European Mathematical Society - springer verlag
https://euro-math-soc.eu/publisher/springer-verlag-0
enPi: The Next Generation
https://euro-math-soc.eu/review/pi-next-generation
<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>
In <em>Pi. A source book</em> the editors L. Berggren, J. Borwein and P. Borwein, assembled a number of reprints that sketch the history of pi, its mathematical importance and the broad interest that it has received through the centuries from the Rhind papyrus till modern times. The last edition (3rd edition, 2004, to which I will refer as SB3) added several papers that related to the computation of the digits of pi by digital computers. Rather than extending this with more recent developments (SB3 was already some 800 pages), it was decided to collect this computational aspect in a new volume. This "<em>The next generation</em>" volume got the rightful subtitle "<em>a source book on the recent history of pi and its computation</em>". Because it extends the papers on digital computation that were added in SB3, the trailing papers of SB3 are reprised here. The papers are ordered chronologically, so of the first 14 papers in this book, 12 were already at the end of SB3.</p>
<p>
It starts with the agm (algebraic-geometric mean) iteration attributed to Salamin and Brent who both published their papers in 1976. It generates two sequences of numbers by iteratively extending the sequences respectively with the algebraic and the geometric mean of the previous numbers. Given appropriate initial conditions, both sequences converge to a common limit related to pi. This method is widely used since these publications of 1976, but the agm idea was actually used already by Gauss and others although not in connection with computing pi. The Borwein brothers discuss a quartically convergent method based on it (1984) and Bailey and Kanada used it to compute millions of decimals of pi (1988). The number of digits computed today has exceeded these computer experiments by many orders of magnitudes and several papers in this book survey the history, and the diversity of formulas and methods and the successive records reached.</p>
<p>
There are, besides the classical methods to compute pi, also several computational methods to generate the expansion of pi. For example, a completely different spigot algorithm computes the decimals of pi one by one but using only integer arithmetic (originally from 1995 and extended in 2006). In a more classical vein is the BBP algorithm (named after the authors Bailey, Borwein and Plouffe) which allows to compute a set of binary (or hexadecimal) digits of pi without the need to compute all the previous ones (1997). This is of course a great help when computing trillions of digits. Of course there are a a number of papers devoted to Ramanujan's notebooks with formulas to compute pi.</p>
<p>
There are also some papers on the proof of irrationality of pi, and of related numbers such as its roots, ζ(2), ζ(3); (i.e. Apéry's constant), Catalan's constant etc. The investigation of the properties of the digits of pi, in particular the normality of pi (still unproved) is discussed and computationally tested. The tests can be nicely visualised using random walks and color coding. Normality means that every possible sequence of <em>m</em> successive digits is equally probable for any basis and for any <em>m</em>.</p>
<p>
The papers are reprinted in their original format, thus with different fonts, lay-out, etc. It happens that the end of a previous chapter or article is still on the first page of the reprint or the start of the next one is on the last page. Even some totally unrelated announcement that appeared at the end of the original journal paper, it is reprinted here unaltered. Just as one would in a pre-digital age collect photocopies of the papers. Nevertheless, the book has an overall name and subject index, which is not obvious in this case. Since the papers come from many different journals (and even some chapters of a book) not all of these papers may be readily available or even known to an interested researcher, or in this case, it may even be a lay person who is interested. Many of the papers have authors that are the main players in the field: David Bailey, Bruce Berndt, and Jonathan and Peter Borwein. As this book was being printed one of its editors, Jonathan (Jon) Borwein, passed away on 2 August 2016. So it was probably too late to add a dedication or a note in this book. This collection he helped to compile and containing several papers that he coauthored, can be considered one of his last gift to the scientific community. </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 collection of reprints of 25 papers discussing pi. Mostly about the computations of its digits and checking the normality. They are ordered chronologically from 1976 to 2015. This is an alternative for yet a fourth edition of <em>Pi. A Source Book</em> by Berggren and the Borwein brothers, the third edition of which appeared in 2004. The computational papers of that 3rd edition are reprised as the initial papers of this volume.</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/david-h-bailey" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David H. Bailey</a></li><li class="vocabulary-links field-item odd"><a href="/author/jonathan-m-borwein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan M. Borwein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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">2016</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">9783319323756 (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">74.19 € (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">521</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li><li class="vocabulary-links field-item even"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</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/gp/book/9783319323756" title="Link to web page">http://www.springer.com/gp/book/9783319323756</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/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</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/11-04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-04</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/11y16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Y16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68q25</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11k16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11K16</a></li></ul></span>Tue, 23 Aug 2016 07:39:47 +0000Adhemar Bultheel47119 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/pi-next-generation#commentsA Primer on Scientific Programming with Python (5th ed.)
https://euro-math-soc.eu/review/primer-scientific-programming-python-5th-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>
While teaching, computers have become a very useful tool. For mathematics in particular, the analysis is often used and illustrated by actually computing something, an equation that can be solved analytically or numerically, integrals and derivatives can be evaluated, models for physical, chemical, or biological systems are used for simulations, etc. Popular languages for symbolic computing are maple and mathematica, and when it it becomes numerical, certainly matlab, and for more advanced problems one may use fortran, C, or any of its variants and there are many other languages around that are used for a first course in computer programming, although to all of them are equally suitable for mathematical applications. There is a fuzzy boundary between the use of such a program just for the illustration of a mathematical definition or algorithm on one side and on the other side learning to program, i.e., where the computer science aspect rather than the mathematics is the main goal. The latter does not need mathematics and can be illustrated with typical computer science algorithms like sorting, or string manipulation, or searching in a list, etc. In a mathematics educational context, it is clearly most efficient to combine both and illustrate the programming aspects with familiar mathematical examples and at the same time illustrate how programming can push the mathematical simulation beyond the limit of computation with pen and paper. That combination is the choice made in this book. And of course in the background there is the pure technical aspect of learning the semantics and the syntax of the language too, which in this case is Python.</p>
<p>
Python is certainly one of these languages which, besides of allowing to program the solution of mathematical problems, is also fit for use as a first introduction to computer science aspects. It is particularly popular because it is freely available, it can be used interactively (style matlab, maple and the likes) as a scripting language and it has all the advanced features that one would expect from a modern computer language. An extra advantage is that it can easily be linked with software written in other languages.</p>
<p>
The latter is in important aspect because computer programs to perform standard tasks or solve numerical problems are around since the middle of the 20th century and many powerful packages are available. Part of the success of a (new) computer language will depend on the fact that it runs for free on many different platforms, that the learning curve is not too steep, and that existing software can still be used. Python obviously is a language satisfying these criterions and moreover it has native constructs like objects and classes, so that also from a computer science viewpoint, it is interesting to introduce the student to something more than just while or for loops and if-then-else constructions.</p>
<p>
This is already the fifth edition of the book, so it has been polished for a number of years. Although Python 3 has been around for a while, Lantangen has chosen to still keep his 2.7 version, basically to keep the large amount of software what is already available for version 2. One has to check every line of the existing software to migrate to version 3 because some features behave differently in Python 2 and Python 3. There is however an automatic tool to convert from Python 2.7 to Python 3.5. Hence the choice that Lantangen made here is not a strong restriction, but I am sure the migration will be made in a future edition.</p>
<p>
It is a textbook with many examples and many exercises, printed on 945 glossy pages. It requires a table or a desk to read because 2.2 kg is not easily manageable in your hands or on your lap. It is not intended to be a first course in programming, nor is it a first course in numerical mathematics. The reader should at least be familiar with the basics of programming. As Lantangen states in the introduction, it is the main intention to learn the student to <em>think</em> as a programmer should so that he or she can produce programs in a quicker and more reliable way. What Lantangen does not mention, perhaps because it is obvious, is that most effort will go into learning the syntax and the behavior of the computer language Python, or Python 2.7 to be more precise, but remarks usually indicate when there is a difference with Python 3.</p>
