European Mathematical Society - 42C40
https://euro-math-soc.eu/msc-full/42c40
enWavelet Analysis on the Sphere: Spheroidal Wavelets
https://euro-math-soc.eu/review/wavelet-analysis-sphere-spheroidal-wavelets
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Wavelets are a powerful tool in many fields, such as numerical analysis, signal and image processing or data analysis. As the title suggests, the main objective of this book is to present the construction of wavelets for functions defined on the sphere.</p>
<p>The book is organised in 6 chapters. The first one is an introduction to the topic and a guide throughout the next five chapters. In chapters 2,3, and 4, the authors study some mathematical tools useful in wavelet analysis. First of all, they present the notion and basic properties of orthogonal polynomials. They also introduce three different methods for their construction: Rodrigues formula, recurrence rules and orthogonal polynomials as solutions to ODEs. Then they apply these results to the construction of classical families of orthogonal polynomials such as Legendre, Laguerre, Hermite, Chebyshev and Gegenbauer. In the third chapter, they present the homogeneous polynomials and their interaction with harmonic analysis on the sphere. They begin with the study of the spherical harmonics in $\mathbb{S}^{2}$ as the solutions of the Laplace equation and then they develop the general theory. Chapter 4 is devoted to the study of special functions, such as beta, gamma, Bessel or Legendre functions among others. They detail their definitions and properties and provide some graphic illustrations as well. </p>
<p>The last two chapters focus on wavelets. First, the authors study wavelets related to orthogonal polynomials such as Chebyshev, Gegenbauer or Cauchy wavelets; next they concentrate on spherical wavelets. Finally in Chapter 6, they show some examples of wavelets' applications to numerical solutions of PDEs, integrodifferential equations, image and signal processing and time-series processing.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Blanca Fernández</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/sabrine-arfaoui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sabrine Arfaoui</a></li><li class="vocabulary-links field-item odd"><a href="/author/imen-rezgui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Imen Rezgui</a></li><li class="vocabulary-links field-item even"><a href="/author/anouar-ben-mabrouk" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Anouar Ben Mabrouk</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/walter-de-gruyter-berlin-boston" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Walter de Gruyter Berlin-Boston.</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-11-048109-9</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">144</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li></ul></span>Fri, 30 Nov 2018 10:27:20 +0000Blanca Fernández48886 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/wavelet-analysis-sphere-spheroidal-wavelets#commentsWavelets. A Student Guide
https://euro-math-soc.eu/review/wavelets-student-guide
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
There are many ways in which wavelets can be introduced, depending on the mathematical knowledge of the student's or readers' background: one may take a linear algebra approach, a signal processing approach, an approach starting from approximation theory, in particular splines, or Fourier analysis, or a general analysis approach, or even use a background from theoretical physics. And there is the digital algorithmic approach versus the continuous analytic vision. There are almost as many ways to introduce wavelets as there are books written with that purpose. The present one, is one I like very much if it is to be used to bring mathematics students at the level of their first or second year at the university into contact with wavelets. If I had to teach such a course, this would definitely be my first choice.</p>
<p>
Not only does it bring the subject in a most suitable and systematic way that, I am sure, mathematics students are used to and probably appreciate most. It is also following some good rules of didactics taking the students by the hand and bringing them to a higher level of understanding, ensuring that at least the bulk of the students does not declutch. A lot of effort is put into taking the rungs of the ladder at just the right pace, not boringly slow or not frighteningly fast, and always placing a chapter in the proper context: what has been achieved, and where do we want to go?</p>
<p>
Faithful to these principles, the first chapter is just a survey, summarizing the content of the whole book, even introducing the Haar wavelet, which being the simplest possible wavelet, does not require much analysis, but it does illustrate the idea. The rest of the first half of the text does not really deal with wavelets at all but introduces vector spaces, inner products, projections, etc., first in $\mathbb{R}^N$, but once this has been explained, all the concepts are shown to be not much different when it is generalized to sequences $\ell^2$ or square integrable functions that form $L^2$. Of course the latter are Hilbert spaces which requires some more advanced elements such as convergence, measure theory, integration, etc. It is not an in-depth analysis of all these topics, but just what is needed to move on is introduced. For example there are some considerations about a basis, density, and orthogonalization in an infinite dimensional space, but the concept of a frame is not introduced. Most of the proofs are included, some parts and some proofs are given as exercises, but again the most difficult ones are left out.</p>
<p>
