathematical beauty, of how things in mathematics work together: that mathematics holds up as a subject and is not just an assortment of tricks for various situations. Hopefully, the trend will turn around and this will not be the last of a dying breed: that others will continue to offer and write expository works on this subject for the benefit of students who wish to continue on to higher mathematics. ... Read more

Book Description

This book is a collection of 375 completely solved exercises on differentiable manifolds, Lie groups, fibre bundles, and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. It is the first book consisting of completely solved problems on differentiable manifolds, and therefore will be a complement to the books on theory. A 42-page formulary is included which will be useful as an aide-memoire, especially for teachers and researchers on these topics.Audience: The book will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics, and some branches of engineering. ... Read more

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This book rocks like Newton and Gauss....
Man, this is a very instructional manual that can help just about anyone having problems learning differential geometry. I give it two tangent vectors up. ... Read more

Book Description

From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role. The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for the reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996. ... Read more

Book Description

In a single volume, the bestselling Transforms and Applications Handbook brings together the most important mathematical transforms frequently used by engineers and scientists. Now in its second edition, all of the material has been updated by the original contributors, with valuable new material added.New in the Second Edition:·Chapters addressing Discrete Time and the Discrete Transforms, Fractional Fourier Transforms, and Lapped Transforms·Complete revisions of the chapters on Mellin, Wavelet, and Hartley Transforms·An expanded table of Laplace transformsEach chapter of the Handbook covers one transform and provides numerous examples and applications demonstrating the use of the transform and its properties. Beginning with a treatment of the delta function and some of the classical orthogonal functions, it details:oFourier TransformsoCosine and Sine TransformsoHartley TransformsoLaplace TransformsoZ-TransformsoHilbert TransformsoRadon and Abel TransformsoTime-Frequency TransformationsoWavelet TransformsoHankel TransformsoMellin Transforms ... Read more

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A best seller in mathematics !
This book contains great materials for engineering, research in mathematics. Every student or profesionnal can find all he needs. A top reference ! ... Read more

Book Description

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Reviews (3)

It's worth your money!
This book is not just for topologists! If you're like me, then you've spent countless nights sans Hatcher's book trying to figure out the fundamental group of a beer can. Look no further, the answers are here!

Be sure to check out the vivid detail Hatcher brings to the Van Kampen theorem. I've not actually read that part myself, as I do not trust german mathematics.

You would not regret if you buy this.
There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko & Novikov for their effort in unifying differential geometry and algebraic/differential topology.
Hatcher's book is intended as one of the series that cover every aspect of the subject. Separate books on vector bundles and K-theory, and spectral sequences respectively, are to appear sometime in the future. Thus this one covers ordinary homology/cohomology and homotopy theory only. His writing style is helpful and user-friendly, not demanding extensive "mathematical maturity".
I am not sure if this is "the" textbook on algebraic topology, but I bet this is among the best ones. You would not regret if you buy this, even when an electronic version is available online (for free) from the author's home page.

The Last Text on Introductory Algebraic Topology
No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's. ... Read more