<p>
When this is used in the context of a course, all the necessary software will have been installed, but a technical appendix explains how to do it on your own machine if necessary, whether it is Linux, Mac, or Windows. Packages like IPython (for interactive use), NumPy (numerics), MatplotLib (matlab-style plotting), sciPython (scientific computing), and possibly Cython (C-expressions) are also used and should be installed too (although Cython is only used in an appendix). The software is absolutely necessary because you can only learn a language by using it. Since there is a lot of software in the examples of this book it is only reasonable that also this code can be downloaded. It is available from the author's github site <a href="http://hplgit.github.io/scipro-primer/">http://hplgit.github.io/scipro-primer/</a>.</p>
<p>
The contents of the book follows the usual steps from elementary operations and formulas, to loops, branching, and input-output operations, meanwhile introducing the different data types (scalars, lists, objects, functions, arrays, dictionaries, strings). Plotting is not so elementary, but some knowledge of plotting with matlab will make it easy by using the MatplotLib package. Classes and working with them (object oriented programming) are less elementary but have now become standard in computer science so this is introduced in the last chapter.</p>
<p>
All the concepts are illustrated using relatively simple examples that are mostly mathematical. An exception to that is the chapter on random numbers, stochastic processes and games. It precedes the last chapter on object oriented programming but does not introduce new programming ideas. It is more applied, rounding up and illustrating all that has been introduced so far. It consists mainly of a collection of less elementary examples. This "rounding up" idea is a feature also used at the end of each chapter, summarizing all the new concepts of the chapter. Following the last chapter is and extensive list of appendices where the mathematical concepts are more important than the software concepts, although still elementary in view of the target readership: Newton iteration, differential equations, even a whole project of a mass-spring oscillator. This project includes everything from modelling the physics till plotting of the result. Another appendix illustrates the use of classes to design an extensive ODE package. There is another appendix on debugging and one on exporting Python code to Cython.</p>
<p>
This book gives a thorough course to learn Python, and yet it is all brought at the level of a first year at the university. The fact that each concept is introduced with an example is essential. It is not a description of the language, it is a description of how the language is used, which is a very natural approach. Of course how well and how quick the student will be mastering Python will mainly depend on how much experimenting she will do. And that is where the many exercises will help to push the student to just do that and learn to avoid the pitfalls of the semantic and syntactic idiosyncrasies of a new programming language.</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 fairly complete treatment the Python programming language. It discusses the different constructs by example, which are typically (numerical) mathematical examples that would perfectly fit into a first years calculus course.</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/hans-petter-langtangen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Hans Petter Langtangen</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">2016</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">978-3-662-49886-6 (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">74.19 € (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">945</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/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</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/gp/book/9783662498866" title="Link to web page">http://www.springer.com/gp/book/9783662498866</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/68-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68 Computer science</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/68n01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68N01</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/67n80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">67N80</a></li></ul></span>Mon, 08 Aug 2016 12:18:02 +0000Adhemar Bultheel47101 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/primer-scientific-programming-python-5th-ed#commentsOpen Problems in Mathematics
https://euro-math-soc.eu/review/open-problems-mathematics
<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 idea to compile this book started in 2014 from an originally casual conversation between the editors about the status of some open problems in mathematics. John Nash was at that time already 86, but he insisted to be actively involved in the selection of the subjects and the preparation of the volume. When the volume was almost ready in 2015, they wrote the preface, and shortly afterwards Nash left for Oslo where he was to receive the Abel Prize. Unfortunately he did not see this book published because he and his wife died in a taxicab accident in New Jersey on the way back home from Norway. Besides a short introduction by Misha Gromov about Nash's scientific ideas, several of the other papers refer to interests and work of Nash and the whole volume is dedicated to him.</p>
<p>
The objective of the book was clear already in 2014 when the idea took off. Since Hilbert formulated his 24 problems in 1900 some of them were solved, some were reformulated in 2000 as one of the Clay Mathematical Institute's Millennium Prize Problems (MPP). Among the most notorious problems that feature on both lists we recognize the Riemann hypothesis. Another holy grail is the P vs NP problem that is more recent and is one of the toughest MPP. All other MPPs are discussed here too except the <em>Poincaré Conjecture</em> (solved by Perelman in 2010) and the <em>Yang-Mills Problem</em> about the mass gap in elementary particle physics. Of course these famous lists do not cover all the open problems since many different lists have been distributed and open problem need not be in any list to be an open problem of course. The editors of this book made a selection of problems they consider important and they have invited experts to write survey papers that give the state of the art of the problem or of the conglomerate of related problems, the historical attempts made to solve them, the methods currently applied, etc.</p>
<p>
The book immediately opens with a strong and extensive survey of 123 pages on the notorious <em>P/NP Problem</em>. The problem was explicitly formulated by Cook and Levin in the early 1970's although its roots are usually assigned to Gödel in 1956. Here we learn that Nash already in 1950 gave a formulation. It has a meta-character since if one could prove that P = NP then it is in principle possible to write a program that formally solves all of the other Millennium Prize Problems too. The paper gives arguments in favor of P = NP and others in favor of P ≠ NP but the general belief is currently that P ≠ NP. Although the Turing machine is an essential element in the precise definition of the classes P and NP, it is not explained (think of any existing programmable digital computer), but the long list of all the different complexity classes is introduced (a glossary is given in an appendix) and theorems are formulated (no proofs) stating what inclusions hold for the respective classes. The larger part of the paper introduces many approaches and concepts of complexity theory like lower bounds, different barriers, oracles, etc. It only shows that so far nobody seems to have a clue on how to tackle the general problem. Some think the problem is just too difficult, but the author, Misha Gromov, is not so pessimistic. Even if the solution is still far away, high up there at the top of the mountain, so much insight is found already in the low vegetation that is being explored just now, that it justifies all the effort invested.</p>
<p>
The other survey papers are somewhat shorter (on average some 20-30 pages) which doesn't mean they are less interesting. Some problems are well known too. The <em>Riemann Hypothesis</em> is one of them and several popularizing books on the subject are available. Alain Connes chose to be somewhat restrictive in his discussion and he describes three approaches to explicit formulas based on geometry (Riemannian, algebraic, and tropical geometry).<br />
The Riemann Hypothesis is related to the distribution of prime numbers and involves zeros of the zeta function. The Generalized Riemann Hypothesis is about the spacing between prime numbers and requires the generalization of the zeta function to Dirichlet L-functions. In the corresponding survey paper this is linked with <em>energy levels of quantum systems</em>. This exposes a very surprising underlying universality that is still unexplained. The paper shows how random matrices is a valuable approach to tackle this problem. With its quantum physics component this paper comes in the neighborhood of the Yang-Mills Theory.<br />
The <em>Birch-Swinnerton-Dyer Conjecture</em> (BSD) is another famous MPP. It asks about rational points on an elliptic curve and the relation with yet another kind of L-functions related to the zeta and the Dirichlet L-functions. The formulation of the problem and some recent results are given.<br />
The <em>Generalized Fermat Equation</em> is again a number theory problem. It is a generalization of the famous Fermat's Last Theorem, solved by Wiles in 1993. Mathematicians have the urgent need to formulate and work on a generalization for every problem they solved. This problem asks for relative prime values $x$, $y$, $z$ and integers $p$, $q$, $r$ satisfying $x^p+y^q=z^r$. This survey concentrates on approaches for the case $1/p+1/q+1/r<1$.<br />
The <em>Goldbach Conjecture</em> is another old number theory problem (every even integer larger than 2 is the sum of two primes). Different approaches used in the long history are briefly reviewed as well as closely related problems and generalizations.</p>
<p>
More algebraic is the number theory problem on <em>discrete logarithms</em>. Note that $\mathbb{Z}/p\mathbb{Z}$, with $p$ prime and excluding zero fprms a multiplicative group. Hence for every $x$, there is some integer $a$ such that $x=g^a$ with $g$ a primitive root modulo $p$. This $a$ is then called the discrete $g$-logarithm of $x$. The problem is important for cryptography since it is difficult to compute this logarithm when $p$ is very large. Some algorithms to compute a discrete logarithm are discussed that link to elliptic curves and finite fields. However, so far no polynomial algorithm is known to compute a general discrete logarithm.</p>
<p>
