Book reviews
https://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenIntroduction to Malliavin Calculus
https://euro-math-soc.eu/review/introduction-malliavin-calculus
<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 Malliavin calculus is an infinite-dimensional differential calculus on Wiener space, that was first introduced by Paul Malliavin in the 70’s, with the aim of giving a probabilistic proof of Hörmander’s theorem. This theory was then further developed, and since then, many new applications of this calculus have appeared.</p>
<p>This textbook provides an introductory course on Malliavin calculus intended to prepare the interested reader for further study of existing monographs on the subject. Moreover, it contains recent applications of Malliavin calculus: density formulas, central limit theorems for functionals of Gaussian processes, theorems on the convergence of densities, noncentral limit theorems, and Malliavin calculus for jump processes.</p>
<p>Recommended prior knowledge would be and advanced probability course.</p>
<p>The book is organized as follows. Chapters 1 and 2, give an introduction to stochastic calculus with respect to Brownian motion. Chapters 3, 4, and 5 present the main operators of the Malliavin calculus. Chapters 6, 7 and 8 are devoted to different applications of the Malliavin calculus for Brownian motion. Chapter 9, 10 and 11 develop Malliavin calculus for Poisston random measures.</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"> </div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-nualart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Nualart</a></li>
<li class="field-item odd"><a href="/author/eulalia-nualart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eulalia Nualart</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</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">9781107039124 1107039126 9781107611986 1107611989 </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">236</div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/49-calculus-variations-and-optimal-control" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49 Calculus of variations and optimal control</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/stochastic-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stochastic Analysis</a></li>
</ul>
</span>
Mon, 20 May 2019 08:40:10 +0000MarFenoy49364 at https://euro-math-soc.euEssential Discrete Mathematics for Computer Science
https://euro-math-soc.eu/review/essential-discrete-mathematics-computer-science-0
<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>When I accepted to write this review, I did not pay much attention to the endorsements appearing in the back cover, because after all they look simply as ads trying to sell the book. Now, after reading the whole book, I looked back at them and I must say that I fully agree with their statements, so I also want to convince all of you to get this book, read and enjoy it as much as I did in the past weeks. </p>
<p>The authors have managed to pack in about 400 pages all the material that I would expect in a discrete mathematics textbook at this level, including all of the following:</p>
<p>- Proof techniques and induction,<br />
- Sets, relations, and functions,<br />
- Propositional and quantificational logic,<br />
- Directed and undirected graphs,<br />
- Finite automata and regular languages,<br />
- Order notation and complexity of algorithms,<br />
- Basic combinatorics,<br />
- Recurrence relations,<br />
- Discrete probability, and<br />
- Modular arithmetic and public key cryptography.</p>
<p>Since other textbooks cover similar material in more than one thousand denser pages, it is clear that the treatment cannot be so exhaustive and some other things have to be excluded (for example, the number of proposed exercises in each chapter of the reviewed textbook is smaller than in other textbooks on the same subject), and hence the contents faithfully correspond to the “essential” idea in the title. The book provides a comprehensive introduction to really all the essential concepts that a student of Computer Science must learn in order to understand the foundations of the discipline. </p>
<p>The textbook is written as to be self contained, requiring little algebra and calculus in the way of prerequisites, but this does not mean that it forgets about mathematical rigor; on the contrary, it begins by introducing proof techniques and then emphasizes mathematical reasoning throughout the whole development (as the authors insist at the beginning, almost all results are proved in detail), together with enough intuition to make the explanations very clear. This is also accomplished by including computing motivations which, in addition to helping the student to understand the concepts, provide a global view of the subject that otherwise can be seen as a set of disconnected pieces. Furthermore, the text is very well written and I have only found three errata and a misplaced definition in 400 pages, which is quite difficult to accomplish. The first author is already very well-known for, among other things, excellent textbooks such as “Elements of the Theory of Computation” with C. Papadimitriou, but here he has put at work all his experience combined with the software engineering expertise of the second author, and both of them together have managed to produce a remarkable textbook that is a pleasure to read; for this reason, it is also perfect for self-study at any level. </p>
<p>Although the contents of this textbook do not fit exactly the syllabus of the discrete mathematics course in my school (where more logic is included in this course at the same time that probability is taught in another mathematics course, while automata and algorithm complexity deserve a separate course each), I have already told my colleagues teaching the discrete mathematics course that this textbook should be included as supplementary material, and at the same time I have ordered some copies for our library. Because, as I said at the beginning of this review, I want to share with everybody my enjoyment of this excellent textbook.</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"> </div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/harry-lewis-and-rachel-zax" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Harry Lewis and Rachel Zax</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">9780691179292</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">US$ 75.00</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">408</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/13651.html" title="Link to web page">https://press.princeton.edu/titles/13651.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><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="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68rxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Rxx</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68r01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68R01</a></li>
</ul>
</span>
Wed, 15 May 2019 15:32:26 +0000narciso49349 at https://euro-math-soc.euFrom Strange Simplicity to Complex Familiarity
https://euro-math-soc.eu/review/strange-simplicity-complex-familiarity
<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>
By now science has found some partial explanations about when our cosmos started and how it evolves, and so the next fundamental question to answer is: How did life come about here on Earth, and what is life anyway? We have come a long way in answering the cosmological questions by a better understanding of the laws of physics. Mathematics has certainly been an excellent tool in this exploration. So if life started in a world governed by these laws of physics, then there should be also laws of physics that explain how it emerged in an environment without life. We are still a long way from solving that mystery, but living cells have been analysed up to a nanoscale, and we know already a lot, but the enormous complexity of the chemistry of life and how it seemingly counteracts the second law of thermodynamics by (re)producing well structured organisms, makes us wonder how the laws of physics apply in these situations. Manfred Eigen, winner of the 1967 Nobel Prize for Chemistry, has collected in this book his vision on this matter, which ranges from the simplicity of the most elementary (sub)atomic particles to the complexity of living cells. The book was originally published in 2013, but since Eigen passed away early 2019, Oxford University Press reprinted the book in paperback.</p>
