Book reviews
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Book reviews published on the European Mathematical Society websiteenThe secret formula
https://euro-math-soc.eu/review/secret-formula
<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 "secret formula" is a method found by Tartaglia to solve cubic equations. The 16th century priority fight between Tartaglia and Cardano's student Ferrari over this formula is well known. A brief summary: Niccolo Tartaglia, an abbaco master in Venice, was challenged in a mathematical duel by Antonio Fior, and solved the questions, all reducible to solving cubic equations, in a couple of hours. This made him instantly famous, since Luca Pacioli had stated that, contrary to the quadratic equation, there did not exist a general formula for cubic equations. Tartaglia did not want to reveal his secret formula, probably to keep up his fame and to have an advantage in future duels. Gerolamo Cardano however wanted to include the formula in his book on algebra that he was preparing. Only after a lot of nagging, Tartaglia discloses to him his method in verse form, subject to strict secrecy. When Cardano and his student Ludovico Ferrari discovered that the formula was known in unpublished work from Scipione dal Ferro, they considered the promise to Tartaglia not binding any more, and the method was included in the book with proper reference. This resulted in a public fight between Tartaglia and Ferrari publishing insulting pamphlets back and forth.</p>
<p>
Niccolo Tartaglia was born in Brescia around 1500. His last name is often believed to be Fontana, but Toscano claims that there is no solid proof for that. Niccolo preferred to use his nickname Tartaglia, i.e. the stammerer, because that is what he did since at the age of 12, a French soldier's sabre mutilated his jaw and left him for dead during a reprisal attack of the troupes of Louis XII on his home town. Against all odds, his mother could keep him alive. Later he became an abbaco teacher. That means that he instructed and applied the practical use of numerical calculation using the newly introduced Hindu-Arabic numerals as described in Fibonacci's <em>Liber Abbaci</em>, rather than the impractical Roman numerals. This practical kind of mathematics was needed in commerce for bookkeeping or for example to converse different measures or currency. It was quite different from the geometry of Euclid's <em>Elements</em> that was taught at an academic level.</p>
<p>
The Renaissance habit of having public challenges and scientific duels had some historical background. A number of questions were formulated by the challenger to be solved within a certain time span. It was a gentlemen's agreement though that no questions should be asked that the challenger was not able to solve himself. The one that was challenged then reposted with a set of questions for the challenger, and the winner of the duel was the one who first solved all the problems first or who solved most of the problems. Tartaglia received in 1530 two questions by Zuanne de Tonini da Coi, and that surprised him because the problems reduced to the solution of a cubic equation, which was claimed to be impossible by Pacioli. So he assumed that da Coi did not know how to solve it either. Nevertheless he started thinking about the problem, and obviously found a solution, at least for some cases. It should be noted that what we write in modern notation as $x^3+ax=c$ and $x^3=bx+c$ were considered to be two different types of equations. Because the terms represented (positive) quantities, (often lengths with a geometric interpretation), they could not be zero or negative. Only positive coefficients were allowed, which made it difficult to switch terms to the other side of the equal sign.</p>
<p>
To explain the state of the art of algebraic manipulation, Toscano sketches in a second chapter the history of how algebra came to Europe from the Babylonian Plimpton 322 tablet and the Egyptian Rhind papyrus to Al-Khwarizmi's Algebra book (<em>The Compendious Book on Calculation by Completion and Balancing</em>) in which is explained how to solve equations without explicitly switching terms to the other side of the equal sign. It would have been simpler if negative numbers were allowed and if our symbolic notation was used. Although the latter was once promoted by Diophantus of Alexandria (3rd century), the habit was lost over time. It is only because of the 16th century events described here that our modern notation and manipulation came about.</p>
<p>
The next chapter is describing the 1535 duel with Fior that started Tartaglia's fame. Antonio Fior challenged Tartaglia with problems that all reduced to cubic equations and Tartaglia, who had figured out how to do it since da Coi's questions, gave the answers in a few hours long before Fior could solve one of Tartaglia's problems. Whether Fior was able to solve the problems himself, is not clear since he kept begging Tartaglia to reveal his method, although he claimed that the method was explained to him by "some mathematician" 30 years ago. This was most likely dal Ferro since Fior was his assistant.</p>
<p>