The remaining MPPs that are discussed include the <em>Navier-Stokes Problem</em> and the <em>Hodge Conjecture</em>. Navier-Stokes is about the existence and smoothness of the solutions of this equation of fluid dynamics. The problem is briefly recalled and linked to related problems. The Hodge Conjecture is a problem of algebraic topology. Although the paper is introductory, it requires some knowledge of topology for better understanding.<br />
The <em>Novikov Conjecture</em> is another topology related problem. It says that higher signatures for smooth manifolds are homotopy invariant. The conjecture has intimate links with geometry, operator algebras and representation theory.</p>
<p>
Other problems discussed include the <em>Plateau Problem</em> about minimal surfaces (the spontaneous shape of a soap film attached to a wire frame). Many generalizations of the simple basic problem are described and progress of the last 100 years is surveyed.<br />
Two problems are attributed to Erdős. The <em>Erdős-Szekeres Problem</em> where it is conjectured that for $n>3$ one needs at least $N(n)=2^{n-2}+1$ points in general position in the plane to make sure that you will always find among them $n$ points that form a convex $n$-gon. Higher dimensional analogs all still unsolved. The <em>Erdős unit distance Problem</em> is asking how many pairs one my find among $n$ points in the plane that are at a distance 1 from each other. When such points are vertices in a graph connected by an edge, graph theory can be a trail to solve it and its generalizations.<br />
More graph theory is used in the <em>unknotting problem</em> (how to compute the knottedness of a knot?) and in two generalizations of the 4-color problem proposed by Hadwiger. The <em>Hadwiger's Conjecture</em> saying that every graph can be <em>t</em>-colored or has a subgraph that can be contracted to a complete graph with <em>t</em>+1 vertices. And the <em>Hadwiger-Nelson Conjecture</em> which also relates to one of the Erdős problems mentioned before. It asks for the chromatic number of the plane, that is the smallest number of colors needed to color the points in the plane such that no two points with the same color are at a distance 1 apart.</p>
<p>
It was the intention of Nash to write himself a survey on cooperative game theory. Unfortunately this paper never got written, but Eric Maskin wrote three pages on the topic to formulate an open problem in this context.</p>
<p>
A general observation with almost all problems discussed in this book is that even though the problem is posed in one subdomain, its solution often requires application of a completely different domain. For example the Riemann Hypothesis started from a prime number problem, but ended up to be the formulation about zeros of a complex function, which can be tackled for example via algebraic geometry. Sometimes one has to invent and explore a completely new branch of mathematics to make some progress. The book contains an enormous load of information about just 17 of the open problems in mathematics, their history, relationships, generalizations, and approaches taken. Although the papers are written independently, some of the problems are related so that a global name or subject index might have been an appreciated feature. The aim of the surveys is usually to reach a general mathematical public, but the papers are not always easy reading if one is not familiar with some basics of the problem. They all have a very extensive list of references that can be very useful if you want to start digging somewhat deeper. Needless to say that you will not easily find a solution. The problems are hard and have resisted many different attacks for a longer time. The good point is that there are still open problems (and there are many more 'out there') and if they are hard, then they usually generate new mathematics as long as they are not solved. This is a wonderful book whose papers give us a guided tour along the heroic battle fields of mathematics. </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>
The history, generalizations, and approaches taken in attempts to solve 17 open problems in mathematics are described in invited survey papers that should bring the reader to the current state-of-the-art. Seventeen such open problems were selected among which five of the Millennium Prize Problems. The selection was made by the editors. John Nash passed away shortly before the volume was ready, but he actively contributed to it and it reflects his interests. It is also a partial tribute to him. </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/john-forbes-nash-jr" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Forbes Nash Jr.</a></li><li class="vocabulary-links field-item odd"><a href="/author/michael-th-rassias" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">michael th. rassias</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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">2016</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">978-3-319-32160-8 (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">137.79 € (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">556</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><li class="vocabulary-links field-item odd"><a href="/imu/combinatorics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Combinatorics</a></li><li class="vocabulary-links field-item even"><a href="/imu/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li><li class="vocabulary-links field-item even"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</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/gp/book/9783319321608" title="Link to web page">http://www.springer.com/gp/book/9783319321608</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/00-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-02</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/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li></ul></span>Mon, 08 Aug 2016 11:59:12 +0000Adhemar Bultheel47098 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/open-problems-mathematics#commentsMy Search for Ramanujan
https://euro-math-soc.eu/review/my-search-ramanujan
<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>
Amir Aczel, the author of several popular science books, was the one who had the idea of writing this book. However, he passed away in November 2015 shortly after his book <em>Finding Zero</em> about the history of our Hindu-Arabic numeral system and his quest to find and save the stone which has the earliest known engraving of the symbol zero. Ken Ono is a number theorist who tells here the story of Ramanujan and discovers many parallels between Ramanujan's obsession for mathematics and Hardy's efforts that brought Ramanujan to the center of the mathematical community of his time and similar incidents in his own life and the life of his parents.</p>
<p>
Ken Ono's father Takashi Ono, was a young Japanese mathematician, in the post WWII period when he was picked up by André Weil, who was then at the IAS in Princeton. Takashi came with his wife to Princeton for what originally was planned to be a short period, but they decided to settle down and stay in the US, despite the initial hostile racist attitude of Americans towards the `Japanese enemy'. Takashi became a respected mathematics professor who spent all his time doing mathematics, while his wife, according to Japanese tradition (the marriage was arranged), was running the household almost invisibly. They had three children of which Ken was the youngest. The three boys were raised as 'tiger children': even performing at the top of their ability was not good enough. To his father, education was like solving a mathematical equation in which there was no place for affection. Ken was mathematically gifted and was therefore predestined to succeed his father. If he did not excel above his schoolmates, he was reproached to make his father to shame. Unable to earn the praise of his father, Ken rebelled, neglected his studies and developed a passion for bicycle racing instead. He eventually dropped out of high school and left his parents to live with his older brother in Canada. He got the consent of his parents because he argued that Ramanujan, much admired by the father, also was a dropout.</p>
<p>
At this point, the story is interrupted by a short biography of Ramanujan. How he was obsessed by mathematics. Not professionally educated, he filled notebooks with mathematical identities, inspired by the goddess Namagiri. He is desperately trying to find a job to make a living for him and his much younger wife that was appointed to him by an arranged marriage. He is finally picked up by G.H. Hardy and brought to Cambridge, UK. Together they overcame initial racism, although it drove Ramanujan at some point to an attempted suicide. Hardy could transform Ramanujan's genius into mathematical successes, but Ramanujan's health problems forced him to return to India where he died at the age of 32. Since then many mathematicians had a hard time interpreting and proving the notebooks he left behind.</p>
<p>
Ken was accepted for the University of Chicago, but freed from parental supervision neglected his studies and became a very active member of the Psi Upsilon fraternity and was semi-professional cyclist. Although his mathematical skills were not reflected by his results, his professor Paul Sally recognized Ken's mathematical talent and convinced him to start really working for his BA, in which he succeeded, and he recommended Ken to start a master at the UCLA. Ken passed his qualifying exams to start a PhD and since he is now earning some money, he married his girl friend who studied to be a midwife. He is however still very insecure, questioning himself and his work, which is not the best of the best, so that he still hears his father's voice reproaching him that he is an imposter, not good enough or not trying hard enough. He found at UCLA another guardian angel in the form of Basil Gordon, a professor who helped him on the path of mathematical research. However, when he presented his work on modular forms at a conference where also the 'big shots' were present, his lecture turned out to be a disaster and he got so depressed that 'the voices in his head' almost drove him to suicide. Basil Gordon could talk him out of his depression and his next conference talk was well received. This got him back on track to finish his PhD which he defended in 1993. He got a job at the University at Athens (Georgia) where Andrew Granville was interested in his work. His work related to Ramanujan problems placed in the context of work by Deligne and Serre. All of a sudden this research became a hype after Andrew Wiles announced his proof of Fermat's last theorem, and Ken's career boosted, his work being at the center of world wide mathematical attention. He is now a respected professor and he won several prizes for his mathematical contributions. The voices in his head finally became voices of approval. At some point, while he is staying at the IAS he could arrange a reunion of his father and André Weil who was 95 at the time. All's well that ends well and all thanks to Ramanujan who has influenced his life and guided him when in deepest need at crucial moments.</p>
<p>