<p>
What Eigen explains is an interdisciplinary approach (requiring physics, chemistry, biology, cosmology) and it is all founded on mathematical principles. So there is a lot of mathematics involved, but he tells a story for everyone. This means that he avoids the necessity to understand the technical details ofr the mathematical (and chemical) formulas while it is still perfectly possible to follow the story. In fact there are not that many formulas. Instead he provides a lot of illustrations and background speckled with some personal anecdotes. Still, there are some framed Vignettes that provide a summary of more technical or mathematical details but that can be easily skipped if the reader is not interested. More extensive technical discussions are deferred to several appendices. Each chapter has also a long list of references for further study. These are obviously from before 2013, so that the most recent publications are not included. But Eigen is rather 'state of the art' for 2013 concerning his own domain which concerns the processes in physical chemistry of living organisms. He has tried to include some of the (in 2013) more recent insights which requires him to dig somewhat deeper into the material in the last two chapters. The technical details are however not the most important ingredient of the book. He just wants to provide in his story some answers to the questions that everybody asks sooner or later: what is life, where does it come from, and how does it continue, and will it evolve? And the story to tell is a long one. It is compiled in five chapters of ten sections each. The title of each of these sections is a question that he wants to answer. Some of the questions seem trivial to answer but when digging deeper into the foundations, the answer turns out not to be so easy.</p>
<p>
The first chapter is dealing with matter and energy. Since Einstein, we know that mass and energy are related, but still, this concept is not easy to grasp because mass is something that can be touched while energy is not. How and where is the transition? What do we touch when we touch something material? If we go to the finest nanoscale, it becomes more and more fuzzy where mass ends and energy starts. We can discuss subnuclear particles, but how far can we subdivide matter? Is our universe discrete or eventually continuous? What is the ultimately smallest quantity? Questions to be discussed with relativity theory and quantum mechanics. Since the smallest and the largest are related by mathematical inversion, some answers are sought in cosmology, and it sparkles the hope for an eventual theory of everything. The Big Bang is a one-time-only experiment that we cannot repeat in a laboratory. Our observations of the remainders can learn us something about the processes happening at the smallest scales in extreme conditions. Thus the cosmos approaching infinity can be related to the ultimate small that approaches zero, which basically means that zero remains as unreachable as infinity. Mathematically the reader is introduced in this chapter to four (and higher) dimensional spaces, symmetry and groups, and the Maxwell equations. Eigen takes his time to give the reader at least a general idea of these concepts.</p>
<p>
In the second chapter about energy and entropy, Eigen relates the microstate of particles to the macrostate of the whole system. Does a system eventually evolve to a steady state, and is then the system in perfect balance? Entropy is something that we can observe in our everyday life, and it is the central theme of this chapter: the different historical definitions of Clausius, the Gibbs paradox, Pauli's scaling, and of course thermodynamics, the Carnot cycle, and Boltzmann. How does entropy relate to `order' and `time', which is what most people think of when entropy is mentioned. The mathematics required in this chapter is mainly probability theory and distributions.</p>
<p>
But entropy is also related to information as Shannon has introduced it. This relation is the subject of chapter three. Here the reader is confronted with bit strings, coding theory, Hamming space, Fermat's little theorem, cryptography, and Markov processes. In the end even Gödel's theorems and a Turing machine are discussed when it is investigated how much information there is available in mathematics. How much information is contained in our genes and how much of this information does evolution pass on to next generations. Can we measure information and is this then `temperature'? An example of the puns that Eigen hides in the text, is the title of one section of this chapter reading `How far is it from Shannon to Darwin?' Of course this refers to the relation between information and evolution theory, but he starts by answering the question with 14288 kilometers, which is the distance from Shannon airport (Ireland) to Darwin's seaport (Australia).</p>
<p>
Evolution seems to be only possible because of the enormous complexity of the information that is provided by chemistry. This complexity of the information is the subject of chapter four. The complexity of an enormous universe that a cosmologist is faced with is nothing compared to the complexity that a chemist needs to face concerning the formation of different molecules. So how does nature handle this complexity? That is where the different levels of complexity are defined, culminating into the classics like the P = NP problem, the travelling salesman problem and the towers of Hanoi. This kind of mathematics is required to describe the process of evolution and also how genes will pass on information. Since there can be errors, this may cause mutations. One of his own contributions in this context is the generalization of bit-sequences in Hamming space. He described the kinetics of a self-organisation of replicating quaternary sequences by considering them as high-dimensional hypercubes, evolving in a Hilbert space. Scattering, random walks, diffusion processes, mutation, bifurcation, and fractals, they all enter the discussion when Eigen explains how evolution can take place.</p>
<p>
The last chapter, also the most extensive one, is about complexity and self-organisation. In other words, the chapter closest to the chemical dynamics of life. As in the previous chapter, we come closer to Eigen's own expertise and hence more physical chemistry is involved. To answer the question `What is life?' is not simple. Clearly, it is not a matter of structure, but it is rather a functional matter of how the cells are organized in their collaboration. Moreover for life to emerge, there must be some physical law that controls the complexity of the system. Even the simplest life form demonstrates an overwhelming complexity. Local approximations can be described by a system of linear differential equations. Mutation can take place during the exponential growth until some saturation occurs. The replica that will survive in this evolution process are the ones with the largest eigenvalue. Non-linearities are introduced by feedback loops causing hypercycles in autocatalytic systems when a cell is invaded by a virus. Errors occur in the replication process within a certain threshold and at some instant in the process that can lead to phase transitions This phase transition has to be understood in information space, not in the structure of the replica. The autocatalytic reaction and phase transition are important concepts in population dynamics that define evolution in a Darwinian sense. Attempts have been made to construct laboratory machines to simulate these chemical dynamics but they are still easily outperformed by nature. Eigen compares the evolution of life with the cosmological evolution of matter starting at the Big Bang and that eventually collapses into a black hole.</p>
<p>