But Tartaglia was now also approached by Cardano, first through his publisher, who wanted to include Tartaglia's method in his book on algebra. When Cardano invited him later to Milan, Tartaglia finally disclosed his method after Cardano had sworn not to publish it. Toscano explains the rhyme that Tartaglia used to summarize how to solve the two forms of the cubic equation mentioned above and how a third form is reduced to one of those. When Cardano's book <em>Ars Magna sive de regulis algebraicis</em> was published in 1545 it was a big success and historians consider it as the beginning of modern mathematics. It contained, besides Ferrari's solution for the fourth degree equation, also the method for the cubic equation with proper reference to Dal Ferro and Tartaglia. Tartaglia was however furious and he published his <em>Quesiti et inventioni nuove</em> containing his account of what has happened, alongside some insulting remarks about Cardano. This was published one year after Cardano's book, while he had neglected for many years to publish his method himself. Cardano had accepted a long-cherished physician's position and had left mathematics teaching, so Ferrari took up the defence of his supervisor and there was quite some verbal abuse in public pamphlets exchanged between Ferrari and Tartaglia in subsequent months. This culminated in a duel between both in 1548 in Milan, with Ferrari as victor.</p>
<p>
Cardano's book is important for the history of mathematics because it initiated some ideas that lead to complex numbers. On the other hand, the Cardano-Tartaglia-Fior-Ferrari interaction is a juicy topic that easily lends itself to be discussed in popular science books. So it has been told by many authors, but it is often Cardano who is placed at the center of the story. For example in P.J. Nahin <a href="/review/imaginary-tale-story-%E2%88%9A-1" target="_blank"> <em>An Imaginary Tale: The Story of √-1</em></a> (1998/2010) some time is devoted to this melee and in the novel by M. Brooks <a href="/review/quantum-astrologers-handbook" target="_blank"> <em>The Quantum Astrologer's Handbook</em></a> (2017) Cardano is the main historical character. In the present book, it is basically the same story all over, but told more from Tartaglia's viewpoint. A lot is taken from Tartaglia's own account with many translated quotations in which the mutual scolding in the pamphlets are made blatantly clear. There is of course some background and history of mathematics but Toscano's main focus is the solution of the cubic equation leaving some other work of Tartaglia and Cardano in the shadow. For example Tartaglia wrote a treatise on ballistics and found that the maximum reach was obtained firing in an angle of 45 degrees. The result is correct although it has several mistakes for which Ferrari reproached him later. He also translated Euclid's <em>Elements</em> to Italian (working on this was his excuse for not publishing his formula).</p>
<p>
Toscano has a pleasant writing style (and/or the translation by Arturo Sangalli is smooth). The opener of the books describes Niccolo with his mother and sister lost in the chaos of French soldiers attacking Brescia. Niccolo is hit twice by the sabre of a soldier and left for dead. That is like the opener of a dramatic novel. The attention of the reader is immediately caught. As Toscano unravels the historical development, he makes use of many quotations, which are fortunately provided by Tartaglia himself. This implies that the story is told close to how Tartaglia has experienced what has happened. However Toscano does not hesitate to give some interpretations and place some question marks where appropriate. The yeast of the story has been told already many times, but it has never been told like Toscano does in this book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a translation of the Italian original published in 2009. On the background of 16th century Italy, Toscano describes how Tartaglia has learned how to solve cubic equations, thus winning in a spectacular way a mathematical duel against Antonio Fior. Tartaglia does not want to share his method with others, but eventually he lifts a tip of the veil for Cardano subject to strict secrecy. Cardano publishes it anyway because he discovered that the formula was described in older unpublished work of Dal Ferro. This results in a fierce public pamphlet war between Tartaglia and Cardano's apprentice Ferrari.</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/fabio-toscano" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Fabio Toscano</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">2020</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">9780691183671 (hbk), 9780691200323 (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 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">184</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/9780691183671/the-secret-formula" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691183671/the-secret-formula</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li>
<li class="field-item odd"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
</ul>
</span>
Mon, 15 Jun 2020 09:29:42 +0000adhemar50829 at https://euro-math-soc.euHot molecules, cold electrons
https://euro-math-soc.eu/review/hot-molecules-cold-electrons