In an epilogue he gives an account of his 2005 visit to India and Ramanujan's home, which is as a pilgrimage to him. In 2011 he is invited to be the mathematics advisor for a Ramanujan documentary <em>The man who knew infinity</em> based on Robert Kanigel's biography with the same title that is to be made on the occasion of the 125th anniversary of his birthday. In an afterword we find a number of mathematical problems related to work by Ramanujan and that are easy to explain.</p>
<p>
It is clear that the author, giving this account of his life, has great admiration for Ramanujan and feels deeply indebted to him which shows on almost every page of this book. There is a scent of a mathematical canonization of Ramanujan, but the author is wise enough to keep that under control. His book radiates his love for mathematics and the beauty that can be found in it. It is also shown that, even though Hardy once said that he did number theory because it was the least applicable branch of mathematics, but it turns out that prime numbers are most applicable in modern cryptography and research about Ramanujan's work is even related to string theory which is trying to understand the fundamental building blocks of our universe. And most importantly, the message is that no matter how depressed you are, there is always hope that at some point things may turn for the better. All you need is faith in yourself and in what you are capable of, and at some point, when you need it most, some kind soul that still believes in you will come to your rescue. This may hold for your life as a whole, but it certainly holds for mathematical research. At some point, struggling with a hard problem, we all may have experienced a moment of despair with no way out, but as Ken Ono describes in his book, at some moment, when you least expect it, you suddenly see the light, and the whole puzzle falls into place. And that is what brings the whole excitement and joy of doing mathematics.</p>
<p>
The book is amply illustrated with grayscale images, which are duplicated in color version in a separate section. All technicalities of the mathematics are avoided so that the book can be read by anyone. A subject and person index is missing and might be helpful to recall some sections, but the story is relatively straightforward so that this is not really a defect. The book is based on a true story but it reads like a script for an American movie with a happy ending.</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 semi-autobiography of Ken Ono, who rebelled against austere paternal education, dropped out of high school, but at crucial moments in his life is picked up by people who believe in his mathematical skills. Several parallels are drawn between his family history and Ramanujan's life. He ends up being a respected professor of mathematics whose main expertise is the mathematics that is behind Ramanujan's notebooks. </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/ken-ono" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ken Ono</a></li><li class="vocabulary-links field-item odd"><a href="/author/amir-d-aczel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amir D. Aczel</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">2016</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">978-3-319-25566-8 (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">28.61 € (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">256</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</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/gp/book/9783319255668" title="Link to web page">http://www.springer.com/gp/book/9783319255668</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li></ul></span>Tue, 21 Jun 2016 10:40:52 +0000Adhemar Bultheel47010 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/my-search-ramanujan#commentsDessins d'Enfants on Riemann Surfaces
https://euro-math-soc.eu/review/dessins-denfants-riemann-surfaces
<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 term "dessin d'enfant" (children's drawing) was coined by Alexander Grotendieck (1928-2014) in the 1980's because he was so enthusiastic about the simplicity of the graphs that represented very complex mathematical objects. The graphs may be simple, but to understand the mathematics that they represent is not. So, unless you are familiar with the concept and with algebraic geometry, it might be a somewhat misleading title.</p>
<p>
The text of this monograph grew out of notes taken while the authors were lecturing on this topic. Thus there is a definition of dessins (there are actually three different but equivalent definitions). The necessary (background) material is either briefly reviewed or added in appendices after the relevant chapters. It is however necessary that the reader is familiar with complex functions and a thorough knowledge of group theory. This is not the best place to become familiar with algebraic geometry. Knowledge of graphs and Galois theory might also help. Thus the book is intended for properly trained mathematics graduate students or researchers. The subject is fascinating though and it is connecting several different mathematical disciplines (complex functions, graphs, group theory, topology, Galois theory,...) which makes this domain all the more interesting to work in.</p>
<p>
For the reader familiar with the subject, the book will bundle the material in a nice way, and it will probably suffice to say that the text consists of three parts: (I) giving an introduction to the concepts that play a major role and a survey of the results obtained in the last 40 years; (II) is a study of regular dessins, how they can be constructed and their classification; (III) discussed two applications: an abc-like theorem for the degrees of functions defining algebraic curves and Beauville surfaces.</p>
<p>
For those not familiar with the subject, it requires some more explanation. A complex smooth meromorphic function β(z) defined on a (compact oriented) Riemann surface X (you can think of the Riemann sphere for simplicity) and taking values on the Riemann sphere is a Belyĭ function if it has no critical points outside {0,1,∞}, i.e., these correspond to branch or ramification points of a certain order. So there are in general several preimages of these points. Taking the preimages of the interval [0,1] results in a bipartite graph. The preimages of 0 are colored white (they are the zeros of β), those of 1 are colored black (these are the zeros of 1-β) and the preimages of the interval define edges, each connecting a white and a black point. This graph is a dessin (d'enfants). The preimages of ∞ are left out but if we had included them, there would be one in each of the connected regions defined by the edges of the dessin. If we color these red, then we can generate a triangulation by connecting the red vertices with the black and white ones on the boundary of its region. A triangle face is the preimage of the lower half plane if its boundary has white, black, red vertices in clockwise order, otherwise it is the preimage of the upper half plane (which is a triangulation of the Riemann sphere). The Belyĭ functions are just special cases of more general smooth functions, but transformations can arrange that in the end one has to study mostly those.</p>
<p>
Of course the situation can be much more complicated because Riemann surfaces can have a nonzero genus, or one may want to work with projective coordinates because we are dealing with points at infinity, and the algebraic curves that the functions define can be very complex, etc. On the other hand, there are many different tools to deal with the problem because there are also different ways to define a dessin. There is the connection with groups, which can be explained as follows. The same triangulation we have described for β can be described by functions 1/β, 1-β, 1/(1-β), 1-1/β, and β/(1-β), which just give permutations of the colors of the vertices, and hence a permutation of the edges. Two permutations (the white and black vertices characterize their orbits) generate a transitive group (called the monodromy group). This is a quotient group of a triangle group and the group of isometries of the hyperbolic plane. So that introduces group theory into the picture. Not surprisingly taking into account the nice symmetries of the pictures of the triangulations or dessins.</p>
<p>
The dessins are useful ways to study Belyĭ functions, but more general hypergraphs introduced by Cori in the context of computer graphics is even more appropriate (although these are not the main topic in this book). Belyĭ (1951-2001) was interested in inverse Galois theory (i.e., find information on the absolute Galois group of automorphisms of the closed field of algebraic numbers). The important Belyĭ theorem (1979) says that it is sufficient to study Belyĭ functions (hence dessins) to understand algebraic curves and the corresponding compact Riemann surfaces.</p>
<p>
All these connections and much more is explained in part I. Part II is a discussion of regular dessins, which are dessins with the highest possible symmetry. Analyzing the symmetry requires an intensive use of group theory, which I will not elaborate on here. For genus 0 (Riemann sphere) or genus 1 (torus) there are infinitely many symmetries possible, but for a genus larger than 1, the symmetries are limited. The Hurwitz bound says that the number of automorphisms of compact Riemann surfaces of genus g is bounded by 84(g-1) and the quasiplatonic curves of genera 2,3, and 4 are analyzed in detail. What is said about dessins can be generalized to maps, which are characterized by a compact Riemann surface X, a graph $\cal G$ embedded in X and an automorphism group G. Classification can be done on the basis of each of these, of course taking the genus into account.</p>
<p>
In part III we find an application in the abc conjecture (suggests an upper bound on the the size of the integers a, b and c solving a+b+c=0 in terms of their prime divisors). There is an analogy with the degrees of self covers of prime degree of an elliptic curve defined over a number field that has complex multiplication. The optimal solutions turn out to be exactly the Belyĭ functions. The extension from curves to complex surfaces is a considerable complication. The Beauville surfaces however allow such an analysis, and this is studied in the last chapter of the book.</p>
<p>
