Book Description

Perturbation theory, one of the most intriguing and essential topics in mathematics, and its applications to the natural and engineering sciences is the main focus of this workbook. In a systematic introductory manner, this unique book deliniates boundary layer theory for ordinary and partial differential equations, multi-timescale phenomena for nonlinear oscillations, diffusion and nonlinear wave equations. The book provides analysis of simple examples in the context of the general theory, as well as a final discussion of the more advanced problems. Precise estimates and excursions into the theoretical background makes this workbook valuable to both the applied sciences and mathematics fields. As a bonus in its last chapter the book includes a collection of rare and useful pieces of literature, such as the summary of the Perturbation theory of Matrices.

Detailed illustrations, stimulating examples and exercises as well as a clear explanation of the underlying theory makes this workbook ideal for senior undergraduate and beginning graduate students in applied mathematics as well as science and engineering fields.

Book Description

Classic study discusses number sequence and number language, then explores written numerals and computations in a wide range of cultures. "The historian of mathematics will find much to interest him here both in the contents and viewpoint, while the casual reader is likely to be intrigued by the author’s superior narrative ability."—Library Journal. 282 illustrations. 1969 edition.
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Reviews (2)

A bit stilted and dense, but fascinating nevertheless
Number Words and Number Symbols is a translation of a work originally in German, and it shows. While the discussions of the origins of various cultures and languages' symbols and words for numbers is almost encylopedic in its completion, it can be very stilted reading at times.

If you don't mind sometimes long and verbose reading, then this is certainly one of the best works on the subject out there. I would quibble that Menninger gives the East less coverage than the West, but his comparisons of number words across Indo-European languages is simply fascinating. Also, too are his theories about how the displacement of one language by another in a region can often lead to weird anomalies in number counting.

Why we count the way we do
Simply, the intellectual archaeology of our numerical universe ... Read more

Book Description

The target audience of this book is students and researchers in computational sciences who need to develop computer codes for solving partial differential equations. The exposition is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods, and basic finite difference methods from a computational point of view. The main emphasis regards development of flexible computer programs, using the numerical library Diffpack. The application of Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. Diffpack is a modern software development environment based on C++ and object-oriented programming. The second edition contains several new applications and projects, improved explanations, correction of errors, and is up to date with Diffpack version 4.0. ... Read more

Book Description

Numerical Optimization has numerous applications in engineering sciences, operations research, economics, finance, etc. Starting with illustrations of this ubiquitous character, this book is essentially devoted to numerical algorithms for optimization, which are exposed in a tutorial way. It covers fundamental algorithms as well as more specialized and advanced topics for unconstrained and constrained problems. The theoretical bases of the subject, such as optimality conditions, Lagrange multipliers or duality, although recalled, are assumed known. Most of the algorithms described in the book are explained in a detailed manner, allowing straightforward implementation. This level of detail is intended to familiarize the reader with some of the crucial questions of numerical optimization: how algorithms operate, why they converge, difficulties that may be encountered and their possible remedies. Theoretical aspects of the approaches chosen are also addressed with care, often using minimal assumptions. ... Read more

Book Description

This book is an introduction to level set methods and dynamic implicit surfaces. These are powerful techniques for analyzing and computing moving fronts in a variety of different settings. While it gives many examples of the utility of the methods to a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems. The book begins with a description of implicit surfaces and their basic properties, then devises the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus. Topics include the level set equation itself, Hamilton-Jacobi equations, motion of a surface normal to itself, re-initialization to a signed distance function, extrapolation in the normal direction, the particle level set method and the motion of co-dimension two (and higher) objects. Part III is concerned with topics taken from the fields of Image Processing and Computer Vision. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from unorganized data points. Part IV is dedicated to Computational Physics. It begins with one phase compressible fluid dynamics, then two-phase compressible flow involving possibly different equations of state, detonation and deflagration waves, and solid/fluid structure interaction. Next it discusses incompressible fluid dynamics, including a computer graphics simulation of smoke, free surface flows, including a computer graphics simulation of water, and fully two-phase incompressible flow. Additional related topics include incompressible flames with applications to computer graphics and coupling a compressible and incompressible fluid. Finally, heat flow and Stefan problems are discussed. A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change its topology or develop singularities, will find this book interesting and useful. ... Read more

Reviews (4)

Good Level Set Book For Animation
I bought this book to better understand the level set methods Fedkiw et al use in their computer animation work in fluids. Their papers often gloss over what level set methods are and how to use them (likely due to page limits in conference proceedings). This book does a very good job of getting you up to speed.