This book is a plea for a fusion of different science disciplines instead of the diverging set of isolated specialisations where scientists work in their own niche, often unaware of what even nearby peers are working on. That is not how science evolves, and that is not how our genes work in their own evolutionary dynamical process of replication. And it is nice to see how almost all of these processes are underpinned at least approximately by mathematical equations but that the complexity involved is so tremendous that an accurate simulation of all these processes is far beyond the horizon.</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>
Eigen explains how things evolve in our universe, from the smallest particles to the most complex processes of living organisms. It is a marvellous interdisciplinary survey of how all sciences can collaborate to provide some insights and to partially answer some of the most fundamental questions that men can ask. It is nice to see how mathematics help to understand all the complex dynamics involved and how information in one form or another is the essential element that defines evolution when it is passed on from one generation to the next.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/manfred-eigen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Manfred Eigen</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2013</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">9780198570219 (hbk); 9780198841944 (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">£ 150.00 (hbk); £ 55.00 (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">754</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/from-strange-simplicity-to-complex-familiarity-9780198841944" title="Link to web page">https://global.oup.com/academic/product/from-strange-simplicity-to-complex-familiarity-9780198841944</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/92-biology-and-other-natural-sciences-behavioral-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92 Biology and other natural sciences, behavioral sciences</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/92-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68p30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P30</a></li>
<li class="field-item odd"><a href="/msc-full/81p45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P45</a></li>
<li class="field-item even"><a href="/msc-full/94-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94-01</a></li>
<li class="field-item odd"><a href="/msc-full/92cxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92Cxx</a></li>
</ul>
</span>
Thu, 09 May 2019 17:51:21 +0000adhemar49329 at https://euro-math-soc.euEuler's pioneering equation
https://euro-math-soc.eu/review/eulers-pioneering-equation
<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>Since The Mathematical Intelligencer conducted a poll in 1988 about which was the most beautiful among twenty-four theorems. Euler's equation $e^{i\pi}+1=0$ or $e^{iπ}=−1$ turned out to be the winner, and that is still today largely accepted among mathematicians. Even among physicists this is true. In a similar poll from 2004, it came out second after Maxwell's equations. The subtitle of this book is therefore <em>The most beautiful theorem in mathematics</em>.</p>
<p>This may immediately raise some controversy, not about the choice of the formula, but perhaps about what it should be called: a theorem, an identity, an equality, a formula, an equation,... A theorem or a formula applies but these are quite general terms. The others refer to formulas with an equal sign. The term identity assumes that there is a variable involved and that the formula holds whatever the value of that variable. That applies to Euler's identity, which is the related formula $e^{ix}=\cos(x)+i\sin(x)$. The previous formula appears as a special case of this identity. Wilson calls the former formula and "equation" but the reader with some affinity to the French language would probably prefer to call it an equality because the French équation means it has to be solved for an unknown variable. But all the previous names have been used interchangeably to indicate the formula. Calling it <em>Euler's identity</em> may not be the most correct but it is probably the most common terminology.</p>
<p>Whatever it is called, the description, if not <em>most beautiful</em>, then certainly the qualification <em>most important</em> or <em>most remarkable</em>, would be well deserved. It involves five fundamental mathematical constants: 1,0,π,e, and i in one simple relation. The 1 generates the counting numbers. The zero took a while to be accepted as a number but also negative numbers were initially considered to be exotic. Rational numbers were showing up naturally in computations, but so did numbers like √2 and π. These required an extension of the rationals with algebraic irrationals like √2 and the transcendentals like π which results in the reals that include all of them. The constant e (notation by Euler) relates to logarithms and its inverse the exponential function growing faster than any polynomial. Finally the imaginary constant i = √-1 (which is another notation introduced by Euler) was needed to solve any quadratic equation. This i allowed to introduce the complex numbers so that the fundamental theorem of algebra could be proved. The exponential and complex exponential are essential in applied mathematics. Euler's identity is most remarkable because it relates exponential growth or decay of the real exponential, and the oscillating behaviour of sines and cosines in the complex case.</p>
<p>All these links allow Wilson to tell many stories about mathematics that are usually discussed in books popularizing mathematics for the lay reader. There are indeed five chapters whose titles are the five previous constants and a sixth one is about Euler's equation. He does this in a concise way. The amount of information compressed in only 150 pages is amazing. This doesn't mean that it is so dense that it becomes unreadable. Quite the opposite. Because there are no long drawn-out detours, the story becomes straightforward and understandable. For example the first chapter (only 17 pages including illustrations) introduces children's counting rhymes, compares the names for numbers in seven different languages, and compares number systems: Roman, Egyptian, Mesopotamian, Greek, Chinese, Mayan, and the Hindu-Arabic. The latter was popularized in the West by Fibonacci and Pacioli. There are many illustrations not only of the notation of these different numerals in this chapter, but there are in fact many other illustrations throughout the book. This does not increase the number of pages needlessly because a picture sometimes says more than a thousand words. There are no colour illustrations but colour is not relevant for what they represent.</p>
<p>This is not the first book on Euler's equation. For example Paul Nahin. <a target="_blank" href="/review/dr-eulers-fabulous-formula"><em>Dr. Euler's Fabulous Formula</em></a>, Princeton University Press (2006), which is a bit more mathematically advanced, and a more recent one by David Stipp. <em>A Most Elegant Equation</em>, Basic Books (2017), which has more info about the person Euler. In the current book Euler's name appears frequently but as a person he is largely absent. For most of the five constants, separate popularizing books have been written or they are discussed in a chapter of more general popular books about mathematics, too many to list them here. Wilson refers to some of them in an appendix with a short list of additional literature, conveniently listed by subject.</p>
<p>There is of course mathematics in this book. It would be weird if there wasn't. But there is nothing that should shy away a reader with a slight affinity for mathematics. Some of it can be skipped, but the exponential and trigonometric functions, series, and an occasional integral do appear. The more advanced definitions or computations, are put in one of the eleven grey-shaded boxes distributed throughout the book, so that skipping is easy. Most of the topics are placed in their historical context. For example, the history of the computation of π is well represented, and also the history of the logarithms as they were derived by Napier and Briggs and how they relate is nicely explained. There are some notes to explain how complex numbers can be generalised to quaternions and even octonions, and several examples from applied mathematics illustrate the meaning and relevance of the exponential function.</p>