<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 well known as a popular science writer. Some twenty books he has published since he started at the end of the previous century with a biography of Oliver Heaviside. Most of his books are dealing with topics involving physics, but there is always keen attention given to mathematics. For example he authored books on explicit mathematical topics like <em>An imaginary tale. The story of √-1</em> (1998) and <em>Dr. Euler's fabulous formula</em> (2006).</p>
<p>The present book is like a mathematical textbook for engineering or science students in which all the derivations are given. Nahin uses an historical approach to introduce Fourier analysis, derive the heat equation, and solve it for different geometries and boundary conditions. When applied to a cooling sphere, this illustrates how William Thomson (Lord Kelvin) estimated the age of our planet by computing how a molten sphere cools down to a sphere with a solid crust (that explains the hot molecules of the title). When the equation is solved for a long cable, it explains how electrons travel through the transatlantic submarine telegraph cable (hence the cold electrons).</p>
<p>So there are a lot of formulas and derivations, but it is not a course as it would be written in modern times. It is taken out of a regular university curriculum and it assumes only the basic calculus from a course at a first year science, engineering, or mathematics level. Fourier series and the Fourier transform are developed from basic principles. Nowadays, the heat equation can be solved efficiently using for example Laplace transforms, but Nahin prefers to use essentially the mathematics available to Fourier who solved it in the time domain. Every step is explained to the smallest details. Sometimes the approach is using an engineering style of mathematics. This means that Nahin is just using an insight from the underlying physics to propose a certain method or to justify a certain solution. Infinite sums and integrals are interchanged, postponing to when the eventual result is obtained whether this makes sense or not. The square root of minus 1 is however denoted by the mathematical standard i, and not by j as is customized by the engineering community to distinguish it from electric current which is also indicated by i or I.</p>
<p>This "engineering mathematics" is also what Fourier applied. His original report on the solution of the heat equation in 1807 was criticized by Lagrange and Laplace because he used his formally obtained infinite sums as if they were ordinary functions. It is not until his "new mathematics" was better understood, ten years later that he was taken seriously and was accepted as a member of the French Academy of Science. Chapter 1 is an eye opener to the sort of mathematics that Fourier introduced. It is for example shown how Fourier obtained $\frac{\pi}{4}=\sum_{k=0}^\infty (−1)^k\frac{\cos(2k+1)x}{2k+1}$. This is well known for $x=0$ (Leibniz formula), but there are many other values of $x$ for which this is also true, much to the surprise of Fourier's contemporaries.</p>
<p>In Chapter 2, the Fourier series are derived and it is shown that they are optimal approximations in a least squares sense. Convergence is not proved. Nahin asks the reader to "accept that our mathematician colleagues have, indeed, established its truth". In this way Fourier series, the Parseval identity, Dirichlet's integral, and the Fourier transform are introduced.</p>
<p>Chapter 3 derives the heat equation $\frac{\partial u}{\partial t}=k(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})$ from first principles. When the medium is a long radiating cable, it is essentially one dimensional and a simple solution is found as a decaying exponential assuming a constant energy loss per unit length, not depending on time. The solution of the equation for different geometries and different physical boundary conditions is discussed in the next chapter. It starts with a cooling problem of an infinite slab with finite thickness ($0\le x \le L$) using a separation of variables ($u(t,x)=f(t)g(x)$) as Johann Bernoulli did. This results in an infinite series with terms of the form $\exp(−ak^2t)\sin(bkx)$ which has to satisfy the boundary conditions. Next, the spherical problem is solved. Assuming isotropy for a sphere, it becomes one-dimensional in the radius $r$. This problem was solved by Lord Kelvin when he applied it to a cooling Earth, which however drastically underestimated its existence to 98 million years because he did not know about radioactive decay or tectonic plates. Next is the solution in a semi-infinite medium with infinite thickness. This is the first case of the slab where the thickness $L$ goes to $\infty$. This is an occasion to show how the Fourier series used for finite $L$ migrates into the Fourier transform when $L\to\infty$. The heat equation is also solved for other cases like a circular ring and an insulated sphere These were also discussed by Fourier in his <em>Théorie analytique de la chaleur</em> (1822), although the last one did not result in a Fourier series.</p>