A book for beginners and experts alike
The book introduces the levelset method, starting from the very basics: the idea of implicit functions, discretization, schemes for numerically solving PDEs. It also covers advanced issues like reinitialization, particle methods, levelset for modelling objects of codimension two. Finally, an extensive section on applications from computer vision and computational physics gives the reader an idea of the power and elegance of levelset methods.
The presentation is very readable and precise enough to allow implementing the algorithms described straight away on the computer.
This is a book for beginners and experts, written by a well-known expert in the field (Osher and Seithan wrote one of the most influential papers on levelsets).

An excellent book for beginners and experts alike
The book introduces the levelset method, starting from the very basics: the idea of implicit functions, discretization, schemes for numerically solving PDEs. It also covers advanced issues like reinitialization, particle methods, levelset for modelling objects of codimension two. Finally, an extensive section on applications from computer vision and computational physics gives the reader an idea of the power and elegance of levelset methods.
The presentation is very readable and precise enough to allow implementing the algorithms described straight away on the computer.
This is a book for beginners and experts, written by a well-known expert in the field (Osher and Seithan wrote one of the most influential papers on levelsets).

great book
Exciting material, cool pictures. I love this book! ... Read more

Book Description

Reviews (1)

good for the experienced
hi all,
this book is not appropriate for the beginners to CFD so i don't recommend it for the instructors who are looking for a introduction book. The tensor notation is also not easy to grasp for the not experienced. Besides the language of the book is not clear and not enough to let the reader apply without referring to another book. This is probably because the book has covered a lot of CFD concepts so deep explanation on every item is not provided.
However I would recommend this book for experienced CFD'ers since it covers many concepts and it can be used as a good referring material. ... Read more

Book Description

Multi-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest. ... Read more

Book Description

This book is concerned with the numerical implementation of Finite Element Analysis using the computer program MATLAB, which is very popular today in engineering and engineering education. The book contains a short tutorial on MATLAB as well as a systematic strategy for the treatment of finite element method. The book is directed towards both students and researchers in engineering. Various examples and exercises are provided out of Mechanical Engineering, Civil Engineering, Aerospace Engineering or Materials Science. The book especially stresses the interactive use of MATLAB. Each example is solved in an interactive manner. The accompanying CD-ROM includes at least 50 MATLAB functions specifically written to be used with this book in the form of a MATLAB Finite Element Toolbox. An extensive solutions manual is provided as well, which includes detailed solutions to all the problems in the book for classroom use. ... Read more

Book Description

This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Goedel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the author's web site. The special feature of this book is that it presents a new "hands on" didatic approach using LISP and Mathematica software. The reader will be able to derive an understanding of the close relationship between mathematics and physics. "The Limits of Mathematics is a very personal and idiosyncratic account of Greg Chaitin's entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Greg's ideas on real computers for those who want to try their hand at furthering the theory." ... Read more

Reviews (3)

This book is a piece of junk. Do not buy it.
Years ago I read a fascinating article in Scientific American
"Randomness and Mathematical Proof" by G Chaitin. (Of course
that was back in the good old days before SciAm switched to the
kindergarten market). Anyway, the article was fascinating and
well written. It was followed up by Martin Gardiner in his
"Mathematical Recreations" column, I think. This left me
with a desire to learn more. On the basis of the article and
the excellent reputation of Springer. I bought this book 2
years ago sight unseen.

What a big mistake. I feel cheated every time I see it on my
book shelf! The worst thing is that I simply cannot believe
that a reputable publisher like Springer actually published
this piece of junk.

I also cannot believe the same person wrote the SciAm article
and this book. Chaitin comes across in his book completely
self-obsessed and full of his own importance: comparing himself
(favourably) to K Godel and Einstein.

The book claims to be "a course in information theory". False.
It is not a course in anything and is written by an illiterate.
It has no value as a text book - indeed its value is negative
becasue it will turn readers away.