<p>A minor glitch: Albert Girard (1595-1632) who was the first to have formulated the fundamental theorem of algebra, is called on page 116 a Flemish mathematician, which is strange because the man was born in France, but, as a religious refugee, moved to Leiden in what was then the Dutch Republic of the Netherlands. So I do not think that the characterization Flemish does apply here.</p>
<p>The book does not go deep into the subjects discussed, but I liked it because it is quite broad, touching upon so many mathematical subjects, mainly in their historical context, while readability remains most enjoyable notwithstanding its conciseness.</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 in which Wilson gives a popularizing account about the historical development of mathematics. His guidance is Euler's equality that connects five fundamental constants of mathematics: 1, 0, π, e, and i = √-1. Each of these is an incentive to discuss respectively different number systems, how counting extends to negative numbers and eventually the real numbers, the approximation and calculation of π, different logarithms, and complex numbers.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</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">9780198794929 (hbk); 9780198794936 (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">£ 14.99 (hbk); £ 9.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">176</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936" title="Link to web page">https://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
<li class="field-item odd"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li>
</ul>
</span>
Sun, 14 Apr 2019 07:22:45 +0000adhemar49288 at https://euro-math-soc.euCelestial Calculations
https://euro-math-soc.eu/review/celestial-calculations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>There are several popular science books available about our solar system, astronomy, and even cosmology. Most of them are descriptive. Obviously, whatever we know about astronomy and our solar system depends upon observations, and these have clearly increased drastically since we have satellites and other probes that do the observations from space. That requires however a thorough knowledge of how celestial objects move with respect to each other, while eventually, all the observations are collected on Earth at some particular time and place. It is not difficult to describe with a formula an elliptic trajectory of a planet in a two-body system, but to know at what time it will show above the horizon for an observer on Earth at some particular place and date, requires a careful transformation between different coordinate systems. If you are an amateur astronomer, and you are interested in doing these computations for yourself, this is the book that will teach you how to do that. It is not high precision rocket science, but you will get reasonably accurate results using the computational methods described in this book. Not that it requires highly advanced mathematics. The subtitle "A gentle introduction to computational astronomy" is spot on. All you need is some analytic geometry and trigonometry. The rest is conversion of units and transformations between coordinate systems.</p>
<p>First there is unit conversion, for example AU (astronomical unit which is the average distance Earth-Sun) versus kilometres and miles, but a time is even more disturbing. Conversion of a fractional number of hours into hours, minutes and seconds (HMS) is relatively simple but 24 hours also correspond to a rotation of 360 degrees, so time can also be measured in degrees, arcminutes, and arcseconds, (DMS) and degrees can be expressed as radians. Moreover the time of the day depends on the longitude position (defining the local mean time (LMT)) of the observer and there is daylight saving time (DST) for some countries. The sidereal time refers to our position with respect to the stars while the Earth rotates, which is of course important in astronomical calculations. On a larger time scale there is a calender problem defining the year (Julian vs. Gregorian calendar).</p>
<p>Next problem is the choice of a coordinate system. To define a location on Earth we are used to spherical coordinates with the origin at the centre, and the z-axis though the North Pole, (in the current epoch defined as the direction towards Polaris) and choosing a main meridian (Greenwich). The Earth's equator lies in the ecliptic plane. This system is similar to the celestial sphere (with its own North Pole —which is close to but different from Polaris because of precession— and its own meridian). For trajectories, we know since Kepler that we need elliptic coordinates depending on the anomaly of the ellipse. A planet circles the Sun on an elliptic trajectory increasing speed as it approaches the Sun in its perihelion and it is slowest when it is farthest away in the aphelion. Then one has to realize that the orbital plane of the planet (or any other celestial object) need not be the same as our ecliptic plane The galactic coordinate system refers to the larger scale where the equator plane corresponds to the average plane of our galaxy (the Milky Way). Minor corrections are required for parallax (like observing the Moon from different directions on Earth) and precession (the rotation axis of the Earth circling the celestial North Pole). All this requires careful transformations between the different space-time coordinates.</p>
<p>Equipped with all these formulas and computer algorithms, one can finally start to put them to good use to predict the time and the position of a phenomenon we want to observe. It should be possible to find the position of the Sun, Moon, stars and planets at a certain day and time for a particular place on Earth. For example will Venus rise above my horizon today, and if it does where and when will I see it? In fact the next chapters discuss star rising and star settings, and for our solar system, there is a more descriptive part discussing the Sun, the Moon, and the planets and other objects in our solar system. Man made satellites are like all other celestial objects, but differ in the sense that they are closer and relatively small with respect to Earth.</p>
<p>So one can compute for example your own time for Sunrise and Sunset, solstices and equinoxes, and the angular diameter of the Sun. For the Moon there are similarly formulas to define instances of Moonrise and Moonset, to compute the phases of the Moon, its distance from the Earth, and moments of solar eclipses. Similar computations can be done for all the planets of our solar system. A distinction has to be made for the ones closer to the Sun (interior planets) and those beyond the Earth (exterior planets). Although the basic laws are the same, our satellites need a special discussion. Because they are closer to us, a higher precision is needed, it makes really a difference whether the origin is at the center of the Earth or at the observer's position, their orbit changes much faster, and they are subject to gravity and therefore are regularly repositioned.</p>
<p>All this illustrates that astronomical calculations are not at all trivial. Fortunately the author has made the Java, Python and Visual Basic code available via github at <a target="_blank" href="https://celestialcalculations.github.io/">celestialcalculations.github.io</a>. There is also a chapter with references to books, websites, almanacs and star catalogs with some explanation on how to use them. The extensive glossary with short descriptions of terms and the detailed index is very useful if one is lost in the terminology.</p>