The second half of the book then treats the wavelets. First the Haar wavelet is revised for which all the wavelet concepts are introduced such as a multiresolution analysis and all its properties, the scaling and the wavelet function, the scaling (or dilation) equation. The next step is to lift this to the more general situation of a general wavelet (assuming it has a finite support), the vanishing moments and the smoothness of the wavelet and the orthogonality properties and how all these properties can be formulated as conditions on the coefficients of the scaling equation. In the next chapter all this is made more concrete by deriving, drawing, and analysing the Daubechies wavelets for $N=2$ (for $N>2$ and other families the analysis is much shorter). In a last chapter, the Fourier-domain treatment of all this is discussed. Fourier analysis is again only introduced at a level just sufficient to do the computations, which avoids the deeper analysis requiring the massive body of Lebesgue integration and the subtleties of Fourier analysis.</p>
<p>
This survey illustrates the level of the approach and also the content is purely mathematical, avoiding algorithms, applications, linear algebra, etc. Each chapter is concluded with a long list of exercises (there are about 230 exercises in the whole text). They respect the level of the text and are not trivial nor exceptionally demanding. A remarkable feature of the book is the use of something like ▶ earmarking many sections typeset in a slightly smaller font. They give some extra information or warning, not really essential to follow the flow of the exposition. It is as if the authors whisper some extra information into the ear of the reader while he/she is studying the text. The authors give also several suggestions in the introduction on how a selection can be made from the text to cover a shorter course, and in an appendix they discuss pointers to the literature and they do this chapter by chapter and in particular also for the exercises. I think this is a book perfect for what it is intended to be and it is obviously prepared with great care for precision, level of complication, and it has very good didactical qualities. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is an excellent text to be used when the topic of wavelets is to be introduced to undergraduate mathematisc students. Half of the text forms an introduction to inner product vector spaces and Hilbert spaces. The second half is introducng multiresolution first for the Haar wavelet, then it is generalized and worked out for second order Daubechies wavelets. An elementary Fourier analysis approach is the subject of the last chapter. The booklet contains many exercises, all of a similar theoretical level. Algorithms, and applications are not considerd.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/peter-nickolas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Peter Nickolas</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107612518 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£39.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">274</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/be/academic/subjects/mathematics/abstract-analysis/wavelets-student-guide" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/abstract-analysis/wavelets-student-guide</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/41-approximations-and-expansions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">41 Approximations and expansions</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li></ul></span>Sat, 04 Mar 2017 12:48:37 +0000Adhemar Bultheel47500 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/wavelets-student-guide#commentsWavelet Transforms and Their Applications (2nd ed.)
https://euro-math-soc.eu/review/wavelet-transforms-and-their-applications-2nd-ed
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The original version of 2002 has been thoroughly revised and recent evolutions in the area of wavelet research have been added. The original concept of the book is maintained, i.e., it can be seen as a reference text or as a study book, complete with definitions, theorems, proofs and exercises. There is mathematical rigor, yet abstraction is only allowed when justified.</p>
<p>
The title may suggest that this book is only about wavelets and all the successful applications. However, the 'introduction' is very throughout and after some historical survey, you will find an extensive discussion of Fourier analysis (Fourier transform, Fourier series, DFT, FFT,...) and Hilbert spaces. Of course the latter are important to introduce (bi)orthogonal bases, frames, etc., which are important in wavelet analysis.<br />
Time-frequency analysis comes into the picture with Gabor (both continuous and discrete) and Zak transforms and the Wigner-Ville and the ambiguity distributions (again continuous as well as discrete transforms are analysed). So it is only on page 337 that the wavelets as such show up. The continuous (CWT) and discrete (DWT) wavelet transforms in chapter 6 and multiresolution analysis (MRA) in chapter 7 form the core wavelet chapters. The remaining chapters treat more advanced or recent topics like a p-MRA on the positive real halfline, nonuniform MRA and the Newland harmonic wavelets combining the short-time Fourier transform and the CWT.</p>
<p>
The applications are distributed throughout. The use of Fourier analysis in the solution of ordinary and partial differential equations and integral equations is classical and embedded in the Fourier analysis chapter. Examples of the theory are included in all chapters. The closing chapter is completely devoted to Fourier and wavelet analysis of turbulence. This subject is relatively new and it is more like a survey featuring properties of Navier-Stokes equations and difficulties that one meets when solving them with Fourier analysis. That involves topics like fractals, multifractals and singularities of turbulence. Subsequently solutions were proposed based on wavelet analysis like adaptive wavelets (Farge et al.) and wavelet transformed Navier-Stokes equations (Memeveau et al.)</p>