<p>Chapter 5 starts with a crash course on electrical circuits: resistors, capacitors, inductors and Kirchoff's laws and describing the behaviour of electrons in an electrical field. And lo and behold, the electrons in a one-dimensional semi-infinite induction-free telegraph cable behave according to the heat equation, again an ingenious insight of Lord Kelvin. Solving that equation was a theoretical achievement, producing the cable and letting it sink to the bottom of the ocean was a risky and adventurous enterprise. In this book, that technological adventure is only lingering in the background. A nice account of this adventure can be found for example in the book <em><a target="_blank" href="/review/mind-play-how-claude-shannon-invented-information-age">A Mind at Play: How Claude Shannon Invented the Information Age</a></em> by J. Soni and R. Goodman (2017).</p>
<p>Heaviside also features in the last chapter discussing the evolution after the 1866 Atlantic cable was realized. He added the inductance to the heat equation which turns it into a wave equation (actually the telegrapher's equation describing traveling waves in transmission lines, smartly solved by d'Alembert). That removes the assumed instantaneous action at a distance in the heat equation, which was causing a diffusion of the signal. The parameters of the cable can be controlled to remove that effect and this improved the usefulness of the cable considerably. Nahin ends by discussing the computation of how an arbitrary signal is transmitted. The diffusion however destroys the information during the transmission. This is illustrated by a matlab program that computes this deformation. The short code is given so that you can try it out yourself. The example shows that the signal is unrecognizable, it can still work though for a binary signal since the only information that one needs to detect is whether or not a bit is zero or one. We can also read how Heaviside explained the asymmetry of the transmission time: a message sent from England took longer than a message sent to England.</p>
<p>The sources used by Nahin, and some additional historical notes are listed at the end of the book, organized per chapter. There is no separate bibliography but there is an index that includes references to these notes. He has also one appendix about Leibniz's formula, i.e., how to compute the derivative of an integral if the boundaries of the integral are varying.</p>
<p>The book confirms what is already known from his previous books: Nahin knows how to write a book mixing physics and (a lot of) mathematics and (still) make it readable for a (relatively) broad public (with only some basic mathematical knowledge). The mathematics in this book certainly take the leading role like it does in lecture notes about the solution of differential equations. Nahin takes his time to explain everything and derive things from the very basics. When the mathematics become too involved or advanced, he uses intuition and asks the reader to accept and believe the result. The hard core mathematical mind may have some problems with his "engineering approach", but it works perfectly well for a first introduction. Anyway, from the historical perspective, this approach was used by the people who originally developed the theory.</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>Nahin introduces us through an historical approach to Fourier series, Fourier transforms, and how Fourier used this to solve the heat equation. Lord Kelvin used the heat equation to model the cooling of the Earth and hence estimate its age and he, and others, solved essentially the same equation to model the flow of electrons in the transatlantic telegraph cable.</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">2020</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">9780691191720 (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 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">232</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/9780691191720/hot-molecules-cold-electrons" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691191720/hot-molecules-cold-electrons</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/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial 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/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</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/35k05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K05</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/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li>
<li class="field-item odd"><a href="/msc-full/35s30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35S30</a></li>
<li class="field-item even"><a href="/msc-full/35l05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35L05</a></li>
<li class="field-item odd"><a href="/msc-full/35k57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K57</a></li>
<li class="field-item even"><a href="/msc-full/94c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94C05</a></li>
</ul>
</span>
Tue, 26 May 2020 16:18:52 +0000adhemar50806 at https://euro-math-soc.euTrigonometric Delights
https://euro-math-soc.eu/review/trigonometric-delights
<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 the first two sentences of the preface, Maor writes</p>
<blockquote><p>
This book is neither a textbook of trigonometry —of which there are many— nor a comprehensive history of the subject, of which there are almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences.