<p>It is amazing to realize how complicate computations are for relatively simple observations like for example the daily Sunrise and Sunset. Fortunately the computer code makes this quite easy. One can only be in awe for the calendars and almanacs produced by ancient civilisations without any of our modern insights or equipment. Now this book brings this within the reach of anyone who can deal with simple computer programs easily downloadable and ready to be installed and executed. This book is quite an achievement bringing all this within the reach of a general public. Not only the clear explanation of the technical mathematical background with formulas and graphs of the coordinate systems, but also for the very informative descriptions, illustrated with pictures, of the astronomical objects and phenomena.</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 book describes all the mathematics needed to compute positions at a certain time of the familiar celestial objects. In this sense, it is also a users manual for the computer code that is made available via github. On the other hand, it is also a description of the phenomena and objects in our solar system.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/jackie-l-laurence" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jackie L. Laurence</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">9780262536639 (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">&pound; 30.00 (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">392</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/celestial-calculations" title="Link to web page">https://mitpress.mit.edu/books/celestial-calculations</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/70f15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70F15</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li>
</ul>
</span>
Sun, 14 Apr 2019 07:00:31 +0000adhemar49287 at https://euro-math-soc.euEssential Discrete Mathematics for Computer Science
https://euro-math-soc.eu/review/essential-discrete-mathematics-computer-science
<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>One may wonder whether physical quantities are discrete or continuous. Calculus has certainly helped to model all kinds of physical phenomena, but when digging deep enough, we are confronted with quantum physical aspects that show a more discrete character of nature. When looking at a large scale, then it is generally assumed that our universe is immense but finite. So we might consider the physical world to be a finite and discrete system.</p>
<p>Digital computers are also by definition finite and discrete, but at a much smaller scale than the cosmological universe and coarser than the subatomic level, although modern basic hardware components using nanotechnology are scraping against an atomic scale. The software that runs on this hardware is essentially a sequence of bits describing how bits are transformed into bits, and hence also this is finite and discrete. So one may say that calculus formulates continuous models as approximation of a discrete reality and numerical analysis simulates these models on computers by producing a discrete approximation at a much coarser level of the continuous models.</p>
<p>To describe the behaviour of a computer, it is clear that it can be in many, but eventually, only a finite number of different states. No wonder that students in computer science will have to learn about discrete mathematics. This involves traditionally graphs, Boolean calculus, complexity, and sometimes also algebraic relations and structures like (finite) groups, rings, etc.</p>
<p>In the early days of computer science, it was the playground of engineers and mathematicians who simulated their numerical models. Therefore many of the "discrete mathematics courses for computer science" were given to mathematics or engineering students who wanted to specialize in computers. However computer science is now a mature topic in its own right, and students of computer science may not have the traditional mathematical background of previous generations. This book is taking this observation into account, and starts at a very elementary mathematical level, even explaining proof techniques, while introducing the first concepts of induction, sets, formal logic,... Although starting with a very elementary introduction, not all the chapters are at this introductory level, and at some point require the introduction of calculus elements like limits, infinite series, integrals, and de l'Hôpital's rule. The text is largely self-contained. All the required concepts and notations are defined, and all the statements are proved in full. A computer could be used occasionally to solve the exercises but one can perfectly do without.</p>
<p>The text has 31 relatively short chapters, each followed by an itemized summary and about a dozen exercises. Some of the chapters are related, like for example the ones on probability, but others can be skipped without a problem. The main global topics include formal logic, graphs, automata, and complexity. The pages have a wide outside margin where all the notes and illustrations are included.</p>
<p>To illustrate how the chapters are built up, let's look at the first one which is an illustration of the pigeonhole principle. A simple concept, and yet, it starts with ideas about a mathematical theorem which is all about formulating and proving a general proposition, rather than just a particular case, but it runs up to the fundamental theorem of arithmetic (every positive integer is a unique product of prime numbers) and the proof that there are infinitely many prime numbers. Further proof techniques are illustrated with other (elementary) results from number theory.</p>
<p>The book continues with sets, used to introduce relations and functions, and induction lead to countable sets, but also to strings generated from an alphabet using syntax rules. Venn diagrams are an introduction to propositional logic, Boolean algebra, and logic circuits.</p>
<p>Graphs are a major part and return in several chapters. They are first introduced by describing a computer model as a discrete state space machine that executes an algorithm. This is a strange choice to see a graph appear for the first time, but it may be motivating for computer science students. However, graphs are studied for their own right: directed and undirected graphs, connectivity, trees, and coloring. Previous topics are retraced with finite automata to construct and analyse strings.</p>
<p>Two mathematical intermezzos introduce limits, integrals, series, and big-O and small-o notation to define orders of magnitude and another one defines infinite summations and series expansions. These are prerequisites to deal with counting problems (combinatorics), discrete probabilities (including conditional probability and Bayes's theorem). Complexity theory might have been another application, but that is not included. The two chapters about modular arithmetic and some elements of cryptography conclude the book.</p>
<p>This is a relatively low level introduction to some elements of discrete mathematics for students in computer science with only a minimal mathematical background. It has a pleasant lay-out for studying, but is a bit overweight (bout 1.2 kg) to take it along. The number of exercises is limited but sufficient for the depth at which the subjects are treated. There are many text books that treat the same subjects (most of them at a higher level though), but there are not many that take this particular approach.</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 about discrete mathematics. It starts at a very elementary mathematical level explaining sets, relations and functions introducing the pigeon hole principle, proof techniques and induction, and continuing with principles of formal logic, Boolean algebra, graph theory, automata, combinatorics, and discrete probability.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/harry-lewis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Harry Lewis</a></li>
<li class="field-item odd"><a href="/author/rachel-zax" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Rachel Zax</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">9780691179292 (hbk); 9780691190617 (ebk)</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">USD 75.00 (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">408</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/13651.html" title="Link to web page">https://press.princeton.edu/titles/13651.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/combinatorics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Combinatorics</a></li>