The subject of algorithmic information theory is worth learning
about. But please do look for another book to learn from - any
book except this one.

Belongs up there with Godel and Jantsch...
Chaitin has spent his life working on translating Godel's work (and Turing's too) into a version that is much more generally applicable: the limits of mathematical reasoning. I'm not sure why this book is not more generally known (probably because we don't like to have our balloon of arrogant rationality popped so easily) and read but it should be required for anyone contemplating further education in science.

Over the years the message from mathematics, physics and cybernetics (and AI for that matter) has simply been one of showing all formal forms of Aristotle's logic cannot ever capture the TRUTH.

Chaitin's easy delivery (this book is based on taped lectures) makes the book a gentle read even for those without the math background to fully understand the few equations. I highly reccommed buying a copy and reading it.

Beautiful attempt to understand the nature of mathematics
Greg Chaitin has written a beautiful and entertaining book about his work to understand the nature of mathematics. He describes his results in a highly informal and entertaining way which is still rigorous. Clearly he sincerly wants to _know_ what mathematics is. I wish that all math books were written this way.

Chaitin shows us the fairly simple computer programs he has written to demonstrate theorems about the limits of the axiomatic method. He shows results which are similar to Go"del's and Turing's results and in fact imply them. He has a simple and striking method of arriving at his results. It is all done with both brains and heart.

Chaitin defines a number which represents the probablility that a computer program halts. He shows how this number cannot be computed with a computer program which contains fewer bits than the number itself. Moreover, no set of mathematical axioms can compute this number with more precision than there are information bits in the axioms.

Since no set of axioms can enable us to fully compute the halting probability, and since axioms enable us to write proofs, and since proofs give us the reason why a mathematical statement is true, then there are some mathematical truths which are true for no reason, i.e. they are random.

But I cannot agree with this conclusion. Chaitin, and many others who say similar things, assume that a proof is a reason. But a proof is only a chain of implication. Our faith in the statement, in the light of the proof, rests on our faith in the axioms and logic.

Imagine a culture somewhere that has a mathematics like ours, except they don't have the distibutive property. They could prove some things that we know, but not others. They play around with their computers and discover some facts which they could easily show if they only had the distributive property, but they can't prove them.

Would it be right for these people to claim that these facts are random, that they are true for no reason? I don't think so. Instead, I think that these people should say, and we should too, that there are some things which are true, they are not random, but they are beyond our ability to prove. They are true for a reason, but we just don't see it yet. ... Read more

Book Description

Applied Mathematics: Body & Soul is a mathematics education reform project developed at Chalmers University of Technology and includes a series of volumes and software. The program is motivated by the computer revolution opening new possibilitites of computational mathematical modeling in mathematics, science and engineering. It consists of a synthesis of Mathematical Analysis (Soul), Numerical Computation (Body) and Application. Volumes I-III present a modern version of Calculus and Linear Algebra, including constructive/numerical techniques and applications intended for undergraduate programs in engineering and science. Further volumes present topics such as Dynamical Systems, Fluid Dynamics, Solid Mechanics and Electro-Magnetics on an advanced undergraduate/graduate level.Volume I (Derivatives and Geometry in R3) presents basics of Calculus starting with the construction of the natural, rational, real and complex numbers, and proceeding to analytic geometry in two and three space dimensions, Lipschitz continuous functions and derivatives, together with a variety of applications. Volume II (Integrals and Geomtery in Rn) develops the Riemann integral as the solution to the problem of determining a function given its derivative, and proceeds to generalizations in the form of initial value problems for general systems of ordinary differential equations, including a variety of applications. Linear algebra including numerics is also presented. Volume III (Calculus in Several Dimensions) presents Calculus in several variables including partial derivatives, multi-dimensional integrals, partial differential equations and finite element methods, together with a variety of applications modeled as systems of partial differential equations. The authors are leading researchers in Computational Mathematics who have written various successful books. Further information on Applied Mathematics: Body and Soul can be found at http://www.phi.chalmers.se/bodysoul/. ... Read more

Reviews (1)

Calculus + Linear Algebra in One Series
I wish Amazon would put a text list of the "Table of Contents" for books.