</p></blockquote>
<p>I could not think of a better characterisation of the book than this. All I can add to this description is to give an idea of which kind of topics were selected and what kind of applications have benefited from these developments.</p>
<p>The successive chapters are organised more or less chronologically, starting with a prologue about the Egyptian Rhind papyrus from around 16-17th century B.C. and ending with Fourier series (18th century). There is of course much attention for the history, but what strikes me in particular, is how much attention is given to the etymology of the mathematical terminology. The origin of the words algorithm and algebra is described in several publications as originating from the Arab author al-Khwarizmi and from al-jabr, which is part of the title of his book, but what is the origin of words such as sine, secant, and many other common mathematical words? Maor carefully pays attention to this. He also shows how trigonometry, which originally was about angles like in pyramid building problems in Egypt, were somewhat made more abstract, in a geometric context of triangles by the Greek, but later, it became more and more part of analysis. The sine and cosine were not only tabulated for computational purposes, but they became functions so that now we see x in sin x as a real or even a complex number, not necessarily corresponding to a physical or geometric angle. The original idea of an angle in degrees or radians in the goniometric unit circle has become somewhat obsolete.</p>
<p>But of course it all starts with angles and chords in planar circles for the Greek, and even earlier in astronomy, which is essentially a three dimensional spherical discipline as practised by Babylonians and almost any civilisation of antiquity. This is the subject of the first two chapters. Then appeared tables of goniometric values of what became our basic goniometric functions. This opens the possibility to introduce algebra (goniometric identities) and gradually also analysis (involving series) into the discipline. This helped considerably to discover (actually re-discover) the heliocentric interpretation of our solar system and to measure our own planet by triangulation and those practical problems in turn stimulated the development of associated trigonometric identities in triangles. But before the heliocentric model, the trajectories of the planets required also more general curves than ther circle like epi- and epo-circles which allow an easy description in terms of trigonometric formulas. Then Maor ventures into a period of proper analysis with the Sine integral and many other relations and series expansions, not in the least for the fascinating number π. These were obtained by master minds such as Gauss and Euler. As complex numbers entered the picture, with Euler's fabulous formula, we are fully involved in complex analysis, conformal maps and ultimately Fourier analysis.</p>
<p>This marvelous survey by Maor of some episodes in the historical evolution of mathematics also allows to sketch some biographies of important mathematicians. There are the "usual suspects" from Greek antiquity (including Zeno whose paradoxes are discussed when infinitesimals from analysis are introduced). Also Regiomontanus (15th C.), François Viète (16th C.), De Moivre (17th C.), Maria Agnesi and her "witch" (18th C.), Jules Lissajous (19th C.), Edmund Landau (20th C.) are discussed in somewhat more detail. aot only the history and mathematicians, also the applications are well documented: astronomy, cartography, spirographs, periodic oscillation, music; and there are detailed mathematical derivations of several trigonometric and other mathematical identities, conformal maps, series converging to π, the solution of the Basel problem by Euler, how Gauss showed that any trigonometric summation formula can be represented geometrically, etc.</p>
<p>All these items are treated requiring only some elementary trigonometric formulas. Some of the standard identities are collected in appendices. In another appendix we find Maor's plea to re-introduce the unit circle and the geometric definitions of the trigonometric functions like cos and sin being projections of the circular point on x- and y-axis, etc. instead of the "New Math" approach. Also Barrow's integration of sec x is moved to an appendix. All the chapters are completed with a section containing notes and references to the sources used. There are many useful mathematical graphs and some grayscale images.</p>
<p>This is an interesting mixture of mathematical history, illustrating the evolution and the usefulness of trigonometry throughout the centuries, and on top of that, it gives some mathematical training by deriving formulas and identities that are easily accessible with only some elementary mathematics knowledge. The book appeared originally in 1998 and is here reprinted in its original form as a volume in the Princeton Science Library. So this is a fortunate occasion to bring this great book back under the attention of a broad audience.</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 a reprint in the Princeton Science Library of the original book from 1998. Maor sketches several episodes on the history of mathematics where especially trigonometry was involved from the Rhind Papyrus to Fourier analysis. The history, the mathematicians, the applications, as well as the derivation of mathematical identities are discussed.</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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</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">2020</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">9780691202198 (pbk), 9780691202204 (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 17.95 (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">256</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/paperback/9780691202198/trigonometric-delights" title="Link to web page">https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li>