It is clear from this brief survey that this is a book for the specialists. Theorems are formulated, but proofs are often rather short, or are only outlined. The text is also regularly interrupted by formulations of exercises which ask to prove some intermediate result or apply a definition in a particular situation (some brief hints are added at the end of the book). The book can thus be used as a textbook for a course on this fascinating topic. With a proper preparation, it gives a good entry point to current research. Not the most general situation is always described, but all chapters are completed with many references with pointers in the text that may refer to a more extensive introduction or survey on a particular topic or to more advanced treatments or a connection that is not elaborated in this book.</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 book based on lecture notes for courses given by the authors. They deal with the study of bipartite graphs on Riemann surfaces known as dessins d'enfants, a term coined by A. Grotendieck. These are simple characterizations of certain maps on Riemann surfaces. Knowledge of complex functions, Riemann surfaces and group theory are prerequisites. The book has 3 parts. The first one gives the necessary definitions and recalls preliminaries, the second studies regular dessins for low genera surfaces, having a maximal symmetry and the last part gives an analog of the abc conjecture. It concerns a bound for degrees of Belyĭ functions associated with algebraic curves and surfaces.</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/gareth-jones" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Gareth A. Jones</a></li><li class="vocabulary-links field-item odd"><a href="/author/j%C3%BCrgen-wolfart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jürgen Wolfart</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">2016</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">978-3-319-25566-8 (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">90.09 € (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">273</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/algebraic-and-complex-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebraic and Complex Geometry</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/be/book/9783319247090" title="Link to web page">http://www.springer.com/be/book/9783319247090</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/14-algebraic-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14 Algebraic geometry</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/14h57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14H57</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/11g32" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11G32</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05c10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05C10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05c25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05c25</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/14h45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14H45</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/14h55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14H55</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/20f65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20F65</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/30f10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30F10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/57m15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">57M15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/57m60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">57M60</a></li></ul></span>Wed, 27 Apr 2016 10:10:03 +0000Adhemar Bultheel46903 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/dessins-denfants-riemann-surfaces#commentsDesigning Beauty: The Art of Cellular Automata
https://euro-math-soc.eu/review/designing-beauty-art-cellular-automata
<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>
Cellular automata (CA) are discrete time dynamical systems that consist of a regular grid of cells. Each cell has a finite number of possible states. The state changes from one time instant to the next depending on the current states of the cell and its neighbors following simple rules, uniform over the grid.</p>
<p>
The best known example is the <em>Game of Life</em> (GoL) that was designed by John Conway in the early 1970s. It became extremely popular after Martin Gardner's picked it up in his <em>Scientific American</em> column. The rules are simple. Each cell has two states: alive or dead. A live cell with 2 or 3 live neighbors survives, otherwise it dies, while a dead cell with exactly 3 live neighbors will become alive. The ease of programming and personal computers becoming widespread in those days, made a whole generation play this GoL fanatically. Even though the system is deterministic, the ever changing patterns in successive steps makes it impossible to imagine the configuration several steps ahead, unless the steps are actually computed. During the evolution, some nice patterns can occur like all kinds of oscillators, gliders and spaceships (groups of cells that walk away from the group in a straight line), glider guns (that produce a regular sequence of gliders), and exotic configurations like a garden of eden (a pattern without a predecessor). There are many websites where one can experiment with the GoL today or any of its extensions and generalizations. Some are mentioned in this book. It is even possible to implement a Turing machine with particular configurations of the GoL. The CA also feature prominently in the somewhat controversial book <em>A New Kind of Science</em> (2002) by Stephen Wolfram, the founder and CEO of Wolfram Research.</p>
<p>
But CA were not only used for recreational purposes. They became a subject of research and were applied for the simulation of all kinds of chemical, biological, or social dynamical systems. They can be implemented in one or two or in any finite dimension, the rules can be made stochastic, there can be many states per cell, etc. When adding colors to the graphical representation, some really nice and appealing pictures can be the result. Because of this somewhat unexpected aesthetic side product, also artists became interested, just for the creation of the graphical effects.</p>
<p>
The latter artistic aspect is the one that is the main focus of this book. It is primarily a collection of nice pictures that were generated by CA. Thirty `artists' (that are also mathematicians, engineers, architects, computer scientists,...) contributed to the volume. They briefly situate their pictures in a few lines up to at most two pages and then their graphics are included with extensive captions giving additional information about the particular result, parameters used, etc. Some of them have websites where animations or software is available for the user to experiment. In all cases the reader is referred to publications for further details.</p>
<p>
For one-dimensional CA, the dynamics can be represented in a two-dimensional picture, but for two and three-dimensional CA, one may only give a snapshot at some time instant since obviously the picture should evolve in time (unless it would represent a steady state). Most of them are in color, although sometimes a black-and-white pattern can also be fascinating. It is not a coffee-table book though. The format is like for an ordinary proceedings volume, but with a contents that is more graphical than textual. It is not a mathematics or computer science book either because the emphasis is on the graphics and the underlying theory is only briefly mentioned (the rules are usually rather simple anyway), but there is a list of 175 references to books and papers for further reading.</p>
<p>
What the book clearly shows is that even though CA are so simple, yet so general, very diverse exiting pictures can be the result. The graphics have an appeal similar to the Mandelbrot and Julia sets and their images that were also very popular in the exhibits and photo books by Heinz-Otto Peitgen and Peter Richter in the 1980s.</p>
<p>
The diversity of CA that are represented in this book is too extensive to be enumerated them exhaustively in this review. Some samples: variations on the GoL (like Larger than Life (LtL), Life without Death, enlightened GoL), continuous reaction-diffusion models, toothpick CA (horizontal or vertical line segments —like toothpicks— are added in every step), CA on grids in hyperbolic geometries or on spheres or on hexagonal grids or Penrose tilings, asynchronous CA, CA with memory, ... And we see applications such as prime generators, ecological examples, piston motion, chemical reactions, seismic simulation, Turing machines,...</p>
<p>
Thus the book should inspire the scientist to present his or her work in a pleasing and graphically attractive way, and it should invite artists to explore the possibilities to be creative with CA. Fact is that CA are an attractive and simple concept that, because of the nonlinearity, can lead to unexpected amazing and fascinating results. Of course there are also difficult questions to ask and conjectures to make that are very difficult to prove, but these issues are certainly not the topic of this book.</p>
<p>
That CA can not only inspire visual artists, but also the composers of music can be heard on a CD <a href="http://synthesist.net/music/anathem/"><em>Iolet</em></a> by David Stutz that contains several mathematically inspired compositions. His piece <em>Cellular Automaton</em> is sung by a choir, where each member acts like a cell in a CA and changes his singing depending on what he hears his neighbors sing. The choir leader regularly injects new patterns to start from. It all sounds like a choir of Buddhist monks or Tuvan throat singing. The idea is based on the SF novel <em>Anathem</em> by Neal Stephenson.</p>
<p>
If you are not familiar with CA, this is a survey that can get you started to explore a catchy subject. Surely you will be tempted to search the Internet and check out some of the available applets and graphical user interfaces to try some experiments for yourself. Beware not to get hooked too much playing around and getting hypnotized by the graphics.</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 basically a collection of examples of nice pictures that resulted from research on cellular automata. Some technical background is given, but the reader needs to consult the literature for the details.</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/andrew-adamatzky" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andrew Adamatzky</a></li><li class="vocabulary-links field-item odd"><a href="/author/genaro-j-mart%C3%ADnez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Genaro J. Martínez</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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">2016</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">978-3-319-27269-6 (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">52,99 € (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">201</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/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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/gp/book/9783319272696" title="Link to web page">http://www.springer.com/gp/book/9783319272696</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/37-dynamical-systems-and-ergodic-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37 Dynamical systems and ergodic theory</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/37b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37B15</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/68q80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Q80</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/92c99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92C99</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 21 Mar 2016 07:04:02 +0000Adhemar Bultheel46820 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/designing-beauty-art-cellular-automata#comments Mathematics in Everyday Life
