Book reviews
https://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenThe Master Equation and the Convergence Problem in Mean Field Games
https://euro-math-soc.eu/review/master-equation-and-convergence-problem-mean-field-games
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In Mean Field Theory (MFT) one studies models for decision making in a large population of $N$ interaction agents. The agents are supposed to have only marginal impact so that the behaviour of one (average) agent is determined by its own state and by a distribution that describes the states of all the others while these are varying in time. The states of the agents are thus described by a stochastic differential equation (SDE) according to which the system evolves in time to some steady equilibrium state. The solution of this optimal control problem is described by a continuous Hamilton-Jacobi-Bellman equation expressing necessary and sufficient conditions for optimality of a value function that at every instant of time determines the optimal behaviour of an agent, depending on the state of the system. Thus the mathematical disciplines involved are mainly control theory and (stochastic) differential equations. Mean Field Games (MFGs) thus assume actually a continuous set of infinitesimal agents rather than a large set of discrete players. Since around 2006-2007, where the framework was properly formulated the subject has boomed. These models were originally introduced because of their applications in economics but later also many other applications in engineering and informatics emerged, and it became an applied mathematical topic that deserves further analysis in its own right, independent of the concrete application.</p>
<p>
In a finite Nash differential game (NDG), the state variables are also functions of a continuous time, but there is a discrete set of $N$ players. If $N$ is not too large, then such an $N$-Nash system has been investigated and its Nash equilibrium is well understood. To find an equilibrium state, there is a set of coupled differential equations that describes the time evolution of the individual value functions for the $N$ agents. The value function of an agent evolves in time depending on the time-varying states of all the agents in the system and this defines the instantaneous behaviour of the agent. Given the value functions at some horizon time $T$ for all the agents, the system has to be solved backward in time, and once these value functions for the individual players are known, one can compute their individual trajectories forward in time, where there is some individual noise added for every player as well as some global common noise for the whole system. This is how an $N$-Nash differential game is solved. One would hope that the equilibrium of an <$N$-Nash system tends to an equilibrium for the corresponding MFG with a continuum of players as $N$ tends to infinity. The purpose of this book is to prove that under appropriate conditions, in some sense, this is true for a large class of MFGs,</p>
<p>
Instead of analysing the complicated Nash system with $N$ really large, the authors use as their main approach the <em>Master Equation</em> (ME), which is an MFT concept imported from systems in physics and chemistry. This ME describes the expected asymptotic behaviour, and thus avoid the complexity of the huge Nash game. This reduces the system of infinitely many differential equations to just one stochastic differential equation whose solution is a trajectory for some average value function $U$. This $U$ depends on time $t$ and the state of the system. According to MFT the state is split into the state $x$ (a $d$-dimensional vector) of the individual agent (because of symmetry it does not matter which one) and a distribution $m$ characterizing some average state of all the other players. It is then proved that under appropriate conditions the value function $v$ of an individual player (any player) of an $N$-player differential Nash-system converges as $N\to\infty$ to the equilibrium solution provided by the Master Equation at a rate of $1/N$. Also the state trajectory $X$ of any player convergences in a probabilistic sense like $1/N^{\frac{1}{d+8}}$ to the associated asymptotic trajectory, (solutions of the McKean-Vlasov SDEs using the $U$ previously obtained).</p>
<p>
This project started with the intenton of writing a paper, but by making it somewhat self-contained and because of the generality of the result, it grew out into a book. Here "self-contained" needs to be understood as "for someone familiar with Nash systems" since it requires some knowledge of the subject that is silently assumed. If this knowledge is not present, then some extra reading of the cited references will be required to understand the details. For example, the first chapter is a rather extensive introduction to the problem and the concepts used, it gives a summary of the results to be proved and surveys how this is structured in subsequent chapters with some guidelines on what to read, depending on the knowledge and the interest of the reader. Equation (1.2) and (1.3) on page 4 describe a Nash system as</p>
<p>
\begin{eqnarray*}<br />
&&-\partial_t v^{N,i}(t,\boldsymbol{x})-\sum_{j=1}^N\Delta_{x_j} v^{N,i}(t,\boldsymbol{x})-<br />
\beta \sum_{j,k=1}^N \mathrm{Tr} D^2_{x_j,x_k} v^{N,i}(t,\boldsymbol{x})\\<br />
&&+H(x_i,D_{x_i},D_{x_i}v^{N,i}(t,\boldsymbol{x}))+<br />
\sum_{j\ne i} D_p H(x_j,D_{x_j}v^{N,i}(t,\boldsymbol{x}))\cdot D_{x_j} v^{N,i}(t,\boldsymbol{x})=F^{N,i}(\boldsymbol{x}),~~~~(t,\boldsymbol{x})\in[0,T]\times (\mathbb{R}^d)^N,\\<br />
&&v^{N,i}(T,\boldsymbol{x})=G^{N,i}(\boldsymbol{x}),~~~\boldsymbol{x}\in(\mathbb{R}^d)^N,~~~i\in\{1,\ldots,N\},<br />
\end{eqnarray*}<br />
and the individual trajectories of the agents by<br />
\[<br />
dX_{i,t}=-D_p H(X_{i,t},Dv^{N,i}(t,\boldsymbol{X}_t))dt+\sqrt{2} dB_t^i+\sqrt{2\beta}dW_t,~~t\in[0,T],~~i\in\{1,\ldots,N\}.<br />
\]</p>
<p>
It is explained that the first system is the Nash system considered with $v^{N,i}$ the unknown value functions depending on $\boldsymbol{x}=(x_1,\ldots,x_N)$, $H$ is the Hamiltonian, $\beta\ge0$ is a parameter, and $T\ge0$ is the time horizon. The second describes the optimal trajectories $X_{i,t}$ of the states of the players, with $B_t^i$ individual noise, and $W_t$ some common noise (both are Brownian motions). This is about all the explanation given, so that a reader unfamiliar with the subject will have some problem already on page 4. Although the statement is more fully introduced in chapter 2, and there is an extra appendix with explanation about derivatives with respect to random variables, there is no further explanation about the notation <$D_p$ or $\Delta_{x_i}$ or about an expression for the Hamiltonian (except that it is related to the cost that agent $i$ has to pay). However, if one is familiar with the basics, then the introduction of the main results and the proofs are well explained. For the MFG system, and the Master Equation, similar expressions are introduced, except that the state is split up as $\boldsymbol{x}=(x,m)$ where $x$ is the state of an (infinitesimal) individual and $m$ the distribution of the state of all the others. Since $m=m(t)$ is time-varying, also its evolution over time has to be monitored separately. Thus we get a similar but somewhat different set of equations for the MFG and the ME. Since these require derivatives with respect to the distribution $m$, the appendix is needed to explain this concept in more detail. The ME is an essential element in this book, and although known in other fields, it can be slightly different as it is applied here. So care has been taken to explain it in the present situation also on an intuitive basis in the introductory chapter. This is helpful to assimilate the subsequent chapters.</p>
<p>
The analysis holds under several restrictions that are carefully explained with many links to the literature. For example boundary problems for $t\in[0,T]$ are avoided by assuming periodic (in time) solutions, $F$ and $G$ satisfy some monotonicity condition, $H$, $F$, and $G$ are supposed to be smooth enough. Some of these restrictions can be removed or generalized but they are maintained here mainly for simplicity. Although it is not explained, the current approach is potentially useful for numerical implementation. Also the full proof in chapter 3 of the existence of the equilibrium for the MFG assumes the first order case ($\beta=0$ and thus no common noise). The second order system ($\beta>0$ with common noise) is further explored in chapters 4 and the ME in chapter 5, with the eventual convergence proof in chapter 6.</p>
<p>
This book appears as a volume in the <em>Annals of Mathematics Studies</em> and it is a major contribution to the state of the art in MFGs which is a must read for researchers in the field. It seems like several preliminary versions of this text were previously made available on the Web. There are few typos, which is an achievement for a book with that many formulas. Even with that many formulas and technicalities, the book is still quite readable, because the authors use the book format (and not a more compact paper format) to explain all their steps carefully. Because of its structured approach, it could be used as a textbook for an advanced course on the subject.</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 authors give a proof that under certain rather general conditions the equilibrium solution for a Nash differential game with <em>N</em> players converges when <em>N</em> goes to infinity to the equilibrium solution of a mean field game (i.e., with a continuum of players). The approach taken is by introducing and analysing the so called Master Equation for the system. This basically reduces the system of <em>N</em> coupled differential equations to one stochastic differential equation describing the situation for an average player because the state of the system is characterized by the state of that average player while the state of all the others is described by a distribution.</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/pierre-cardaliaguet" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Pierre Cardaliaguet</a></li>
<li class="field-item odd"><a href="/author/fran%C3%A7ois-delarue" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">François Delarue</a></li>