<li class="field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li>
<li class="field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li>
</ul>
</span>
Sat, 16 May 2020 14:47:19 +0000adhemar50783 at https://euro-math-soc.euDIRICHLET SERIES AND HOLOMORPHIC FUNCTIONS IN HIGH DIMENSIONS
https://euro-math-soc.eu/review/dirichlet-series-and-holomorphic-functions-high-dimensions
<div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even"> </div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/Report%20Defant%20and%20others%20book.pdf" type="application/pdf; length=172655" title="Report Defant and others book.pdf">Dirichlet Series and Holomorphic Functions</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/andreas-defant" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andreas Defant</a></li>
<li class="field-item odd"><a href="/author/domingo-garc%C3%ADa" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Domingo García</a></li>
<li class="field-item even"><a href="/author/manuel-maestre" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Manuel Maestre</a></li>
<li class="field-item odd"><a href="/author/pablo-sevilla-peris" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Pablo Sevilla-Peris</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-1-108-47671-3</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">707</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/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</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/30-functions-complex-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30 Functions of a complex variable</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/30b50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30b50</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/46g20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46g20</a></li>
<li class="field-item odd"><a href="/msc-full/46g25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46g25</a></li>
<li class="field-item even"><a href="/msc-full/32axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">32axx</a></li>
</ul>
</span>
Fri, 15 May 2020 17:07:07 +0000Jose M. Ansemil50782 at https://euro-math-soc.eu Linear Algebra and Optimization with Applications to Machine Learning. Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning
https://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra
<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 "with applications" in the title of the book should be read as "applicable" because it provides the fundamental mathematics, but it does not explicitly treat the applications that are mentioned. In this volume I, these fundamentals are basically linear algebra, and in volume II it is promised to cover optimization. I can imagine though that there is some interaction like for example the important subject of linear programming. Except for a few illustrative examples, the applications themselves are supposed to be covered in other courses or textbooks. There is a bit of statistics, which I think is also important for the applications mentioned, but probability and statistics as well as calculus is assumed known, but anyway, whatever is needed from these is recalled briefly.</p>
<p>Since this volume introduces the fundamentals with applications in mind, it is in some sense similar to a first course in linear algebra that I have been teaching to engineering students for many years. This volume has also some numerical procedures and even matlab codes. Those I taught in a separate numerical course. The procedures discussed include Gaussian elimination, Cholesky factorization, QR decomposition, eigenvalue and SVD algorithms, Krylov and Lanczos methods, but it skips the basic numerics of rounding errors, error propagation, numerical stability, and the more analytic problems such as numerical quadrature, differential equations, zero finding, etc. The latter also have a definite link with linear algebra, but clearly including all applications of linear algebra is an interminable task.</p>
<p>I wrote my own lecture notes, not satisfied with the existing books that were not providing the desired abstraction and that spent too many glossy pages on the introductory level with many examples and applications. In many ways my notes were very similar to the material covered here, which is why I like this book so much. But since I was trying to cover as much as possible in the most efficient way restricted by the amount of credits that were assigned to the course, my notes were much more concise. This is quite different in this book since, at its introductory level, it is almost encyclopedic and it is like the text I would have liked to write if I were not restricted by a time limit to cover all the material. For example, I spent some time explaining that finite dimensional real vector spaces and linear maps can be treated in an isomorphic way by discussing $\mathbb{R}^n$ and matrix algebra, and then I could just do matrix algebra. Not so in this book. The abstract vector spaces remain present throughout the book. Infinite dimensional vector spaces are a problem because there you need infinite sums and convergence, which require topology to define convergence, and the maps are operators. The authors here maintain some elements of function spaces but certain analysis aspects are not really covered in detail. But otherwise almost all the proofs are fully written out.</p>