<p>
The exponential development and success of wavelet analysis is due to a simultaneous alertness and collaboration between mathematicians, engineers, and theoretical physicists. So there are several approaches to the topic. On avarage, the continuous transforms are the favorites of mathematicians and physicists, the engineering applications in (digital) signal and image analysis and the numerical solution of functional equations are favoring the discrete transforms. The former often emphasize the analysis aspect, the latter often find that they are better off with a computational (linear) algebraic approach. This book tries to keep a good balance between both, but I believe the authors have a reasonable bias towards the former camp of the CWT. Although the discrete transforms are all presented and given proper attention, the continuous transforms come up first to pave the way for the discrete versions. The applications are more the applications in a mathematician's vision: functional equations, analysis of turbulence (emphasis on analysis), while engineers would call subjects like for example image and signal processing and compression or the computational and numerical aspects of the equation solvers, as true applications. Those engineering kind of applications is not exactly what you should look for in this book. On the other hand, all the elements for practical (also engineering) applications are introduced. However as much as the text keeps far away from unnecessary abstraction like 'harmonic analysis on locally compact groups', it keeps away from computer programming too.</p>
<p>
The book is an up to date reference work on univariate Fourier and wavelet analysis including recent developments in multiresolution, wavelet analysis, and applications in turbulence. The systematic construction of the chapters with extensive lists of exercises make it also very suitable for teaching.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a revision of the original text of 2002. The main theme are wavelets, but it has an extensive introduction about Fourier analysis, Hilbert spaces, Gabor transforms, and Wigner-Ville distributions in time-frequency analysis. Worth mentioning are relatively new topics such as an MRA on the positive real line, Newland harmonic wavelets, and the final chapter which gives a survey about the use of wavelets to analyse turbulence.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/lokenath-debnath" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lokenath Debnath</a></li><li class="vocabulary-links field-item odd"><a href="/author/firdous-shah" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Firdous Shah</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8176-8417-4 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">84,79 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">568</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/birkhauser/engineering/book/978-0-8176-8417-4" title="Link to web page">http://www.springer.com/birkhauser/engineering/book/978-0-8176-8417-4</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/42axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42Axx</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65t60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65T60</a></li></ul></span>Wed, 04 Feb 2015 14:47:13 +0000Adhemar Bultheel46001 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/wavelet-transforms-and-their-applications-2nd-ed#commentsA Mathematical Odyssey. Journey from the Real to the Complex
https://euro-math-soc.eu/review/mathematical-odyssey-journey-real-complex
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
It may happen that mathematicians are confronted with the question what the heck they are doing all day except for computing pi since the Greeks had everything written down and there is nothing to be invented. One and one is two and that's all there is to it. With this book Krantz and Parks try to give a nontrivial answer to counter this opinion. Just like Homer's Odyssey describes the adventurous return of the hero Odysseus, presumed dead after the Trojan war, yet eventually slaying Penelope's suitors, this Krantz and Parks book reports on some astonishing successes of mathematicians in the past century, showing that mathematics is far from being dead. With these fourteen examples, the authors illustrate that mathematics is still very much alive and that it has great influence on our lives. Here is the list of the topics.</p>
<p>
<em>The four-color problem</em>. Since its original formulation in 1875 many well known mathematicians had tried to find a proof that 4 colors were sufficient to color any map so that no two neighboring countries would have the same color. It was with a computer-assisted proof by Appel and Haken in 1976 and after many corrections and an acrimonious debate about this being a mathematical proof or not that in 1981 the Math Department of the University of Illinois acclaimed that <em>Four Colors Suffice</em>. By now this kind of proof has been accepted and it was applied in other situations, but long after 1981 alternative proofs were obstinately looked for.</p>
<p>
<em>Mathematics of finance</em> is approached with a long history starting with the origin of interest, compound interest, before arriving at stock markets with the Dutch East India Company. Next, the reader is introduced to financial terms like derivative (a future contract), forward, option, arbitrage, call option, and most importantly how to determine the price of, for example, a call option. Several stochastic models existed already but the grand entrance was for the Black-Scholes equation for which Scholes and Merton got the Nobel prize in 1997 (Black died in 1995). However as the 1987 crash illustrated, models can and need to be improved, which is an ongoing topic of investigation for many mathematicians currently employed in the financial sector.</p>