<li class="field-item even"><a href="/author/jean-michel-lasry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean-Michel Lasry</a></li>
<li class="field-item odd"><a href="/author/and-pierre-louis-lion" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">and Pierre-Louis Lion</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">9780691190709 (hbk); 9780691190716 (pbk); 9780691193717 (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">£ 136.00 (hbk); £ 62.00 (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">224</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/books/hardcover/9780691190709/the-master-equation-and-the-convergence-problem-in-mean-field-games" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691190709/the-master-equation-and-the-convergence-problem-in-mean-field-games</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/control-theory-and-optimization" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Control Theory and Optimization</a></li>
<li class="field-item odd"><a href="/imu/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</a></li>
</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/91-game-theory-economics-social-and-behavioral-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91 Game theory, economics, social and 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/91a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91A80</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/49-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49-01</a></li>
<li class="field-item odd"><a href="/msc-full/49n75" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49n75</a></li>
<li class="field-item even"><a href="/msc-full/91h15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91H15</a></li>
</ul>
</span>
Mon, 21 Oct 2019 14:15:00 +0000adhemar49831 at https://euro-math-soc.euThomas Harriot
https://euro-math-soc.eu/review/thomas-harriot
<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>
As a young mathematician and astronomer, Thomas Harriot (c. 1560-1621), was hired by Sir Walter Ralegh, to train the captains of the ships ready to cross the Atlantic and claim some territory for England in the New World. Ralegh was a poet, politician, and frequenter of the court of Queen Elisabeth I. The faith of the two man was entangled since. A first expedition brought two native Americans back to England and Harriot learned their language and invented a phonetic alphabet to write it down. He even crossed the ocean himself to visit their home land during a second expedition. After his return, he wrote the only book published during his life: <em>The Briefe and True Report of the New Found Land of Virginia</em> (Virginia named after Elisabeth, the virgin queen). He continued working for Ralegh as a land surveyor. Later Harriot got employed by Henry Percy, 9th Earl of Northumberland. With observations and tedious computations he improved his skills as an astronomer, a mathematician, and a physicist.</p>
<p>
Elisabeth, the short tempered virgin queen, depended on trusted advisors like Ralegh and Percy, but when Ralegh got married without her consent, she considered it treason. The period was turbulent with many political en religious tensions for example with the catholic Queen Mary of Scots, and hence also with France and Spain resulting in naval battles. Privateering was a lucrative pastime for the crew of the ships that were exploring the New World. Political compromises and treason, spying and conspiracies, were common games for Queen Elisabeth and her successor King James. Take on top of that the emergence of the scientific revolution of the 17th century, and the exploration of the North and South Americas, and Arianrhod has all the ingredients to write a thrilling and adventurous novel about it, and so he did. Only it is all based on true and well documented facts.</p>
<p>
Arianrhod has found a good balance between explaining the mathematical and astronomical work of Harriot, and sketching what happened on a political and personal level of the main characters that he describes in the turmoil of events at the end of the 16th and the early 17th century. These main characters whose fortunes and misfortunes are told, are besides Harriot mainly Ralegh and to some extend also Percy but there are many others as well. Too many to keep track of if this were a fiction novel, but real life is not that simple. A name list in the appendix with one line description per name might have been welcome for a reader not so familiar with the period, its politics and its science.</p>
<p>
It is characteristic of many biographers that they bring an idolatrous glorification of their subject. Somehow this struck me in this book too. It is clear that Arianrhod describes Harriot and Ralegh as the worthy heroes with almost sacred virtues. For example the attitude that Harriot and Ralegh have towards the native Americans, considering them as friends and treating them with respect, leaving their dignity is unusual for that time. The devotion of Ralegh for his queen, even when she locked him up in the Tower of London for many years (as she also did with Percy) is outspoken. His continued attempts to colonise parts of North and South America to flatter the Queen (which both turned out to be disastrous) and the noble way he behaved on the scaffold when he was beheaded under King James I are almost beyond human limits.</p>
<p>
Harriot fell under the bad faith of his employers when they were accused of treason, and that shone on him and his work and ideas were scrutinized for possible atheistic elements. He was even imprisoned for a short while on the charge that he had cast an horoscope on King James during the Gunpowder Plot. However, he always tried to stay somewhat in the shadow, concentrating on his scientific work, and so he escaped most of the misfortunes that befell on his employers and could have been his faith too. This is one element of excitement, but the whole book is a thriller: will the pioneers survive crossing the ocean, will they survive in the midst of unknown tribes, will the prisoners of the Tower be executed, will the catholic Gunpowder Plot or the courtier's Main Plot against King James be a success, will the war at sea with Spain be won, will the money be raised for yet a new expedition,... all components that can bring some tension in an engaging story told with brio. That is why, besides Harriot, Ralegh, and Percy, many other (mostly political and scientific) characters are staged in this complicated interplay of intrigues. And Arianrhod is not the first to use these elements in a book. He suggests that Shakespeare has used some of the events described here as scenes in his plays.</p>
<p>
When it comes to Harriot himself, these parts of the book are mainly about his scientific achievements. The reader is instructed about how Harriot explained the seamen what they should know to find the position of his ship, and how they could compute it in an efficient way. We learn about his study of the loxodromic curve and the equiangular spiral, how he unravelled the secret of the Mercator projection, how he measured the acceleration of free falling objects to study gravitation and how he computed the trajectory of a canon ball (that was before Newton formulated his laws of motion). Harriot studied the precession of the Earth and the Gregorian calendar, the refraction of light and gave an explanation for the rainbow. We read about his atomistic views, his exploration of probability theory, and of course his astronomical computations and his study of sunspots. He produced a map of the moon and he had some correspondence with Kepler. He was also one of the first to simplify algebra by introducing symbolic notation: he used letters for variables, he had a notation for exponents, a variant of our equal sign,... Arianrhod places all his discoveries in context, sometimes going back to Greek antiquity or by discussing contemporaries or scientists who came after Harriot who discovered the same things independently, and whose name became attached to these results.</p>
<p>
It is a shame that Harriot did not publish more because that would certainly have given him a reputation comparable to Galileo, Kepler, and cartographers and mathematicians of his time. When at the age of sixty he seemed to be ready to start rounding up his work and publish it, he got health problems. It might have been a kind of cancer that started with his nose that finally killed him. It is not unlikely that it was caused by excessively smoking tobacco. Allegedly he, and his employer Ralegh, are responsible for introducing pipe smoking in England. He made his testament and asked some friend to order his notes and publish them posthumously. However not much came out of that. His notes got lost until they were rediscovered at the end of the 18th century, but again, it took a long time before it was recognized that Harriot had a lot of results that we now know by the name of other scientists, while Harriot had these already much earlier. Slowly Harriot's achievements were realized by historians analysing his notes that are now fully digitized and made available through the <em>European Cultural Heritage Online</em>.</p>
<p>