https://euro-math-soc.eu/review/mathematics-everyday-life
<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>
This is not the first book on mathematics in everyday life. Some of there are truly recreational and are addressing a general motivated layperson like for example <em><a href="/review/raising-public-awareness-mathematics">Raising public awareness of mathematics</a></em> (E. Behrends et al) or <em>Figuring it out - Entertaining encounters with everyday math</em> (N. Crato), and there are many other examples. Some are more recreational and collect tricks, games, history and anecdotes, some others embed the mathematics in a novel where the story becomes more important. At the other side of the spectrum you find the true lecture notes that include many applications and examples.</p>
<p>
The present book is different. It is intended to be a book with everyday illustrations of applications of mathematics intended to accompany introductory mathematics courses for a beginning university student. Thus it contains the application part, outside of the volume with the (more) theoretical lecture notes which will normally contain all the definitions, theorems, and proofs. It is somewhat similar to <em><a href="/review/everyday-calculus-discovering-hidden-math-all-around-us">Everyday Calculus</a></em> (by O.E. Fernandez) although the latter is just illustrating and does not provide exercises for the student as in Haig's book which is also broader, not restricted to only calculus. The content is organized by application domain (finance, economics, dynamics, sports, social sciences, gaming, and gambling). The mathematics involve a cross section of what the students will get in traditional classes (calculus, differential equations, probability, linear algebra, combinatorics,...). The applications usually have just one mathematical component i.e., as a rule they don't mix different mathematical disciplines. Each of the seven chapters give many examples that are worked out and each one ends with a rather extensive set of exercises for the student to solve. Most problems are numerical or computational, occasionally an exercise asks for a proof. They stay within the same complexity of the examples given in the chapter. No solutions are provided though. For the reader who is interested in further reading some references are provided. Besides an appendix with general useful mathematical facts at the end of the book, some chapters also have an appendix attached which elaborates somewhat deeper on a technical matter.</p>
<p>
To give an idea of the applications covered, I will give some examples from the different chapters. The first chapter has financial applications that are mostly involved with interest computation. Annual percent rate (APR), compound interest, the 72 rule (it takes about 72/p years to double the capital with an interest rate of p), investing, loan, taxes,...</p>
<p>
Differential equations in the second chapter are derived but are restricted to first and second order. Solution methods are analytic, not numerical. Models are given for physical problems, but also the Lotka-Volterra model for predator-prey simulation.</p>
<p>
The sports and games chapter is rather extensive and covers almost many different sports (tennis, rugby, snooker, darts, athletics, golf, soccer) but also tournament design. For example the optimal place in the rugby field to kick the ball towards the poles, where to hit a snooker ball, the chance of scoring a soccer penalty depending on the chances of the player aiming left or right and the keeper diving left or right, what is best during golf: consistency or a flamboyant game with risky shots? etc.</p>
<p>
Chapter 4 on business applications involves stock control, delivery of goods, human resource management, check digits, promotion policy, investment and profits,... Several examples involve linear programming problems, so the simplex method to solve such problems is explained.</p>
<p>
Among social science applications we find voting techniques, the Arrow paradox, the Simpson paradox, the problem of false positives in medical applications, how to measure social inequality in a population, etc.</p>
<p>
Also TV games problems are considered, discussing questions such as when to make a highly rewarding risky rather than a less rewarding save decision, of course the classic Monty Hall problem is one of them,... Several British TV shows are scrutinized in this way. Most will not be familiar to non-British readers, but the situation is explained and the problem (usually involving probabilities) is clarified before a solution strategy is given.</p>
<p>
The last chapter involves gambling: lottery, roulette, horse racing, and card games. Of course probability is here the main mathematical ingredient. For readers who are particularly interested in this chapter, I can recommend to read more on these sports gambling strategies in <a href="https://people.cs.kuleuven.be/~adhemar.bultheel/WWW/EMS/r143.php">L.A. Math: Romance</a> (J.D. Stein) although there it is framed in a bit more "playful" environment.</p>
<p>
The book could be appealing to non-student-but-motivated-hobbyists, but I guess there are lighther and more amusing alternatives that are better suited. The content invites to really work on the topics that were presented. So for students and their instructors, this gives a wealth of ideas for practical sessions that are intended to work with the theory from the mathematics courses in a practical environment.</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 exercise book illustrating the application of mathematics in everyday life that goes along with mathematics that are taught to beginning university students. The body of the text consists of examples and their solutions organized in chapters by application domain, and followed by a number of exercises per chapter. </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/john-haigh" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Haigh</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">2016</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">978-3-319-27937-4 (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"> 42.39 € (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">170</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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/gp/book/9783319279374" title="Link to web page">http://www.springer.com/gp/book/9783319279374</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/97d50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97m10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97M10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97u40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97U40</a></li></ul></span>Sun, 13 Mar 2016 13:54:33 +0000Adhemar Bultheel46797 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-everyday-life#commentsLinear Canonical Transforms
https://euro-math-soc.eu/review/linear-canonical-transforms
<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 Fourier transform rotates the time-frequency content of a signal over 90 degrees from the time axis to the frequency axis. Already in the 1970s it was observed that certain optical systems rotated the signal over an arbitrary angle, which became known as the fractional Fourier transform since it it acts like a fractional (i.e., a real) power of the Fourier operator. The fractional Fourier transform is thus depending on this rotation angle forms a one-parameter family of transforms. <br />
Around the same time, quantum physicists realized that certain systems could transform the position and momentum as a vector into their new values leaving the quantum mechanics invariant. All such transforms were obtained by multiplying the position-momentum vector with a unimodular matrix. In the scalar case such a matrix depends on 3 free parameters. In a general N-dimensional space we have a 2 by 2 block matrix that form a real symplectic group Sp(2N). These transforms are the essence of the family of linear canonical transforms (LCT).<br />
When in 2D optics the vector of position and momentum of the optical ray is considered, the same kind of transformation can be used, and the fractional Fourier transform like many other (fractional) transforms can be seen as a special case of the LCT. The reader less familiar with the subject should be warned that these fractional transforms are not directly related to the equally important and equally flourishing domain known as fractional calculus which studies fractional derivatives and fractional integrals.</p>
<p>
Since the early days many papers appeared on all kinds of fractional transforms, and even several books, among which a basic one on <em>The Fractional Fourier Transform</em> by Haldun M. Ozaktas, Zeev Zalevsky, and M. Alper Kutay in 2001. There the emphasis was on definitions, mathematical properties and computation and their applications in optics and signal processing. The LCT is already there but it is not the main focus. Two of the authors of that book are now also editor of the present one. One could think of it as the LCT analog of their fractional Fourier transform book, but less extensive, and it is not a monograph. Several experts are contributing to the present book. It gives an up-to-date overview of the many aspects of the LCT. As it appears as a volume of the <em>Springer Series in Optical Sciences</em>, there is an understandable bias towards the optical viewpoint and applications, with less emphasis on the quantum physics.</p>
<p>
The fifteen chapters are subdivided into three parts: (1) Fundamentals, (2) Discretization and computation, (3) Applications. All the aspects are covered: operator theory, theoretical physics, analysis, and group theory in the early chapters, discrete approximations and digital implementation in the second part, and it ends with some applications in the last part.</p>
<p>
In the first part one gets the basics in some 100 pages: some history and of course the definition and properties, the kernels when written as an integral transform, all the types and special cases, and the effects of the transform in phase-space. The eigenmodes are as important as the Gauss-Hermite eigenfunctions of the Fourier transform. So there is a separate chapter dealing with the eigenfunctions. Also uncertainty principles play an important role when it comes to sampling theory for these transforms. The first part is completed with two chapters covering extensively the optical aspects of the LCT.</p>
<p>
The second part deals with the computational aspects. While the fractional Fourier transform resulted in a rotation in the time-frequency plane, the LCT will result in oblique transforms. It is then important to obtain some analogs of bandwidth, sampling theorems, and degrees of freedom in the signal when one wants to come to an implementation of a fast discrete LCT transform. Analyzing these effects is related to a decomposition of the LCT in a sequence of elementary transforms which boils down to a sequence of chirp multiplications and Fourier transforms. Several possibilities are proposed to come to a fast digital LCT implementation, although no software is provided. The approach taken is from a signal processing viewpoint. One chapter gives an alternative that is based on optical interpretation. That alternative approach relies on coherent self-imaging, known as Talbot effect which describes Fresnel diffraction of a strictly periodic wavefront. Knowing that Fresnel diffraction is a special case of LCT. Therefore various generalizations of self-imaging in the wider LCT context can also lead to a practical implementation of discrete LCT.</p>