<p>
<em>Ramsey theory</em> named after F. Ramsey (1903-1930) may be a bit less known. Ramsey was convinced that absolute chaos did not exist. A typical question to answer is "How many elements of some structure must there be to guarantee that a particular property will hold?" For example P. Erdős formulated the following variant "How large should <em>N</em> be so that in a room with <em>N</em> people, at least <em>k</em> people are mutually acquainted or at least <em>k</em> are mutually unacquainted?" Of course the idea is to find the smallest number <em>N</em>. The inconceivably large Graham's number (powers of powers of powers...) is an upper bound for an equivalent problem in graph theory: If <em>N</em> points in the plane are connected with either a red or a blue line (nodes are acquainted or not), how large should <em>N</em> be so that either <em>m</em> of them are red or <em>n</em> of them are blue? The Ramsey number <em>R</em>(<em>m</em>,<em>n</em>) is the smallest value of <em>N</em>, and is, except for some simple cases, only known to be in some interval.</p>
<p>
<em>Dynamical systems</em> are better known in a wider audience, via the popular Mandelbrot and Cantor sets, fractals, the Lorentz attractor, and the butterfly effect. Visual effects and chaotic effects that can astonish and fascinate the observer. A three dimensional Mandelbrot set, the Mandelbulb, is used an an illustration on the cover of the book.</p>
<p>
<em>The Plateau problem</em> of minimal surfaces spanning a closed curve was originally posed by Lagrange in 1760, but it became popular and was named after the Belgian J. Plateau (1801-1883) who observed that soap films were good approximations to such surfaces. Plateau made several observations on curvature and the angle between soap bubbles where they meet. A formula was derived by Enneper and Weierstrass and Costa, Hoffman and Meeks designed a minimal surface that did not intersect itself. A beautiful shape that has inspired several artists. One of the first Fields medals were awarded to J. Douglas for his new approach for the solution of the problem, although an obscure but ill understood paper by Garnier preceded his. Proofs of Plateau's observations were given by J.E. Taylor in 1976 and generalizations of the problem are still under investigation.</p>
<p>
<em>Non Euclidean geometry</em> arises when we leave the 5 Euclidean postulates. Exploring this possibility was a consequence of the efforts to show that the last postulate (only one parallel line is possible through a point outside a given line) was independent of the others. It stimulated J. Bolyai and N. Lobachevsky to come up with hyperbolic geometry where it is possible to draw at least two distinct parallel lines through a point. The consistency was only proved by Beltrami in 1868. It was B. Riemann who revolutionized the way we now think of geometry and Riemannian geometry is what was used by Einstein. Calabi's conjecture of the existence of some nice Riemannian metric on a complex manifold was proved by Yau (1977) which earned him a Fields Medal, and nowadays Calabi-Yau manifolds are an essential tool for theoretical physicists working on string theory.</p>
<p>
<em>Special relativity</em> is another topic that is not too difficult to explain. Einstein got a Nobel Prize in 1921 especially for his discovery of the photoelectric effect. A major campaign was set up to award this Prize also to H. Poincaré (who, among many other things, developed relativity theory parallel to Einstein) but because, as Mittag-Leffer states, the Nobel committee "fears mathematics because they don't have the slightest possibility of understanding anything about it", Poincaré never got it.</p>
<p>
<em>Wavelets</em> revolutionized Fourier analysis and they have a remarkable track in mathematics. Morlet, a geophysicist, came up with an alternative for the windowed Fourier transform, and with the help of Grossmann, a theoretical physicist, developed a theory of frames. Y. Meyer, a mathematician specialized in harmonic analysis, got hold of their papers and this was the start of a new approach to the domain. I. Daubechies later developed an orthogonal set of wavelets, which can only be described by an algorithm. The time between the mathematical formulation and the extensive applications in engineering (e.g. jpeg compression code) is remarkably short as compared to many other mathematical ideas. This is a relatively short chapter, discussing more the history of Fourier analysis while it is rather short on the wavelet stuff itself.</p>
<p>
<em>RSA encryption</em>, named after Rivest, Ahamir and Adleman, is another widely used application. Most probably your browser uses RSA whenever you open a https website, where personal or private information is exchanged. The RSA code is freely available and has never been compromised in the 30 years that it has been around. It is remarkable that it completely rests on an old and simple idea of prime numbers. In particular the difficulty to find large prime factors is the fact that makes it work.</p>
<p>
<em>The P/NP problem</em> follows immediately from the previous topic. P indicates the class of problems that can be solved in polynomial time, i.e., the execution time is a polynomial in the size n of the problem. NP however stands for nondeterministic polynomial, which means that it can be checked in polynomial time that a given solution is indeed solving the problem. It is generally believed that the class NP is strictly larger than P, but that has not been proved as yet. The chapter elaborates extensively on automata, Turing machines, and formal languages.</p>
<p>