To conclude: this is a marvelous book because of the engaging way it is told, very much unlike a dull biography with an enumeration of facts. Moreover it is also well documented by additional material to be found in the last 100 pages of the book. There you can find a number of graphic illustrations that are needed to understand some of the mathematics that are discussed. These are moved to the appendix probably to allow the reader to skip some of the mathematics if he or she is not interested. Many of these extra pages are filled with notes that explain some background or give the origin of a quote or a justification for a statement in the text. Of course the list of primary and secondary sources used are there too, and a well stuffed index of names and subjects. A warmly recommended read about England in the period of Shakespeare, shortly before Newton, with in the background a turbulent dance of politics, when war could still be avoided by marriage, but fighting over colonizing the Americas, and over religious controversies never ceased. On this canvas Arianrhod paints the bubbling emergence of the Scientific Revolution to which Harriot was a silent contributor.</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 biography about the life and work of Thomas Harriot (c.1560-1621), an English astronomer and mathematician. Because he did not publish much, most of his work has been hidden for long time but since the legacy of all his notes was rediscovered, historians of the 19th century and later have found that his work preceded in several ways results by mathematicians like Galileo, Kepler, and even Newton. This book makes these insights available for a general public.</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/robyn-arianrhod" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robyn Arianrhod</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">9780190271855 (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">£ 19.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">376</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/thomas-harriot-9780190271855?cc=be&amp;amp;lang=en&amp;amp;" title="Link to web page">https://global.oup.com/academic/product/thomas-harriot-9780190271855?cc=be&lang=en&</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>
<li class="field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
</ul>
</span>
<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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
<li class="field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li>
<li class="field-item even"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li>
</ul>
</span>
Mon, 14 Oct 2019 09:05:19 +0000adhemar49811 at https://euro-math-soc.euMath Art
https://euro-math-soc.eu/review/math-art
<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>
Mathematicians consider some mathematics to be beautiful, and there has indeed been scientific research measuring that mathematicians showed increasing brain activity in the frontal cortex when seeing mathematical formulas. This brain activity is similar to what is observed when people see a beautiful painting or listen to music. So there must be some link between mathematics and art. Several mathematicians are known to be also great musicians of produce visual art and most mathematicians have a soft spot for a certain kind of visual works of art that have some mathematical flavour. The <a href="http://bridgesmathart.org/" target="_blank">Bridges Organization</a> promotes the interaction between mathematics and visual art, music, architecture, and it organizes an annual conference around these ideas.</p>
<p>
Several books are available in which mathematics is linked to art. Some are coffee table books, mainly consisting of pictures, others are philosophical essays with some illustrations. This beautifully illustrated book by Ornes is about the "mathematical art" of nineteen contemporary visual artists. Its format is a careful balance between a sketch of the artist, a short discussion of his or her work, and an easily accessible introduction to the "mathematics behind the art". Some of the artists were once represented at these Bridges conferences and some are at the MoMath museum. For obvious reasons Ornes is mainly interested in visually pleasing work, with a strong mathematical background. Most of the artists are still living and Ornes is quoting some of them, which shows that he has interviewed or at least spoken with these artists.</p>
<p>
Divided into four parts, Ornes discusses 19 artists and their work. A telegraphic survey of the contents and the list of artists is given at the end of this review. Some of the artists are professional mathematicians, but others are just inspired by mathematics. The selection of the art is quite diverse. It can be monumental sculptures, weaving, computer generated curves, quilts, 3D-printed objects, wood carving, or crocheting. The work presented is selected to serve the purpose of this book. The artists can have other work of a quite different nature, or it can be early work and they may have moved more recently to a different kind of work. The appendices about the mathematics that served as an inspiration is diverse as well. There is pi and phi (the golden ratio), the Fibonacci numbers and primes, the Pythagoras theorem, set theory and infinity, geometry with classical Platonic and Archimedian solids, fractals and non-Euclidean geometry, topology and the Moebius band, space filling curves and tilings, computer science with complexity theory, algebra with symmetries and groups, and more. An appendix is linked to one artist, but there are cross references to other artists as well. Clearly the selection of topics and artists is very diverse, but this is only a very small section from a vast domain showing a growing interest for this kind of interaction between mathematics and art.</p>
<p>
The size of the book is nearly square (9 x 9.5 inches) and it is printed on glossy paper. So it can serve as a coffee table book but it has more to read than it has to see. The cover is black with a white design by Bathsheba Grossman. I could not find the reference in the book for the cover picture (although all other pictures are properly credited). The picture is actually a dodecahedron based design for a <a href="https://www.materialise.com/en/mgx/collection/quin-mgx">lamp</a> that is 3D printed by Materialise. It is also an illustration on Grossman's <a href="https://en.wikipedia.org/wiki/Bathsheba_Grossman" target="_blank">Wikipedia page</a> (3 Sept 2019). Grossman has also a Klein bottle opener, i.e., an operational bottle opener in the shape of a Klein bottle.</p>
<p>
I like the book very much. Unlike some other popularizing math books, it literally illustrates the beauty of mathematics, and makes this beauty accessible to non-mathematicians. Hopefully they will be triggered by the beauty of the pictures, to also read the mathematical appendices, which are written at a level that can be read and understood by anyone.</p>
<p>
To conclude, a quick summary of the 19 cases that are collected in four parts.</p>
<ul><li>
Part 1: <em>Making sense of the universe</em>.
<ul><li>
The art of pi - <em>John Sims</em>, who among other work, produces quilts like coloured QR codes where colours are defined by the digits of pi.</li>
<li>
Geometry in motion - <em>John Edmark</em> designs objects that require dynamics, and here one should consult <a href="https://www.johnedmark.com" target="_blank">his website</a> to understand and appreciate his work. The mathematics here deals with the Fibonacci numbers and the golden section.</li>
<li>
The proof is in the painting - <em>Crockett Johnson</em> has paintings that are inspired by graphical proofs of the Pythagoras theorem.</li>
<li>
One to one to infinity - <em>Dorothea Rockburne</em> produces abstract art, sculptures and installations, that draw inspiration from set theory. The mathematical appendix discusses set theory and different orders of infinity and gives a proof that there are infinitely many primes.</li>
<li>
The many faces of geometry - <em>George Hart</em> makes sculptures by weaving several identical components together that shapes Platonic solids. The mathematics is about regular and classical polyhedra and their stellations.</li>
</ul></li>
<li>
Part 2: <em>Stranger shapes</em>
<ul><li>
Space and beyond - <em>Bathsheba Grossman</em> makes sculptures that are periodic minimal surfaces or projections of the 120-cell in 3D space.</li>
<li>
The consequences of never choosing - <em>Helaman Ferguson</em> has monumental sculptures like an umbilic torus decorated with a Peano space filling curve. This and other space filling curves are discussed in the appendix.</li>
<li>
The tangled, torturous universe of fractals - <em>Robert Fathauer</em> produces fractal organic sculptures. Fractals are introduced in the appendix but is continued in the next case.</li>
<li>
The mystical and the mathematical - <em>Melina Green</em> focusses on the Mandelbrot set and generates an image of the set that suggests the shape of a Buddha.</li>
<li>
The equations of nature - <em>David Bachman</em> is a topologist and his art was originally the result of describing nature by equations and then generate artificial objects that look very natural. More recently his work visualizes more abstract ideas. The appendix is discussing topology.</li>
</ul></li>
<li>
Part 3: <em>Journeys</em>
<ul><li>
The wandering mathematician - <em>Robert Bosch</em> produces a piecewise linear Jordan curve that is denser at some places which, from a distance, gives the impression of a grey-scale reproduction of for example the Mona Lisa of whatever other image one cares to choose. The construction of the curve is based on a traveling salesman algorithm which is discussed in the appendix together with the P versus NP problem.</li>
<li>
The curves in the machine - <em>Anita Chowdry</em> is inspired by the Lissajous curves and produces some steampunk instrument to generate such complex curves.</li>
<li>
The algorithms of art - <em>Roman Verostko</em> (born in 1929) has embraced the first computers and designed algorithms to produce graphical art. The appendix discusses some elements from complexity theory and quantum computing.</li>
<li>
Projections - <em>Henry Segerman</em> has work inspired by stereographic projection, producing a Riemann sphere that is the projection on the sphere of for example a regular grid in the plane.</li>
</ul></li>
<li>
Part 4: <em>(near) Impossibilities</em>
<ul><li>
Following yarn beyond Euclid - <em>Daina Taimina</em> is known for her crochet work representing hyperbolic geometry. The appendix explains and illustrates rather well hyperbolic geometry with the Poincaré disk or half plane models.</li>
<li>
Bounding infinities - <em>Frank Farris</em> produces symmetric images and transitions in wall paper groups using deformed photographs as a stamp. Some of his work is discussed in <a href="/review/creating-symmetry-artful-mathematics-wallpaper-patterns" target="_blank"> <em>Creating symmetry</em></a>.</li>
<li>
Connections - <em>Carlo Séquin</em> is a computer scientist who produces complex large mathematically inspired sculptures. The appendix discusses symmetry and group theory.</li>
<li>
Math and the woodcarver's magic - <em>Bjarne Jespersen</em> is a wood carver who produces a wooden object where a sphere is capture inside a polyhedral structure that, unable to take it out or put it in. The appendix is about tessellations that cover the plane.</li>