Sorry about the length: Amazon does not allow postings for each volume in a series!

Volume 1: Derivatives and Geometry in R^3
1. What is Mathematics?
2. The Mathematics Library
3. Introduction to Modeling
4. A Very Short Calculus Course
5. Natural Numbers and Integers
6. Mathematical Induction
7. Rational Numbers
8. Pythagoras and Euclid
9. What is a Function?
10. Polynomial Functions
11. Combinations of Functions
12. Lipschitz Continuity
13. Sequences and Limits
14. The Square Root of Two
15. Real Numbers
16. The Bisection Algorithm for f(x) = 0
17. Do Mathematicians Quarrel?
18. The Function y = x^r
19. Fixed Points and Contraction Mappings
20. Analytic Geometry in R^2
21. Analytic Geometry in R^3
22. Complex Numbers
23. The Derivative
24. Differentiation Rules
25. Newton's Method
26. Galileo, Newton, Hooke, Malthus and Fourier

Volume 2: Integrals and Geometry in R^n
27: The Integral
28: Properties of the Integral
29. The Logarithm log(x)
30. Numerical Quadrature
31. The Exponential Function exp(x) = e^x
32. Trigonometric Functions
33. The Functions exp(z), log(z), sin(z) and cos(z) for z e C
34. Techniques of Integration
35. Solving Differential Equations Using the Exponential
36. Improper Integrals
37. Series
38. Scalar Autonomous Initial Value Problems
39. Separable Scalar Initial Value Problems
40. The General Initial Value Problem
41. Calculus Tool Bag I
42. Analytic Geometry in R^n
43. The Spectral Theorem
44. Solving Linear Algebraic Systems
45. Linear Algebra Tool Bag
46. The Matrix Exponential exp(xA)
47. Lagrange and the Principle of Least Action
48. N-Body Systems
49. The Crash Model
50. Electrical Circuits
51. String Theory
52. Piecewise Linear Approximation
53. FEM for Two-Point Boundary Value Problems

Volume 3: Calculus in Several Dimensions
54. Vector-Valued Functions of Several Real Variables
55. Level Curves/Surfaces and the Gradient
56. Linearization and Stability of Initial Value Problems
57. Adaptive Solvers for IVPs
58. Lorenz and the Essence of Chaos
59. The Solar System
60. Optimization
61. The Divergence, Rotation and Laplacian
62. Meteorology and Coriolis Forces
63. Curve Integrals
64. Double Integrals
65. Surface Integrals
66. Multiple Integrals
67. Gauss' Theorem and Green's Formula in R^2
68. Gauss' Theorem and Green's Formula in R^3
69. Stoke's Theorem
70. Potential Fields
71. Center of Mass and Archimedes' Principle
72. Newton's Nightmare
73. Laplacian Models
74. Chemical Reactions
75. Calculus Tool Bag II
76. Piecewise Linear Polynomials in R^2 and R^3
77. FEM for Boundary Value Problems in R^2 and R^3
78. Inverse Problems
79. Optimal Control
80. Differential Equations Tool Bag
81. Applications Tool Bag
82. Analytic Functions
83. Fourier Series
84. Fourier Transforms
85. Analytic Functions Tool Bag
86. Fourier Analysis Tool Bag
87. Incompressible Navier-Stokes: Quick and Easy

This book/series has some problems; however, the attempt to wed Mathematical Analysis, Numerical Computation, and Application AND to also talk about Calculus and Linear Algebra at the same time -- makes these three books worth a look.
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Book Description

Remarkable progress has recently been made in the development and application of quantum trajectories as the computational tool for solving the time dependent Schrodinger equation. Analogous methods for stationary bound states are also being developed. The purpose of this book is to present recent developments and applications of quantum trajectory methods in the broader context of the hydrodynamical formulation of quantum dynamics. While many chapters of the book deal with Lagrangian quantum trajectories in which the velocity matches that of the probability fluid, other chapters deal with what will be termed post-Lagrangian trajectories. There are also many state-of-the-art topics covered that are unique to this book. On the pedagogical side, a number of sections will be accessible to students who have had at least one course in quantum dynamics. There is also considerable material for advanced researchers, and chapters in the book cover both methodology and applications. ... Read more