<li class="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="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68rxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Rxx</a></li>
<li class="field-item odd"><a href="/msc-full/05-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-01</a></li>
</ul>
</span>
Sun, 14 Apr 2019 06:50:12 +0000adhemar49286 at https://euro-math-soc.euModern Fortran explained. Incorporating Fortran 2018
https://euro-math-soc.eu/review/modern-fortran-explained-incorporating-fortran-2018
<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"> </div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/Recension-Fortran_0.pdf" type="application/pdf; length=115654">Recension-Fortran.pdf</a></span></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/michael-metcalf" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael Metcalf</a></li>
<li class="field-item odd"><a href="/author/john-reid" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Reid</a></li>
<li class="field-item even"><a href="/author/malcolm-cohen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Malcolm Cohen</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-19-881189-3, 978-0-19-881188-6</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">522</div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li>
</ul>
</span>
Wed, 20 Mar 2019 11:52:09 +0000Angel Felipe49214 at https://euro-math-soc.euThe Mathematical World of Charles L. Dodgson (Lewis Carroll)
https://euro-math-soc.eu/review/mathematical-world-charles-l-dodgson-lewis-carroll
<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>
Charles Lutwidge Dodgson (1832-1898) is the real name of Lewis Carroll, the author of <em>Alice's Adventures in Wonderland, Through the Looking Glass</em> and of other such books. He wrote these for Alice Liddell, the daughter of Henry Liddell, the dean of Christ Church in Oxford and he became forever famous as `the man who wrote <em>Alice</em>' and for a long time this was the only way he was remembered. Yet he was also a mathematician with very original ideas and a creditable photographer. These aspects were only properly recognized since the 1980s. His diaries and his collected publications were becoming available since the 1990s and with that material, more studies of his non-fiction work was possible. This book gives a survey of what is known so far.</p>
<p>
The book starts with a (short and in some respect only partial) biography, concentrating on the mathematician in him. Introduced by his father (a country parson) to mathematics as a young child, he excelled in this subject at school. Later, just like his father, he studied at Christ Church in Oxford and got a master degree in mathematics. While studying he gave private lessons to students. He was ordained a deacon when he was 29, but never became a priest. When Liddell became the dean of Christ Church, Dodgson was appointed as 'Master of the House' which gave him a reasonable income but also a large teaching load. He starts publishing his pamphlets as aids for teaching, and some work on the evaluation of determinants. He took on photographing as a hobby. He became rather good at it, using it as an art form, rather than as just a way to catch reality. Urged by Alice Liddell to write up the stories he told during their boat trips, he started working on <em>Alice's Adventures</em> which was published under his pen name Lewis Carroll. Teaching was his main occupation besides his writing and the photography. He became (reluctantly) Curator of the Christ Church Common Room for ten years when he was 50. This was a time-consuming burden requiring management and many decisions to take. In that context he could use his already existing interest in voting systems. He died a week before his 66th birthday.</p>
<p>
The next chapters are written by specialists and discuss in more detail Dodgson's contributions to different mathematical subjects. The first one deals with geometry. Dodgson was teaching geometry following Euclid's books as it was usual in those days. However there were some new ideas, among others by Sylvester, criticizing Euclid's approach. Euclid's arguments were not always waterproof in a mathematical sense, and there was the emerging hyperbolic geometry of Lobachevsky. Dodgson was defending however Euclid in his booklet <em>Euclid and his Modern Rivals</em>. The discussion about the parallel postulate involved either infinitely long lines or infinitely small quantities. So he tried to replace it by a finite alternative using an "obvious" property about areas outside and inside an hexagon inscribed in a circle. He thought of non-Euclidean geometry as nonsensical.</p>
<p>
In the next chapter on algebra, it is explained what his condensation method for the evaluation of determinants is. This may be his most useful original contribution to mathematics. He was also opposed to the name 'matrix' with the meaning we give it today. He called it a 'block' because a matrix refers to the mould, rather than the object, which is the mould filled with numbers. To denote the elements in a matrix, which we denote by a letter with two indices, he had his own strange notation. His work on determinants was published but (like many of his mathematical publications) remained largely unnoticed for a long time.</p>
<p>
Logic has been one of Dodgson's favourite subjects and he wrote several texts about it. In those days, as we still have today, there existed several proposals for an approach to, and the notation of, formal logic. The one from Boole was, and still is, a very useful one. Dodgson adhered the formal approach which was not easily accepted by classical logicians, and required a lot of dispute. He developed several tools to deal with logic problems: a method with diagrams, he had his own formal notation, a method of trees, and an algorithm to solve syllogisms. Bertrand Russell once said about Dodgson' work that it was brilliant but largely useless.</p>
<p>
Dodgson also wrote several texts on voting systems. The problem of cycles (where each candidate can win from the next candidate in a cyclic way) was known several centuries before, but Dodgson was probably not aware of that. He detected it on his own and made proposals for a correct voting system, for assigning seats to parties in a political election, or for a correct outcome of a tournament. He wrote about these problems in terms of game theory, an approach that John Nash would bring to a culmination only much later.</p>
<p>
It is clear that the author of the <em>Alice</em> books would also be interested in recreational mathematics, puzzles, riddles, and games. Many examples are discussed as well the way in which Dodgson solved them. He also had techniques to remember dates and numerical data, and techniques to check divisibility which could be smuggled into a number game.</p>
<p>
It is strange that Dodgson was so little recognized for his mathematical work. Most of his, sometimes original, approaches were only discovered at the end of the 20th century. He was not the best salesman for his results. Perhaps he didn't take it seriously enough, and maybe he was too quarrelsome (sarcasm indeed happened sometimes), or perhaps he was rather obscure when sticking to his own notation and ideas. He had for example his own symbolic notation for the trigonometric functions. He also was stammering a bit, but that did not seem to be a serious hinder to him. Moreover his contributions are very diverse. He did not have a single field in which he became the renowned expert and finally perhaps he was also in the shadow of Lewis Carroll. The legend goes that Queen Victoria, charmed by reading <em>Alice's Adventures in Wonderland</em>, asked for his next book and promptly received his <em>An Elementary Treatise on Determinants</em>. This story is however a hoax. He may not have received during his life the recognition by mathematicians he deserved, but nevertheless this book has an extensive chapter discussing his mathematical legacy in geometry, trigonometry, algebra, logic, voting, probability, and cryptology as it became clear only recently.</p>
<p>