<li>
The possibilities - <em>Eva Knoll</em> uses many different media to express herself among which weaving where some relative prime repetition of patterns creates some extra pattern on top of the underlying one. The appendix gives a discussion of algebra and all its different meanings in mathematics.</li>
</ul></li>
</ul></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 richly illustrated book discussing the relation between mathematics and art by describing the work of 19 contemporary visual artists and explaining the mathematics that is behind their artwork.</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/stephen-ornes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Ornes</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/sterling-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sterling Publishing</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">978-1454930440 (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 (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">208</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.sterlingpublishing.com/9781454930440/" title="Link to web page">http://www.sterlingpublishing.com/9781454930440/</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/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</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/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li>
<li class="field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Mon, 09 Sep 2019 14:07:48 +0000adhemar49707 at https://euro-math-soc.euThe Tenth Muse
https://euro-math-soc.eu/review/tenth-muse
<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 opening of this novel tells the story of the tenth muse who refuses to be a muse like her nine sisters. She wants to sing her own song, rather than be an inspiration for others, and as a consequence, all her powers are taken away from her and she has to live as a human. She is reincarnating as the women that stood out in the course of history as an artist, philosopher, or, in the case of this novel as a mathematician, who had to stand their ground in a world dominated by men.</p>
<p>
The whole story is told by its main character Katherine, an older successful mathematician, who is reflecting upon her childhood when she grew up in the 1940-50's in Michigan where she showed to be highly gifted with mathematical skills. The main part of the novel is happening while she is entering the university and later as she is working towards a PhD on the (fictional) Mohanty problem that is supposed to be a main opening towards the proof of the Riemann hypothesis. An old (again fictional) Schieling-Meisenbach theorem produced by mathematicians from Göttingen in the 1940's seems to be an essential element to solve the problem but it is also a key element in the novel. Moreover she is going through an identity crisis, and is looking for her, probably German, roots, These two reasons make her decide to spend some time in Bonn (Germany) where she hopes to find what happened to her parents around the time that she was born. Eventually, at the end of the novel, it turns out that Katherine has turned away from (analytic) number theory and has found new challenges in the realm of dynamical systems which has applications in many diverse applied sciences, which is the opposite of what Hardy claimed about number theory.</p>
<p>
The author Catherine Chung has a mathematical degree from the University of Chicago but she received her Master of Fine Arts from Cornell University. For the mathematics in this novel she found some inspiration in popular science books and had some help from friends to check the mathematical content. And there is indeed a lot of mathematics mentioned. As a child, Katherine has to sum the numbers 1 to 9 and immediately comes with the answer 45 using the same technique as it is told that Gauss did by pairing numbers symmetrically in the sequence. She is however not praised for her ingenuity but punished instead because she thinks this is all too obvious and does not want to write down anything. In this case the teacher is a women, but for the rest of the novel, the bad guys are all men. She later goes to college where she is betrayed by her best friend. When they both hand in the same answers to their assignment problems, she is automatically accused of plagiarism while it happened the other way around. More depressing affairs happen to her again and again, even her thesis advisor, with whom she has an affair, disappoints her, and it are always men who do this to her. She is the underdog in all situations: she is not only a women, she is Chinese-Caucasian in the Midwest, is on bad terms with her step-mother, she turns out to be Jewish as well, and she is trying to make a career in science, typically dominated by men. How many stereotypes can be bestowed upon one person. And it is not only happening to her, it happens to most of the women in this novel. When during the war Japanese soldiers threaten to kill the boys of a Chinese family, the daughter is sold to spare the life of the boys. Men think to own a woman and that she can be given away. Even if it is done with the best of intentions, to safe the woman, it still is disrespectful.</p>
<p>
The reason for reviewing this novel here is that the main character is a mathematician, so there is necessarily some mathematics involved. Obviously the Riemann hypothesis appears, but also the zeta function, the Hilbert problems, and the Boltzmann equation show up but without technical details, and some short sections are included about Hypatia, Emmy Noether, Sophie Germain, Sofia Kovalevskaya, Mary Mayer, Ramanujan, Turing, and what happened in Göttingen during the war also plays a role. In a postscript Chung mentions 30 more names of real mathematicians that make a short appearance in the novel: from Gödel and Poincaré to Selberg and Weyl. The link with fiction is made via a fictional Schieling-Meisenbach theorem. Presumably this theorem can be used to solve the equally fictional Mohanty problem (Chandra Mohanty is in real life a professor at Syracuse University defending transnational feminism). What is well described is the urge of a researcher to find a solution for his or her problem in a competing environment where it is important to be the first to publish and one has to be careful not to share too much before, while collaboration is necessary. As a woman or a student this makes you vulnerable because you will always be in the shadow of the co-author. The reader is however not really informed about the details or technicalities of Katherine's research or of most of the mathematics involved, there are only simple descriptions of what topology is about, or some similar descriptions of other topics. Nevertheless, it is instructive for an outsider to learn that when somebody can solve the Mohanty problem for even integers, then there is immediately the challenge of solving the problem for the more difficult case of odd integers. This is how mathematics is a never ending story. The more we know, the more there is left to investigate. Of course the novel is not about the mathematics, but about Katherine who happens to be doing research on a mathematical topic.</p>
<p>
Mix all these issues: mathematics, romantic involvements, treason, the atrocities of the war, gender and race discrimination, and some Buddhist symbolism and Roman mythology and it seems like an overdose that is impossible to concur in one fictional character. And yet, Chung wonderfully succeeds in making it somehow acceptable. It is an engaging story with many twists that keep surprising the reader and that pushes the reader forward, eager to find out what will happen next. There are a lot of mathematical issues that are not revealing much, but still it is remarkable how much of (popular) mathematical issues have been smuggled in by Chung. All of these will be easily assimilated by many readers who would otherwise not be interested in picking up a book about popular mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a novel about a female mathematician who tries to stand her ground in a world dominated by males. She is like the tenth muse who wants to sing her own songs rather than be the inspiration for others. She also is looking for her identity and in her quest she spends some time in Germany where she learns the truth about her parents and the faith of scientists and mathematicians from Göttingen in the 1940's.</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/catherine-chung" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Catherine Chung</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/harpercollins-ecco" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">HarperCollins /Ecco</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">978-0-0625-7406-0 (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">USD 26.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">304</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.harpercollins.com/9780062574060/the-tenth-muse/" title="Link to web page">https://www.harpercollins.com/9780062574060/the-tenth-muse/</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li>
</ul>
</span>
Mon, 09 Sep 2019 14:01:29 +0000adhemar49706 at https://euro-math-soc.euThe Mathematics of Various Entertaining Subjects volume 3
https://euro-math-soc.eu/review/mathematics-various-entertaining-subjects-volume-3
<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 third of the biennial MOVES (Mathematics of Various Entertaining Subjects) conferences was organized in 2017, as usual in the MoMath museum in New York. This book contains its proceedings. Also <a href="/review/mathematics-various-entertaining-subjects-volume-2" target="_blank">volume 2</a> was reviewed here earlier. The editors, the publisher, and the concept of the previous volumes did not change. The overall theme of the conference was this time <em>The magic of math</em> but that can of course include about anything. There are four parts, each containing three to six papers: (1) Puzzles and brainteasers, (2) Games, (3) Algebra and number theory, (4) Geometry and topology. I will pick some examples to give an idea about the contents.</p>
<p>