<p>
The application part illustrates how LCT can be used to solve certain problems like deterministic phase retrieval, analyzing holographic systems, double random phase encoding, speckle metrology and quantum states of light.</p>
<p>
This book is a most welcome addition to the literature. The subjects discussed appear in quite diverse contexts and it is therefore difficult to get the same overview as it is presented here. The general approach is the same as was used in the fractional Fourier book that I mentioned above, but it is less thorough. Moreover, since the different chapters are written by different authors, and because they highlight different aspects, the notation is not always strictly uniform, but that should not be hindering too much. The part on the algorithmic implementation is rather detailed but no software is provided, so it is a challenge for computer scientists to design an optimal implementation so that it can become standard software in signal processing packages.</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>
The linear canonical transform is a phase space transform with roots in optics and quantum mechanics. This is a collection of survey papers written by renowned specialists that give a state-of-the-art of the many aspects of the linear canonical transform with an emphasis on the optical interpretation and applications. The quantum mechanical aspects are less present. It covers the basic mathematical and optical aspects, the possible implementation of a fast discrete algorithm, and several of the possible optical and signal processing applications.</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/john-j-healy" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John J. Healy</a></li><li class="vocabulary-links field-item odd"><a href="/author/m-alper-kutay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">M. Alper Kutay</a></li><li class="vocabulary-links field-item even"><a href="/author/haldun-m-ozaktas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Haldun M. Ozaktas</a></li><li class="vocabulary-links field-item odd"><a href="/author/john-t-sheridan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John T. Sheridan</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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">2015</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">978-1-4939-3027-2 (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">137,79 € (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">264</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/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li><li class="vocabulary-links field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</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/gp/book/9781493930272" title="Link to web page">http://www.springer.com/gp/book/9781493930272</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/78-optics-electromagnetic-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">78 Optics, electromagnetic theory</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/78-04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">78-04</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/42a38" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A38</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65r10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65R10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/94a12" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94A12</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/94a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94A20</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81v80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81V80</a></li></ul></span>Tue, 09 Feb 2016 07:03:55 +0000Adhemar Bultheel46705 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/linear-canonical-transforms#commentsThe Real and the Complex: A History of Analysis in the 19th Century
https://euro-math-soc.eu/review/real-and-complex-history-analysis-19th-century
<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>This is a history book on the development of mathematics in the 19th century. Each chapter is built up around one or a few mathematicians. First a short bio is sketched, often embedded in the political context of their time, but the more important part is where it is shown what theory the person has developed, in what context it was done, hence why he (there are unfortunately no she's) did it. It shows that even the big shots of mathematics that contributed to the tremendous expansion of mathematical knowledge in the 1800s were searching, sometimes uncertain or even mistaken. With several of the emerging concepts that we are now familiar with today still being shaped and reshaped, these were first formulated, not taking care of the tiny details, which needed a revision or even rethinking the concept later on. It was also the century in which analysis came to the foreground as one of the main mathematical topics next to geometry and algebra that had dominated before. Analysis became a working tool that was used in other domains of mathematics, it became essential in modeling physical phenomena and it was intensively used to solve many applied problems. However, mathematics gradually evolves towards a more abstract subject and is developed more independently from the applications. How all this came about becomes clear by reading this book.</p>
<p>The book is written as a textbook on the history of mathematics, and hence it is assumed that the reader has attended some analysis courses: real and preferably also complex analysis. There are also (few) end-of-chapter exercises which are clearly pointing to history students. That means that some topics are suggested to investigate and to report on them in an essay. There is also an end-of-course chapter giving advise of how to choose a topic for an essay and what kind of content it should be given (the mathematical technicalities are less important, as long as they are correct, but concentrate on why and how mathematical concepts grew into the ones that we know today).</p>
<p>The organization of the chapters is more or less chronological. One may recognize three parts: the first one sketches the situation in the early and the first half of the 19th century; the second part deals with the middle of the century, when complex analysis comes more to the foreground; and the third part is then announcing the transition to the 20th century, the foundations of mathematics are questioned, set theory, the real number system, topology all push mathematics into a more abstract framework.</p>
<p>Often the mathematician's findings were written down in books that grew out of lecture notes, which of course forced them to reflect upon the foundations of what they were teaching and these books were obviously very influential. The concepts of course still survive, but we would not always be satisfied with the way they were originally described.</p>
<p>The initial setting is made in the first three chapters with Lagrange, Fourier and Legendre. It is clear that these were interested in developing new ideas, not worrying too much about the basics or the finer details. Lagrange struggled with the foundations of analysis, his approach being basically algebraic without infinitesimals. Fourier had proposed his trigonometric series, and Legendre's contribution was to set up a theory of elliptic integrals. These elliptic functions form a recurrent topic in the next chapters as it was further developed by Abel who was the first to place them in the realm of complex functions, as did also Jacobi, Gauss, Liouville, and Hermite. In fact, Gray uses them as a kind of case study that extends over a large part of the book. Cauchy of course was influential on many other domains as well: continuity, series, differentiation and integration, and complex functions. He was the first to introduce more structure and rigor in real analysis. Equally productive was the master calculator Gauss with contributions on integration and complex analysis. These chapters span approximately the first half of the century. Gray interrupts here the development to give a reflection on what has been achieved so far.</p>
<p>The next half of the century starts with a new topic: potential theory (Green, Cauchy, Dirichlet). Riemann is given somewhat more attention in several chapters with his Riemann function, elliptic functions (the bread crumbs in this historical expedition), but of course also complex analysis and his conformal mapping theorem. There is also a discussion of how his work was received by his contemporaries. An alternative for Riemann's geometric function theory was provided by Weierstrass who initiated the concept of an analytic function. Here Gray reflects again on the past chapters. Still real analysis, and certainly complex analysis had not reached the rigor that we are used to.</p>
<p>This rigor was only starting to develop in what is the third part of this book. For example, it was only noted 20 years after its original formulation that Cauchy's theorem stating that the sum of (infinitely many) continuous functions was continuous did not always hold true. Only then, it was realized that functions could be much more exotic objects than the smooth curves that they were originally thought of. Mending Cauchy's theorem led to the concept of uniform convergence, non-differentiable and non-integrable functions (Bolzano, Cantor, Schwarz, Heine, Dini,...). This entailed Lebesgue's integration theory and the unavoidable rigorous definition of real number system, set theory, and topology.</p>
<p>It is also interesting to see what triggered the development of mathematics. Fourier used his series in heat diffusion problems, Legendre used elliptic integrals to study the mechanics of a pendulum, potential theory grew out of electromagnetic problems. However with rigor came also abstraction and, although still applicable, mathematics itself became the driving force for its own expansion.</p>
<p>The text is illustrated with portraits of the key mathematicians and where appropriate, plots to visualize some of the interesting functions or graphs are included. In several appendices, we find some translated papers (Fourier, Dirichlet, Riemann, Schwarz), and some more technical mathematical ones on series of functions and their convergence and on potential theory. Obviously a list of references (many of them being the original publications that were discussed, while others are more recent historical studies) as well as a mixed index of names and subjects are indispensable in a book like the present one.</p>