The book ends with a complete bibliography listing all his publications and a list of references for additional reading. With that this book completes a survey of the life and work of a (perhaps somewhat dull) mathematician that is in many ways the opposite (but in as many ways also the complement) of the light-hearted Lewis Carroll that authored the books witnessing of such a rich fantasy dedicated to a little girl called Alice. A man well known as Lewis Carroll, yet perhaps too long underestimated as Charles Dodgson. This nicely edited book with many illustrations and written by experts on the subject will certainly help to turn the tide by adjusting the image and allow you to form a proper opinion about Charles Dodgson.</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 book starts with a biography of Charles Dodgson, the mathematician, best known for his books like <em>Alice's Adventures in Wonderland</em> under his pen name Lewis Carroll. Its main purpose however is to discuss his work and influence as a mathematician.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li>
<li class="field-item odd"><a href="/author/amirouche-moktefi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amirouche Moktefi</a></li>
<li class="field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">9780198817000 (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">£29.99</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">288</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/the-mathematical-world-of-charles-l-dodgson-lewis-carroll-9780198817000" title="Link to web page">https://global.oup.com/academic/product/the-mathematical-world-of-charles-l-dodgson-lewis-carroll-9780198817000</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li>
</ul>
</span>
Wed, 20 Mar 2019 09:56:06 +0000adhemar49213 at https://euro-math-soc.eu99 Variations on a Proof
https://euro-math-soc.eu/review/99-variations-proof
<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>
From the title, you might think that the book is showing different proofs of a crucial theorem that has been reinvented many times and hence has been proved in many different ways. The Pythagoras theorem could be a good candidate with many published proofs, but still it would be hard to find 99 different ways. No, the author wants to illustrate that there are many different styles of communicating mathematics, in particular of giving a proof of some theorem. What the theorem is, is not essential here. The choice of the author is to prove some totally unimportant statement: If $x^3-6x^2+11x-6=2x-2$, then $x=1$ or $x=4$. The cubic equation has indeed a double root at 1 and a simple root at 4.</p>
<p>
The idea of the book is inspired by <em>Exercises de style</em> (1947) by Raymond Queneau, a member of the experimental writers group <em>Oulipo</em> (Ouvroir de littérature potentielle). What "styles" could one think of? There is certainly a tradition of doing mathematics that is inherited from historical mathematical centres in a country. There is definitely a difference between some papers written in a German or a French tradition. When looking at history, then proofs by Euclid or by Newton are certainly different and they are not at all like computer (assisted) proofs. Digging into history along this line of ideas will result in many proofs of the same statement looking very differently. One would come a long way, but still, 99 variations is a lot. Another source for differentiating is the way things are represented: graphical (geometric or other), colours, prefix or postfix notation, hand waving, oral discussion, e-mails, encrypted, sign language,... and there are some fun variations. In this way Ording arrives at 99 proofs. In fact, there is also a proof numbered 0, which has the statement of the problem with a proof omitted, which is indeed a way in which a lemma is often formulated in current mathematical papers. This is nicely represented by the cover of the book which has a ten by ten square grid of black squares, except for the first one which is white. The grid fills almost the whole cover which is possible since the shape of the book is almost square too.</p>
<p>
Most of the proofs are short and take only a few lines. On the back of the page, Ording gives some comments, for example where he got his (historical) inspiration or how to read the proof (for geometric constructions) or explaining the notation, the symbols, or the language, etc. For example the psychedelic proof is just a black-and-white fractal plot of the attraction basins of Newton's method. Such a "proof" obviously requires some explanation. These backside notes give also cross references to related proofs in the book. Exceptionally a proof takes more than one page. The comments by Ording turn the book into a fragmented survey of many historical mathematical factoids. Even if some of the proofs are abstract and incomprehensible for the layman, it can still be considered a popular science book about the peculiarities and trivia of mathematics. We meet proofs as dialogues formatted after one of the oldest Chinese texts on mathematics, but also a dialogue as it would develop at a tea party in a mathematics institute, or a proof can take the form of a screenplay featuring Cardano and Tartaglia, etc. There is a proof in the form of a parody of the collaborative discussion modelled after the Polymath projects by Yitang Zhang and Terrence Tao to work on the twin prime conjecture. Fermi was famous for his proofs scribbled "on the back of an envelope" and that one is represented too with a hand written proof on the back of the page. There are proofs as given during an exam, or during a seminar, or on a blackboard, proofs discussed in a referee report, in a patent application, in blogs, in a newspaper article, preprints on arXiv, a tweet by Cardano (if he would have been able to twitter), ... Among the more surprising ones are the ones using a music score, colour spectra, origami, a slide rule,... In fact most of them are surprising and/or amusing.</p>
<p>
The book can be safely considered as one produced by constrained writing as an oulipo author would produce it, and indeed as Queneau did in 1947, not for mathematical proofs, but for some surrealistic short story. The influence of constrained writing is for example obvious in a verbal proof using only monosyllabic words. There are many more hidden trouvailles like the proof called "Ancient" (in Babylonian cuneiform signs) as opposed to the one called "Modern" (with high level of abstraction). These are not placed next to each other but the first is numbered 16 and the second gets the opposite number 61. It must also have been great fun inventing parodies for styles like doggerel, paranoid, mystical, or social media. But it is more than just fooling around. There is always some rationale behind the way it is presented.</p>
<p>
I love this book. It is so much fun to read, and there are many double layers to be discovered. It is temping to invent some extra variations of your own. It is so much more than a book about mathematics. It is indeed creative writing under constraints. I will pick it up regularly and I am sure there will be more hidden gems to be unravelled that are easily missed in a first reading.</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 in using different mathematical styles to prove a theorem about the solution of a particular (otherwise not important) cubic equation. It is inspired by, and written in the style of, the constrained writing embraced by the French oulipo writers group, in particular the book <em>Exercises de style</em> (1947) by Raymond Queneau was a main source of inspiration.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/philip-ording" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Philip Ording</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-15883-9 (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">24.95 USD</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">272</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/13308.html" title="Link to web page">https://press.princeton.edu/titles/13308.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li>
<li class="field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
</ul>
</span>
Wed, 20 Mar 2019 09:50:58 +0000adhemar49212 at https://euro-math-soc.euImagine Math 6
https://euro-math-soc.eu/review/imagine-math-6
<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>