The six papers in the part <em>Puzzles and brainteasers</em> usually start by challenging the reader with a number of problems for which the solution is given at the end. An example: a variant of a well know prisoner's hat riddle can be formulated as follows: in a set of prisoners each prisoner gets either a red or a yellow hat. They see the colour of the hats of all the others but they do not know the colour of their own hat. They have to answer simultaneously either the colour of their own hat or pass. If one guesses the wrong colour or if all pass, their sentence is extended. If one guesses correctly and none is wrong, they are freed. What should be their strategy (supposing they want to be free)?. To solve this problem the colours are represented as 0 or 1. Each prisoner can compose the 0-1 string of the ordered set of prisoners, except for his own bit. This can be treated like an error correcting code problem. There exists a set such that the probability that the binary string belongs to that set is much larger than that it belongs to the complement of that set. So the problem is reduced to computing a Hamming distance. The solution is explained in elementary steps, but it eventually ends with explaining Hamming codes, finite fields, parity check matrix, Hamming distance, etc. Another guessing problem involves random walks and hidden variables. So, this shows that what starts as a puzzle or game, will eventually lead to the introduction of some mathematics, which is the set up of almost all the contributions in this book.</p>
<p>
In the <em>Games</em> section we find five papers. Intriguing questions, sometimes with surprising answers, can be asked. Take for example the following ones. What are the chances to have a winning row, column of diagonal in a Bingo game (American style) played with 5 x 5 cards with or without the central square free? How to code and count different Tsuro cards? How difficult is it to loose a checkers game? (Suppose you want your son to win without violating the rules.) Here is another problem involving probability: Each player has to move in turn a random number of steps along a path of squares and the target is to end exactly at the end square. If they don't, they have to turn back on the path until they have made the required number of steps. How many moves are needed on average to finish? How many times is a square visited? Several versions exist of another game called "Japanese ladders" (also known as "Ghost Leg", or with several other names). It is a challenging problem to find a strategy to play these games by adding rungs or legs and win. The mathematical equivalent is to decompose and manipulate permutations as a succession of adjacent transpositions. Answering all these questions in the Games section, involves combinatorics, probability, symmetry, graphs, etc.</p>
<p>
The mathematics required in the part on <em>Algebra and number theory</em> is obvious from its title. The first long paper is by Persi Diaconis and Ron Graham. Diaconis is a well known mathematician and magician and he was an invited speaker at the 2017 MOVES conference. The contribution is about the magic of Charles Sanders Peirce, known to be the father of pragmatism. Peirce's 1908 paper <em>Some amazing mazes</em> is difficult to read, so here Diaconis and Graham analyse the principles that support one of the most complicated card tricks ever, each of these principles can be inspiration for some card trick in its own right. The mathematics involved is interesting as well. The analysis contains for example an implicit proof of Fermat's little theorem. Other papers in this section also involve symmetry, groups, modulo arithmetic, and graphs, to solve games like Khalou, or puzzles like KenKen.</p>
<p>
The three papers in the last part are grouped under the title <em>Geometry and topology</em>. One paper is about flexagons, knots, and twisted bands (of which the Moebius band is the simplest example). The second is an interesting graph problem. Consider a regular triangular grid graph covering the plane with cities located on some of the vertices. What is the shortest set of railroad tracks that allows to reach every city from any other city if the railroad can only move along grid lines? This problem is originating from a board game called TransAmerica, but it is also a practical problem to wire the lights on the grid points of the glass dome of the Dalí museum in St Petersburg, Florida which is constructed as a triangular metal grid frame filled with 1100 triangular windows of approximately equal size (although no two are identical). The last paper in this section is about the incredible amount of ways in which a set of Lego blocks can be connected. Just 8 simple jumper plates (one notch at the top and three slots at the bottom) allow for 393314 different compilations. What is the entropy, i.e., hat are the asymptotics of $\frac{1}{n}\log N(n)$ as $n$ becomes large and $N(n)$ is the number of ways to compile $n$ elements?</p>
<p>
As this incomplete survey illustrates, this is a mixture of fun and serious mathematics where professional mathematicians, computer scientists, and enthusiastic gamers and puzzlers can meet. Recreational mathematics has grown out of its infancy and there are some tough results that can be proved and some serious challenges that can be formulated as open, yet unsolved, problems. Some games and puzzles may have been invented by smart amateurs for recreation, solving the puzzle, winning the game, or computing your chances, has become a topic that often requires some mathematical training. Anyone from amateur to professional will be fascinated by the diversity of challenges and solutions proposed. Not the highest level of mathematical abstraction is needed, so with some elementary knowledge the book can we assimilated, but still, it requires a certain willingness to wade through all the mathematics, which is intended to be an essential part. But mathematics is fun and the book is playful and accessible. Moreover, as Bhargava writes in his foreword: isn't most, if not all, mathematics recreational?</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 contains proceedings of the third biennial MOVES conference (Mathematics of Various Entertaining Subjects) organized in 2017 at the MoMath museum in New York. The papers do not only present the games and puzzles and their fun-aspect, but they connect them to the underlying mathematics that is not always elementary. This is recreational mathematics taken seriously as a mathematical subject.</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/jennifer-beineke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jennifer Beineke</a></li>
<li class="field-item odd"><a href="/author/jason-rosenhouse" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jason Rosenhouse</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/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">9780691182575 (hbk), 9780691182582 (pbk), 9780691194417 (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"> £ 97.00 (hbk), £ 40.00 (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">352</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/14228.html" title="Link to web page">https://press.princeton.edu/titles/14228.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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li>
</ul>
</span>
Mon, 09 Sep 2019 13:51:55 +0000adhemar49705 at https://euro-math-soc.euIntroduction to Topology
https://euro-math-soc.eu/review/introduction-topology
<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">Francisco Gallego Lupianez</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/review.pdf" type="application/pdf; length=62545" title="review.pdf">review-Topology-Singh</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/tej-bahadur-singh" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tej Bahadur SINGH</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" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</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">978-081-13--6953-7</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">472</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/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li>
</ul>
</span>
Thu, 05 Sep 2019 12:37:01 +0000Francisco49693 at https://euro-math-soc.euHumble Pi
https://euro-math-soc.eu/review/humble-pi
<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>
Matt Parker was a math teacher who has set himself as a life goal to be a math and science communicator. He is well known as a guest in tv-shows, by his stage shows and Numberphile videos on Youtube, and as the author of <a href="/review/things-make-and-do-fourth-dimension" target="_blank"> <em>Things to make and do in the fourth dimension</em></a> (2014). The current book got the subtitle "<em>A comedy of maths errors</em>", and a slapstick comedy it is, because of all the things "that went not exactly as planned" because of the "mathematical" errors behind the wrong decisions made. Unfortunately some of the consequences were rather dramatic and caused a loss of lives, money, and energy. Parker warns us in the introduction that he has deliberately left three errors in the book for the reader to detect. Also the cover of the book represents a funny Boeing plane with the wings wrongly attached. That is very unlikely to happen in reality, but unfortunately accidents with planes happen all too often and it is sometimes just a tiny programming or construction error that has caused the death of many passengers and crew in the history of commercial aviation.</p>
<p>
This book is not the first to describe a collection of errors with numbers or mathematical errors. For example, read about math errors made by students in <em>Magnificent Mistakes in Mathematics</em> (2013) by A.S. Posamentier and I. Lehman while J.A. Paulos wrote his book <em>Inumeracy</em> (1988) to warn the man in the street not to be blind for blatant mathematical misconceptions, and more on the informatics side, there is <em>Weapons of math destruction</em> (2016) by O'Neil and the more recent <em>Bits and Bugs</em> (2019) by T. Huckle and T. Neckel. Also Parker includes many software errors in his collection. So the "maths errors" have to be understood in a broader sense. The fact that if integers are represented by one byte or 8 bits and thus can at most be 11111111 or 255 in decimal notation, will cause an overflow problem when 256 is reached much like the dreaded (but anticipated) millennium bug. The 256 overflow problem however has often been overlooked and has been a source of many mistakes and disasters when a count suddenly dropped from the maximal 255 to zero. This caused Twitter, Minecraft, and Pac-man to break when reaching level 256 but it also can turn an X-ray medical instrument into a deadly weapon.</p>