<em>Primality testing</em> is again related to the two previous problems. It was not until the AKS algorithm of 2002 by Agarwal, Kayal, and Saxena who generalized some ideas of Fermat (his little theorem) to get a polynomial time algorithm for primality testing, which places this problem in the class P. In the foundations of mathematics it is explained that a mathematical proof traditionally consists of a sequence of statements, in principle derived from some axiomatic system following some rules of (formal) logic. As seen in the first chapter, nowadays the computer can play an essential and active role in the concept of a proof, by algebraic computation or verifying an extensive set of possibilities. The whole chapter is building up via an introduction to formal logic to Gödel's incompleteness theorem.</p>
<p>
Wiles' proof of <em>Fermat's last theorem</em> of 1995 is one of the last triumphs of mathematics in the 20th century. An introduction is given to polynomials over finite fields, elliptic curves, the Taniyama-Shimura-Weil conjecture, and Frey curves, to explain how Wiles, by proving Serre's form of the Frey conjecture actually proved Fermat's last theorem.</p>
<p>
<em>Ricci flow and Poincaré's conjecture</em> are connected in the proof by G. Perelman that he published in three papers in the years 2002 and 2003 on arXiv. The conjecture dates from 1904 and says that every 3-dimensional surface on which a closed curve can be continuously deformed to a point is homeomorphic to a 3-sphere. It was one of the <em>Clay Millennium Problems</em>. It is of great importance because it relates to relativity theory and the shape of our universe. Hamilton introduces differential equations generating Ricci flows that can be considered as geometric evolution equations that result eventually in the geometric objects predicted by Thurston in his geometrization program to describe a possible classification of all n-dimensional manifolds. The mathematical community did not accept Perelman's proof immediately. Certainly too long to his taste and, deceived in the mathematical establishment, Perelman retired from the Steklov Institute and from mathematics. He was awarded the Clay Millennium Prize and the Fields Medal which he both declined.</p>
<p>
Although there is some relation between some of the 14 subjects, each chapter can be read independently. The text is intended for the layman, but some knowledge and affinity with mathematical concepts is advised to help assimilate the material. Some average undergraduate mathematics from secondary school should suffice, and to really enjoy the texts, the reader should not have an aversion of mathematics. The chapters differ a bit in style and extent. Sometimes it starts with an extensive account of the very early history to introduce the topic, some are more mathematical than others, but they obviously serve the same purpose. The treatment is usually not very deep. It's just enough to enlighten the readers and give them an idea of what the topic is about and by whom and how the problem was eventually solved, and what impact it may have on society. It is also made clear that as mathematics advances, the problems can only be solved by applying interdisciplinary techniques. That obviously requires much more collaboration and hence ease of communication between mathematicians to finally crack the problem. Every chapter ends with a short list of references, but it is noteworthy that these are every time preceded by a section called "A Look Back". Why this is done is not very clear, but it seems to collect the crumbles of what has not been mentioned before. Most often that section contains some historical remarks or side stories about some of the main players in the theory or some future perspectives or generalizations. All in all, a book that is in the tradition a Krantz's previous books, which gives the non-mathematician an idea of what mathematicians get so passionately involved with and how that has resulted recently in successes.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
By describing 14 recent success stories of mathematics, the authors give the non-mathematician an idea of what mathematicians do for a living, what kind of problems they are looking at, and how their joint efforts eventually resulted in successes.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/steven-krantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven Krantz</a></li><li class="vocabulary-links field-item odd"><a href="/author/harold-r-parks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Harold R. Parks</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4614-8938-2 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">35,69 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">397</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/mathematics/history+of+mathematics/book/978-1-4614-8938-2" title="Link to web page">http://www.springer.com/mathematics/history+of+mathematics/book/978-1-4614-8938-2</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/68p25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P25</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/91g70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91G70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05D10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/37f45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37F45</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/53a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11y16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Y16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83c15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83C15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03f03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03F03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/53c44" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53C44</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/57r60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">57R60</a></li></ul></span>Wed, 13 Aug 2014 06:06:30 +0000Adhemar Bultheel45671 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematical-odyssey-journey-real-complex#comments