<p>It is very well known that history is of utmost importance and that we should learn from it. However, it is astonishing how fast even recent history is completely forgotten. This book certainly learns something to students but also to professional mathematicians, something that is too often neglected. Mathematics develops not by adding some epsilon improvements to an existing sequence of definitions, properties, theorems and proofs. The majority of the papers that are published are in that vein of thinking. With each one the boundaries of our mathematical knowledge crawls a bit forward. But what is actually progressing mathematics is the exploration of an unknown mine field. The explorers that venture there are the ones whose names will be printed in bold face in future history books. This book learns that these explorers of the past may have felt uncertain, made mistakes, and even used a trial and error approach. Only when the right track has been flagged, the road can be paved with the proper rigor needed to move on. So this book is not only an interesting read for the students who (have to) study it, but equally valuable for professional mathematicians. This is if they are prepared to take the time and reflect on the not so distant past of their beloved subject to which they want to contribute.</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 textbook for a course on the history of mathematics. In three parts Gray sketches the evolution of the shaky emerging of analysis in the beginning of the 19th century, its growth into a more rigorous subject and the extension to complex analysis, and finally how near the end of the century the foundations of mathematical analysis were revised, resulting in the definition of the real number system and new subjects such as topology entering mathematics.</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/jeremy-gray" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jeremy Gray</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">2015</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">978-3-319-23714-5 (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">37,09 € (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">366</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><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</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/gp/book/9783319237145" title="Link to web page">http://www.springer.com/gp/book/9783319237145</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span>Tue, 09 Feb 2016 06:16:38 +0000Adhemar Bultheel46704 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/real-and-complex-history-analysis-19th-century#commentsGalileo and the Equations of Motion
https://euro-math-soc.eu/review/galileo-and-equations-motion
<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 first of the three laws of motion formulated by Newton (1642-1726) says that every object in a state of uniform motion remains in that state unless an external force is applied. This is essentially a reformulation of Galileo's inertia concept. This is sometimes considered to be the start of modern science. Among many other well known things that Galileo (1564-1642) achieved as an astronomer, he also described the law of falling bodies that we know as <em>s = ½at²</em>. The objective of this book is to analyze critically why these results are attributed to him and how Galileo came to these conclusion and how he formulated them. Such an historical study is not trivial. There are of course the texts that were published during Galileo's life, but even more important in this context are some of his unpublished early notes as well as the texts that Galileo, who was blind at the end of his life, dictated to Viviani who was taking care of him. Because of his trouble with the Church, these had to be hidden, even after Galileo's death. Later they were sold in parts and in so doing were dispersed and got nearly lost. Fortunately the Grand Duke of Tuscany, Ferdinando II, could recollected whatever he could find.</p>
<p>
This book is a thorough historical study, mainly addressing the professional historian. To meet the objectives mentioned above, one has to know and understand first what was known before Galileo and second try to understand how Galileo was thinking and how he came to his ultimate conclusions. This is far from a trivial task. It is very difficult with our current understanding of physics and of our solar system to set our brain in a state of understanding that corresponds to the time and circumstances Galileo was living in.</p>
<p>
Boccaletti is therefore zooming in and focussing on this specific task. Thus, this is not a biography of Galileo, and his adherence to the Copernican heliocentric solar system and his problems with the Catholic Church are not explicitly covered and only mentioned in as far as it fits the main focus: the equations of motion. More specifically he restricts the book to the study to dynamics and kinematics deliberately avoiding statics.</p>
<p>
On the other hand, precisely the unpublished notes, known as the <em>De Motu</em> (written in the period 1589-1592) of the young Galileo and the <em>Discourses (Discorsi e dimonstrazioni matematiche intorno a due nuove scienze)</em> (written during 1635-1642) of the mature Galileo are crucial. The first was written during his first appointment in Pisa and discusses the Aristotelian-Archimedean dynamical principles of how bodies sink or float in fluids. The second was written during house arrest and dictated by the old Galileo. Both are important to explore and understand Galileo's mind. They were included in the complete works of Galileo that were only published some hundred years after his death. They were edited, studied, and commented by several people and the interpretation was not always the same. Other important sources are his <em>Le Mecaniche</em> which are lecture notes written during Galileo's teaching in Padua around 1592. A short version of these notes was published posthumously. When he was teaching on the same subject ca. 1598, this resulted in a longer version. Handwritten versions were circulating and rediscovered much later. Another text is the <em>Dialogue</em> (ca. 1630). This is, just as his <em>Discourses</em> written in the form of a dialogue between two characters, one of them being an alter ego of Galileo, and the other one is the one that is asking questions and that is instructed by the first. This form of writing was not unusual during Galileo's time.</p>
<p>
Like in most historical studies there are many quotations, and they are often quite long. They can be either form Galileo's writing or from historians that have studied Galileo. In this book they are placed in the text and they form the backbone of the facts that are being told. To keep things readable though, they are not in the original language but are mostly given in an English translation.</p>
<p>
There are, in accordance with the objective of the book, two parts: the period before Galileo, and the period of Galileo. The first part of course starts with the ancient Greek, in particular Aristotle, and continues with an overview of the Middle Ages and early Renaissance. Attention is paid to the first criticisms on Aristotle by a group at Merton College (Oxford) and another one, known as the Parisian school. Later the Italians joined the discussion in the sixteenth century in particular Tartaglia and Benedetti. Some historians have suggested that Galileo got his ideas directly from Benedetti, but Boccaletti argues that Galileo probably never read Benedetti.</p>
<p>
The second part goes though the manuscripts of Galileo and shows how he gradually criticized the Aristotelian dogma's and developed his own ideas, mainly based on experiments. His favored experiment involved objects sliding or rolling down an inclined plane. The rumors that he would have dropped weights from Pisa's tower is most probably a legend. The inertia principle appears in several of his publications and letters. The dialogues in the <em>Discourses</em> go on for several days, so that they are subdivided into parts indicated as the first day, the second day, etc. Since these came to us in versions that were not published during Galileo's life, and editors later gave their own interpretation, it is not always clear what was really intended. Anyway, whatever relates to Galileo's study of motion of falling objects, or the inertia principle or the parabolic trajectories of projectiles and even some of his notes on the pendulum are brought to the foreground. Galileo also considered motion relative to different reference systems. At the end of the book, this is also briefly explained and why one can read in modern textbooks that the equations of motion remain invariant under 'Galileo transformations'.</p>
<p>
Since it is not the focus of this work, it is not really stressed, but Galileo was also the one who started formalizing things and used mathematics and formulas, much more than what was usual in the Greek tradition. This is a book for historians, if you are interested in a biography or in his astronomical contributions or his dispute with the Catholic Church, one should try to find another book. For an easy reading biography I can recommend <em>Galileo Galileo - When the world stood still</em> by A. Næss (Springer Verlag, 2005).</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>
Newton's first law of motion is the law of inertia: every object in a state of uniform motion remains in that state unless an external force is applied. Other well known laws of motion are that the distance traveled by an object, traveling with a uniform accelerated speed, like a falling object is <em>s = ½at²</em>, and the parabolic trajectory of a projectile. For all of these, Galileo has contributed to their origin. The purpose of this book is to explain how Galileo gradually transformed from a believer in the Archimedian approach to eventually arrive at the new insights of these laws of motion. Boccaletti carefully analyses the available texts to understand this transformation. This is a study mainly addressing the professional historian.</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/dino-boccaletti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dino Boccaletti</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">2015</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">978-3-319-20133-7 (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">105.99 €</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">189</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</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/gp/book/9783319201337" title="Link to web page">http://www.springer.com/gp/book/9783319201337</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/70-mechanics-particles-and-systems" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70 Mechanics of particles and systems</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/70f15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70F15</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Fri, 27 Nov 2015 16:47:00 +0000Adhemar Bultheel46568 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/galileo-and-equations-motion#comments