<p>
Other tangentially mathematical errors are related to the unwanted interpretation that is made by Excel of what you type: the leading zeros of telephone numbers are removed when inserted as a number, if an hexadecimal number contains an E it can be interpreted as a scientific notation for a decimal number, it may think to recognize a date in strings of biological information with names for enzymes like MARCH5 or SEP15, or in strings of the form 6/11, etc. It is obviously a bad idea to misuse excel sheets with many formulas. Parker makes an analysis of a large set of excel sheets that are interconnected by a large set of formulas with an inevitable daunting amount of errors. Calenders have changed in the course of history and this has been the reason why the Russian delegation arrived two weeks late at the Olympic Games in 1908 because in Russia the Julian calendar was still in use while the rest of the world had switched to the Gregorian calendar. The different units for weights or volumes, and distances nearly crashed a plane that fell out of fuel in mid-air or another one was heavily overweighted and just made it to its destination (kg and lb are not the same). But there are of course the genuine mathematical problems when engineers do not make the correct computations (or make last moment changes and neglect to redo the computations) during the construction of a bridge or a building. The video of the Tacoma bridge is legendary, but there are many other bridges that had stability problems like the wobbly Millennium bridge in London. The Walkie-Talkie is a nickname for a London skyscraper with parabolic glass facade that reflected sunlight and melted things and set carpets on fire in the focal point. Parker is also the man behind a petition protesting against the UK road signs pointing to football stadiums with the wrong football logo. The classic football is an inflated truncated icosahedron with 20 white hexagons and 12 black pentagons. The logo consisted completely of hexagons. Rounding and statistical errors have upset the financial markets, and were misused in politics. Non-causal correlations are often used as arguments by activists or pressure groups. It's not all strictly mathematics, but Parker keeps going on and on. It's misunderstanding and misinterpretation galore, and often a sequence of coincidences for which Parker borrows the image of a Swiss cheese from James Reason: there are holes in every level of the security checks, like holes in every slice of cheese, but the holes have to line up to let the error slip past them all. Unfortunately that happens from time to time.</p>
<p>
On the nerdy side: The pages of the book are numbered from 314 down to 0 and then switches to count down from 4,294,967,295 which is 232-1 for the appendix with the list of illustrations and the index, an example of underflow in binary countdown. A similar type of error played an important role in the first disaster example where the time was stored in 4-bytes and all electrical power was shut down in the Boeing 787 plane when the counter reached 2,147,483,647. The index of the book is automatically produced by a code written by Andrew Taylor. The code selects interesting couples of two successive words from the text. References are given to pages and lines where these couples appear, like for example Richard Feynman: 154.95522, 222.00000-223.29851 meaning that the text "Richard Feynman" appears on the bottom line of page 154 and is discussed on pages 222 top line to page 223 line 11 from the top. This allows to figure out that lines are numbered as multiples of about 2985 starting from line 0 at the top. The more precise value 2985.074662686... is probably a conversion into some units that I have not figured out. There is an exception for the index entry "oddly specific" where lines are shown with 13 digits instead of 5. For example the first is pointing to 117.2089552238806 and "oddly specific" appears indeed on page 117, line 8 from the top, which should on other pages correspond to 117.20896. Another funny and deliberately vague entry is "deliberately vague: somewhere between 7 and 10 and maybe 74". There you can indeed find the phrase "deliberately vague" in connection with the hush up of some errors in the system. It is fun figuring out for yourself how the index is produced. More details below after the "Spoiler alert". Whether the title of the book has any relation with the name of the supergroup of Humble Pie of the late 60's is not clear either. In any case, if pi stands for mathematics, then engineers, programmers, and anyone applying mathematics as a humble tool in their great creative design should be aware of the importance of this humble tool that can safe the life of many that should not die because of a miscalculation.</p>
<p>
All of these mistakes with amusing and not so amusing consequences are told by Parker in general in a funny way, even though there are a lot of people dying in this book. Notwithstanding the long list of examples of what went wrong, probably many other mistakes were never discovered or were swept under the rug after investigation, so that the public doesn't even know they ever happened. Newly discovered mistakes create new safety regulations. But new boundaries will always be explored, that is just human's nature, and humans will always make mistakes. We just have to learn to be alert and do the mathematics properly. Read the book! You'll enjoy it! Mathematics is required in life, but not for enjoying this book.</p>
<p>
<strong>Spoiler alert</strong> If you scroll down, you might read things that you prefer to find out for yourself.</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
There is some hint in the "oddly specific" item of the index on how the lines are numbered. That entry refers to places 139.1194029850746 and 117.2089552238806 where the text appears on line 5 of page 139 and line 8 of page 117. Dividing 1194029850748 by 4 gives 298507462687, which is to be the oddly specific value of the distance between lines (multiplied by 10,000,000). The bottom line (line 33) corresponds to 298507462687 × 32 = 9552238805984, which is rounded to 95522. The first (hard cover) edition of <em>Things to make and do in the fourth dimension</em> had no index. But in the paperback edition an index was added, also produced by Andrew Taylor. There the references were given in 3D: first the page number and then the coordinates of a square like on a map: columns are indicated with a letter (A,B,C) and rows with a number (1-5), giving 15 squares per page to approximately locate the place where the item of the index can be found.<br />
Another bit worth noting is that in the hardback edition of <em>Things to make and do</em>, Brady Haran in the acknowledgements was misspelled as Bradley Haran. Parker promised to have the error corrected in the paperback edition. In this book he thanks again Bradley Haran to which he adds "Consider this a sign of my appreciation, mate". Haran popularized the Parker Square, which Parker produced on Numberphile. It was supposed to be a 3 x 3 kind of magic square containing 9 unique squared numbers with the same sum along rows, columns and diagonals, an unsolved problem in mathematics. Parker is however proud to have found one sloppy approximation in which squared numbers are repeated and one diagonal sum fails. This became known as the Parker Square, an inside joke on Numberphile.</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>
In his well known funny way, Matt Parker is collecting an impressive list of mishaps and coincidences where things went wrong because of (sometimes tiny) mistakes made by engineers, programmers, or just by any link in the chain leading to a (near) disaster. In the background it is always a bad number or an incorrect formula or just the wrong logic that is applied. So the "maths" in the subtitle "a comedy of maths errors" has to be understood as broadly as in the expression "do the math". It's pretty obvious that the mistake should not have happened, if someone just had thought about if for a sec.</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/matt-parker" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Parker</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/penguin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Penguin</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">9780241360231 (hbk), 9780241360194 (pbk), 9780141989136 (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">£20.00 (hbk), £9.99 (pbk), £16.99 (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">336</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.penguin.co.uk/books/300640/humble-pi/9780241360231.html" title="Link to web page">https://www.penguin.co.uk/books/300640/humble-pi/9780241360231.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>
<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/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</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/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li>
</ul>
</span>
Thu, 22 Aug 2019 08:23:32 +0000adhemar49658 at https://euro-math-soc.euNever a dull moment
https://euro-math-soc.eu/review/never-dull-moment
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is an accomplished biography of a great mathematician of the XXth century: Hassler Whitney. A biography written from the vantage point of a close friend, also a mathematician. The book mixes with balance life events and mathematical achievements, as well as suitable digressions to contextualize the information given. All in a progressive chronological order cleverly organized, which catches the reader's fancy. We have the accounts of Whithey’s familiar and social circumstances, scientific motivations, academic events, mathematical achievements, various cultural interests... In the face of such an offer, it is advisable a quick global glimpse to all tresors hidden in the 385 pages to find afterwards a personal path to dwell morosely on them. For sure there will be rereadings, but this is only good. This book has many books inside ! Now, this spot is where one quits reading this review and starts reading the book: comments that follow could spoil surprises and enjoyement. The warning made, we proceed.</p>