Springer published the 6 volumes of the preceding series <em>Mathematics and Culture</em> and volumes 1-3 of the current <em>Imagine Math</em> series. After an interrupt for the two volumes <em>Imagine Math</em> 4 and 5, the current book is again a Springer publication. Several of these volumes were reviewed here (just search for <a href="https://euro-math-soc.eu/search/node/Emmer">Emmer</a>). The books contain diverse contributions that reflect the presentations at a recurrent conference that wants to bring together scientists, mathematicians, and artists from different disciplines to discuss the influence of mathematics in our culture in the broadest possible sense. That means that the selection of contributions for this book is again showing a very broad plethora of papers ranging in style from proper mathematical to philosophical and from historical to a collection of pictures or everything else that is considered appropriate. Imagine Math 6 records the proceedings of the conference organized in Venice, Italy, in 2017. It has 24 papers grouped into 8 parts, although the boundaries between the parts are not very strict. What follows is a brief glossary illustrating the diversity of topics.</p>
<p>
The first part is about the Iraqi architect Zaha Hadid (1950-2016) who created many remarkably fluid and jaunty buildings and constructions. Her ideas are continued by the company that still bears her name. The book contains many pictures illustrating her designs. Some topological and fluid dynamical aspects of these shapes are discussed by M. Emmer.</p>
<p>
The second part is about Mathematics and the Media. Mathematical awareness is raised by expositions in the MUSE, the science museum in Trento, and the efforts of Frank Morgan, editor of the Notices of the AMS convincing the authors-mathematicians of the Notices to bring mathematics at a level acceptable for readers that are not mathematicians.<br />
Osmo Pekonen is a Finish mathematician, who is also historian of science. He studied the life of Maupertuis, an 18th century French mathematician who was involved in an expedition measuring by triangulation part of an Earth's meridian in Lapland. The goal was to decide on the shape of the Earth (flattened at the poles or at the equator). Pekonen became so involved that he himself impersonated, on stage and in a film, the character of Maupertuis.</p>
<p>
Music and mathematics have been connected since antiquity. So no wonder there is a part about this involvement. It is illustrated how algebra, topology, category theory, chaos theory, prime numbers, palindromes, etc. can be detected in music styles and scores.</p>
<p>
In the part about applications, we learn that analysis of big data can lead to prediction of earthquakes or crimes.<br />
The proof that by summing the divergent series of all the positive integers can lead to the surprising Ramanujan result -1/12 has been an internet hype for a while. It is used as a pretext to mention similar such cases of paradoxical results.<br />
There is a paper that shows how computer analysis helped to reconstruct a parametrized virtual version of the Arch of Titus (1st-century CE) using some stones found at the archaeological site of the Circus Maximus in Rome.<br />
The mathematical analysis of soap bubbles clusters is another example of semi-applied mathematics.<br />
On a more linguistic level we find an interesting contribution discussing the fact that the formal language of mathematics is only used among mathematicians, and hence is not used by the common people. Even the relatively elementary mathematics of antiquity was essentially for an elite. Common people did not have or were not able to read books. And yet, occasionally, some of the mathematics or mathematical terminology has entered common social language not only today but also in previous centuries.</p>
<p>
Part 5 about visual mathematics, could also be part of the set of applied mathematics papers. The three contributions here deal with design problems: how design students express the concept of "balance" in their projects, how a numerical model is derived from visual information that can be used to simulate the heart function, and a third paper is on the design of gears, with not only the traditional circular ones but also the elliptic and polygonal ones, some nautilus shapes, and even amusingly weird ones.</p>
<p>
Mathematics and art is of course an obligatory part in this kind of books.<br />
One paper is about non-convex star-like (2D and 3D) mathematical objects throughout history.<br />
Another describes the role of the pentagram in the art of the Dutch sculptor Gerard Caris.<br />
And a last one is from the photographer Vincent Moncorgé presenting his project in which he wants to capture with his photos some real-life mathematicians at work in their natural habitat, avoiding the characteristic formulas. This is quite interesting since it may remove the biased public opinion about a mathematician as a weird and unworldly individual.</p>
<p>
Mathematics and physics is the title of the penultimate part. Topology and physics in fluid dynamics has some historical roots (the dichotomy between continuous and discrete), but that is of course also of major importance for quantum mechanics. So these two (fluid dynamics and quantum theory) are the two somewhat related topics of this part.</p>
<p>
The last part is about mathematics and physics. Emmer discusses the book from 2015 <em>Éloge des mathématiques</em> reflecting a dialogue between the French philosopher Alain Badiou, and a journalist Gilles Haéri (also available in English <em>In Praise of Mathematics</em> 2016).<br />
There is a paper showing how Luca Pacioli (447–1517) was influenced by the work of Euclid and another one is computing the dimensions of the heaven, the hell and the purgatory as described in the <em>Divina Comedia</em> of Dante.<br />
The first chapter of Simon Singh's book <em>The Simpsons and their Mathematical Secrets</em> is reprinted.<br />
And finally the work of the Scottish science writer Mary Sommerville (1789-1872) is discussed.</p>
<p>
This brief survey should convincingly illustrate that mathematics has deep roots into our culture. Some more technical contributions may be hard to follow for some readers if not trained well in mathematics, and other readers may be a bit intimidated by the philosophical jargon used in the more contemplating contributions, but the book is broadly readable in general. There are in this volume remarkably many colour pictures included. If you liked the previous books in the series, you will love this too. If you don't know any of the previous books, this volume might be a fine occasion for jacking up your cultural backpack.</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>
These are the proceedings of the Imagine Math conference held in Venice, Italy in 2017. In this recurrent series of conferences the participants discuss the role of mathematics, theoretical and applied, as it is strongly embedded in our culture and how it is intimately related to many art forms in several, sometimes unexpected, ways.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/michele-emmer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michele Emmer</a></li>
<li class="field-item odd"><a href="/author/marco-abate" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marco Abate</a></li>
<li class="field-item even"><a href="/author/eds-2" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-international-publisher" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publisher</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">2018</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">978-3-319-93948-3 (hbk); 978-3-319-93949-0 (ebk)</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">68.64 € (hbk); 55.92 € (ebk)</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">340</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.springer.com/gp/book/9783319939483" title="Link to web page">https://www.springer.com/gp/book/9783319939483</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a66-00a67-00a30-00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A66 00A67 00A30 00B15</a></li>
</ul>
</span>
Wed, 20 Mar 2019 09:43:30 +0000adhemar49211 at https://euro-math-soc.eu