<p>In non-academic matters, the book tells us about Whitney’s love of mountains, photography and music. And a fully proactive love indeed: he was an expert climber, constructed his own cameras and played violin in his own string quattor. Of all of this we get a vivid and appealing record. Each anecdote, each episode, each quotation is juicy and brought into life by Kending with ease and ability. The picture on the cover is a must! </p>
<p>But Whitney is a mathematical figure of his century, and his mathematical achievements are the main theme: graph theory and matroids (in connection with the four color problem), cup product and ring cohomology, embedding theorems, week (?) and strong, singularities of smooth mappings, stratifications… even some "after research" dealings with didactics. Here we find the minute rapport of Whitney discoverings: the triggering questions, the illuminating examples, the key ideas, including technical details and references to study closely. And the illustrations are so, so good! Furthermore the author provides the historical background prior to the moment Whitney plunges to a question and the contemporary atmosphere. An occasion to learn some history of Mathematics! There are also some rich phylosophical explanations, see those on the concept of decomposition. Compulsory to mention now the Notes collected at the end of the book.</p>
<p>Another remarkable feature are the discussions of Whithey’s ways to approach problems, turn around them, find crucial examples, make guesses to new insights and tackle the difficulties towards a happy solution. It is not a coincidence that Whitney’s umbrella is the archetypical counterexample all of us exhibit so often. Also, impossible not to mention the theatric accounter with Bezout "quite a" theorem as staged by Kending. And the adjective happy is not casual: it is distilled from Whitney’s attitude in research and learning. </p>
<p>Also, the author recollects more personal remembrances of the Institute, his experiences there, both academic and social. Seemingly collateral, they take us in emotionally and give a hint of more intimate issues. The way Kending meets A. Weil is a jewel! There is also a chapter on "little things" that mean a lot: a most gracious respite. Special mention deserves too the wealth of graphic stuff: pictures of Whitney, family, friends, other mathematicians, a stereo camera, social events… the picture on the cover, of course.</p>
<p>Thus we end back to the essential: forget this review and start reading the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Jesus M. Ruiz</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 accomplished biography of Hassler Whitney, his life and his mathematics. The author narrates in high detail Whithey’s familiar and social circumstances, scientific motivations, academic progresses, mathematical achievements, various cultural interests... But this is also a Mathematics book (on Whitney's Mathematics), and a good one to learn fundamental questions, illuminating examples, key ideas, with rigorous details and proper references. Also some interesting history of Mathematics. A wonderful work to read attentively and enjoy its many facets.</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/keith-kendig" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Keith Kendig</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/amsmaapress" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">AMS/MAAPRESS</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">ISBN 9781470448288</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">60$</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">384</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://bookstore.ams.org/spec-93" title="Link to web page">https://bookstore.ams.org/spec-93</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/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
<li class="field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</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>
Sat, 17 Aug 2019 16:13:21 +0000JMRz49647 at https://euro-math-soc.euChaos and dynamical systems
https://euro-math-soc.eu/review/chaos-and-dynamical-systems
<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>
At the end of 1960's and during the 1970's, chaos theory was a bit of a hype, and many books appeared on the fun-side of chaos as well as serious mathematical books developing the theory of dynamical systems that could lead to chaotic behaviour. Feldman describes his book as an introduction to the subject that is somewhere in between those two approaches. Chaos, he says, is a phenomenon, not a theory. He of course has to explain concepts and technical terms, but he does it without being too abstract. His approach is more on an intuitive basis, letting the reader be surprised by the results produced, and then asking (and answering) the questions that the reader naturally would ask: Why is it like that? How is this possible? Feldman has a lot of teaching experience and is able to explain everything requiring only a minimum of mathematics (the derivative is necessary because he uses differential equations, but no integrals are used, so that high school mathematics suffice).</p>
<p>
He considers two types of dynamical systems: iterated function systems and differential equations. So explaining what these are (in one dimension) is the subject of the first two chapters (together with definitions of stable/unstable equilibrium, orbit, etc.). It is followed by an almost philosophical chapter about what a mathematical model is supposed to be if it should reflect a physical reality from a world that is ruled by an unrealistic Laplace's demon. In many cases a simplified model is much better to highlight exactly that aspect that one is interested in, rather than confusing it with a minute imitation of every detail that is not relevant. There are many kinds of models that can be used.</p>
<p>
The main purpose of the book starts in chapter 4 where the (discrete) logistic equation is introduced. It describes the evolution of a population. The sensitivity to initial conditions is defined as the butterfly effect (including the Lyapunov exponent) and, depending on the parameters, the system has different limiting behaviour. This moves on smoothly to the next chapter which illustrates that, although the system is deterministic, the sequence that is generated is random (to be distinguished from sequences produced by stochastic systems). The logistic differential equation (which also describes population dynamics, but now with harvesting) is used to illustrate bifurcation with a reference to catastrophe theory, tipping points, and hysteresis. Next chapter fully explores the associated bifurcation diagram with its period-doubling route to chaos and a definition of the Feigenbaum constant. Most surprisingly, this constant turn up again and again in different systems. This is called universality.</p>
<p>
Here the reader may be wondering how this is possible and therefore Feldman makes an excursion to universality in physics and what is called there renormalization. That gives an intuitive idea about a stretching and folding process, which is the key to the self-similarity of the bifurcation diagram. This is used as an explanation for a more abstract setting in which it is stated (and graphically checked) that all the iteration functions mapped by this kind of self similarity mapping, is converging to a universal function which is the attractor. This explains universality, i.e., why many different iteration functions all move to the same asymptotic behaviour. Some remarks are given on phase transition, critical phenomena and power laws. The link with fractals could have been made here, but Feldman avoids this byroad.</p>
<p>
With the Lotka-Volterra differential equations and the Rössler system, Feldman introduces multi-dimensional systems and the phase plane. The bifurcation diagrams of the first part is replaced now by strange attractors. A completely new concept because bifurcation spreads out more and more, the attractor pulls in every trajectory. There is however randomness in the attractor as there was in bifurcation and the stretch and fold operation is also here applicable. The Lorentz attractor is a famous example that is at the origin of the term "butterfly effect". It is less obvious that some form of reverse engineering is possible: just selecting the maxima from the chaotic behaviour, allows to produce the Lorentz map, which reveals the iteration function used. Using the time series with delays of one coordinate, a Poincaré map can be constructed that gives an idea of the strange attractor in phase space, provided the correct delays are chosen.</p>
<p>
At several places Feldman refers to complex systems, which is a field in which many components interact with each other like on the Internet, global climate, etc. This is "dynamical systems brought to the next level" and complex systems are still a subject of intense research. In this book references are made to agent based systems, nonlinearity, emergence, and the limits of universality, concepts that return when studying complex systems. So it should be no surprise that in the concluding chapter, Feldman, besides giving a brief summary of what has been discussed, also points to some links and differences with complex systems.</p>
<p>
With a minimum of mathematics, Feldman succeeds in introducing the reader to the world of dynamical systems and the, almost mythical, chaos that they can produce. He explains that there is nothing magic about sudden bifurcation phenomena and that there is nothing strange about a strange attractor, but that there are simple mathematical explanations for these phenomena. The reader can just be satisfied with his explanations, but Feldman is obviously hoping that some of his readers are genuinely interested in the mathematics, and for them he provided extensive sections where he gives advise about what literature to consult for further details. I could spot a few typos not detectable by a spell checker, but none serious. Some in the style of the footnote on page 151: "...is not a always group".</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 smooth introduction to dynamical systems, bifurcation, chaos, and strange attractors. A minimum of mathematics is required, but ample references to the literature with more mathematical details are provided.</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/david-p-feldman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David P. Feldman</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/princetion-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Princetion 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">9780691161525 (pbk), 9780691189390 (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">35.00 USD (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">264</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/13469.html" title="Link to web page">https://press.princeton.edu/titles/13469.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/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</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/37-dynamical-systems-and-ergodic-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37 Dynamical systems and ergodic theory</a></li>
</ul>
</span>
<span class="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/37-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37-01</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/34-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">34-01</a></li>
<li class="field-item odd"><a href="/msc-full/34h10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">34H10</a></li>
<li class="field-item even"><a href="/msc-full/37d45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37D45</a></li>
</ul>
</span>
Wed, 14 Aug 2019 13:47:02 +0000adhemar49642 at https://euro-math-soc.euHow to Fall Slower Than Gravity
https://euro-math-soc.eu/review/how-fall-slower-gravity
<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>
Paul Nahin is known for his many books written to popularize mathematics, but readers familiar with his work, know that there is always quite some mathematics involved, and it is not always the simplest of problems or computations that he describes. This book is a sequel to <a href="/review/praise-simple-physics-science-and-mathematics-behind-everyday-questions" target="_blank"> <em>In praise of simple physics</em></a> in which he introduces, like in this one, problems from physics with solutions. The MacGuffin for this book, as Nahin writes in the preface, is a letter published in the <em>Boston Globe</em> in which it is contested that in an exam for college placement it is required to know about quadratic equations. The title was <em>Who needs to know this stuff?</em>. Instead of writing an angry answer, Nahin wrote this book to illustrate that mathematics and physics in combination with basic laws of physics can solve real life problems. As a motto for the book, it opens with a problem of Lord Rayleigh from the 1876 mathematical tripos in Cambridge. Those who excelled in these exams had a bright future, whatever they chose as their profession. I doubt that this book will convince the authors of the letter in the <em>Globe</em> that quadratic equations are useful to know for everybody, but if the reader has taken calculus and physics courses at an introductory level, and is intrigued by the power of mathematical physics, then this book will give nice examples of what is possible and it has some challenges for the reader too. This is to illustrate that anyone who had these elementary courses (which is about anybody whatever his or her later profession turns out to be) should in principle be able to solve such problems. Several examples are already proposed in the long preface, which sets the tone for the rest of the book.</p>
<p>
The bulk of the book consists of 26 problems (with variations). Some of these are classic and have appeared elsewhere. If they are simple exercises as in introductory courses, then the reader is mainly on his own to solve the problem. For the more demanding problems, Nahin goes through a discussion leading towards a solution and at the end leaves some challenges for the reader. Such a challenge can be a variation of what has been discussed or a step that has been skipped or a similar problem, or it is asked to write a computer program to check some of the numbers numerically. Usually the challenge is simpler than what he has gone through already. An extensive discussion of the solutions to these challenges are found in the second part. Not all solutions can be found analytically, so sometimes a short matlab code is included if necessary. These programs are not always written in the most elegant programming style, but most importantly they are quite readable and they just do what they need to do.</p>
<p>
To give an idea about what kind of problems are discussed, here are some examples.</p>
<ul><li>
The first one is a classic problem of launching a projectile over a wall. This typically involves a parabolic trajectory and thus the quadratic equation pops up.</li>
<li>
The second is a fun problem, seemingly impossible to solve: before noon, snow starts falling at a constant rate and a snowplow starts clearing a long road at noon at a constant volume per hour. The second hour it travels half as far as in the first hour. When did it start to snow? Hard to believe, but Nahin gives an analytic solution that results in the exact moment (up to the second) that it started to snow.</li>
<li>
There are some problems involving Monte-Carlo simulation because an analytic solution is infeasible, and hence programming a simulation is required.</li>
<li>
On the other hand, there are problems related to combinatorics where straightforward programming is excluded because numbers become too large (unless extended precision is used), and so these require the analytic derivation of asymptotic formulas.</li>
<li>
A classic more involved example is to find the tangential speed and time when someone falls off a slippery log (assumed to be a perfect cylinder).</li>
<li>
A 1967 paper of Nahin is recycled in a discussion of NASTYGLASS. That is theoretical glass that acts as a filter cutting off all electrical power below a certain threshold but leaving intact what is larger. Looking at a nice picture through this glass is supposed to make it ugly (hence the name).</li>
<li>
The problem described by the book's title is about the physics of a raindrop that is accumulating mass as it falls though a humid environment. In its simplest form it will accelerate at only one quarter of the gravitational constant.</li>
<li>
As we progress in the book, the problems become more involved with longer elaborations by Nahin. Some earlier problems return like rocket launch but now launching underwater, and it is explained how <em>Enola Gay</em> could launch the atomic bomb and escape the blast.</li>
<li>
It is shown how to compute ζ(6) using only undergraduate mathematics assuming Fourier series and the Dirac impulse are known (which he supposes to be available at the end of the undergraduate level).</li>
<li>
He also connects ζ(s) to prime numbers and cryptography. This connection can be verified experimentally by computing with a simulation the probability that two (or more) randomly selected numbers are coprime.</li>
<li>
After an excursion via cubic and quartic equations, in the last problem, the quadratic equation turns up again in a model to detect a fault in an undersea cable and in a Wheatstone test bridge.</li>
</ul><p>
In some appendices, extra material is provided about continued fractions, and the problem by Lord Rayleigh mentioned in the beginning of the book.</p>
<p>
</p>
<p>
The different problems can be considered independently in most cases, although there are cross references for some. What Nahin offers is a bit of an unusual mixture of explanations and analysis of physical phenomena and some related problems for the reader. Much is in the style of his previous book <em>In simple praise of simple physics</em>. Whether we should classify all the discussed problems as "practical" in the sense that people are confronted with these in everyday life is highly disputable, but they all do involve basic laws from physics (although somewhat simplified) and it is illustrated (at this elementary level) how mathematics helps a physicist to solve such problems. As Nahin writes in his preface: "Millions of students are enrolled worldwide in calculus and physics courses, and the majority of them will not become mathematical physicists, but this does not mean that they cannot enjoy the power of mathematics making sense of a physical world".</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>
As a sequel to his book <em>In praise of simple physics</em>, Nahin discusses 27 more problems from physics that can be solved using relatively simple mathematics and some elementary physical laws. He leaves some challenges for the reader for which he gives solutions in the second part of the book. The title of the book refers to the problem describing a raindrop collecting more mass as it falls through a humid fog.</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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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">978-0-6911-7691-8 (hbk); 978-0-6911-8502-6 (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 27.95 (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">320</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/13262.html" title="Link to web page">https://press.princeton.edu/titles/13262.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-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li>
<li class="field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
</ul>
</span>
<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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</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/00a07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a07</a></li>
<li class="field-item odd"><a href="/msc-full/65z05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65Z05</a></li>
</ul>
</span>
Wed, 14 Aug 2019 13:42:26 +0000adhemar49641 at https://euro-math-soc.eu