<p>Thanks to LaTeX it is nowadays no problem to produce a professionally looking text. The illustrations however require different tools and producing good quality graphics is a challenge. Clearly the authors of this book had the same problem with graphics that I also had. The text is excellent, but the graphics are definitely of lesser quality. Unfortunately it is not only a problem of how they are generated and reproduced, also they do not always make very clear what they are supposed to illustrate.</p>
<p>What would one expect in a basic linear algebra course for engineering-type students? I think that should include vector spaces and linear maps and how they relate to matrices, the rank of a matrix with range and null space, determinants (in my opinion as little as possible), linear systems with Gaussian elimination, normed and Euclidean spaces, orthogonalization and QR, eigenvalue and singular values with generalized inverses and least squares, and geometric interpretation of all these concepts. All this is extensively discussed in this book. The numerical and matlab algorithms and certainly the iterative methods, I would rather expect in a more specialized numerical course, but it is not completely unexpected that they are found here in this book. More unexpected are the following: A discussion about the Haar wavelet with some signal and image processing; the chapter on linear systems is introduced with a discussion about the computation of interpolating Bézier curves; and the computation of the matrix exponential, important for the solution of differential equations, is discussed to some extent. There is also an extensive discussion of groups (SU(2), SO(3), and quaternions. This is important for robotics. Finally, there is much material related to graphs: graph Laplacians, clusters, and graph drawing. The final chapter about polynomial factorization and the Jordan form is less elementary and not always found with the same detail in a basic course. So there is a lot of material, and I can imagine that one wants to make a selection. No advise is given by the authors and it would be difficult anyway since successive chapters rely on previous ones. The authors have earmarked only some sections that they considered to be more advanced and these can be skipped initially.</p>
<p>To conclude, I can definitely recommend this extensive book on linear algebra that is both self contained and thorough and mostly remaining at a basic level. I have always considered linear algebra a basic tool in many applications. The applications mentioned in the title are computer vision, robotics and machine learning but they are not really discussed. However the Haar wavelet is a hint to image and signal processing, robotics are related to the study of the quaternions and the rotation groups, the linear algebra and the graphs are useful for a lot of applications, thus also for machine learning. To come to the actual applications though will need extra material, continuing the basics given here, but also some extra material for example statistics, and optimization (the latter is promised in volume II). Every chapter has a list of exercises (no answers are provided). There is a bibliography which lists mostly books and an index (12 pages). With a book of this size, the index can never be extensive enough. I tried to look up some topics that were not listed. On the other hand, I noted separate entries for "Jordan block" and "Jordan blocks", which makes no sense of course. I know from experience that collecting an index in an artisanal way is, just like generating good graphics, a time consuming task that needs patience and many iterations.</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 extensive and thorough introduction to linear algebra that includes some extras like wavelets, Bézier curves, groups of rotations, quaternions, and applications in graph theory, that are of particular interest for applications in computer vision, robotics, and machine learning.</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/jean-gallier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean Gallier</a></li>
<li class="field-item odd"><a href="/author/jocelyn-quaintance" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jocelyn Quaintance</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</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">2020</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">9789811206399 (hbk), 9789811207716 (pbk), 9789811206412 (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">GBP 175.00 (hbk), GBP 85.00 (pbk), GBP 70.00 (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">824</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.worldscientific.com/worldscibooks/10.1142/11446" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/11446</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/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li>
<li class="field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li>
</ul>
</span>
<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/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix 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/15axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15Axx</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/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li>
<li class="field-item odd"><a href="/msc-full/65d19" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D19</a></li>
<li class="field-item even"><a href="/msc-full/65t05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65T05</a></li>
</ul>
</span>
Fri, 10 Apr 2020 06:07:24 +0000adhemar50668 at https://euro-math-soc.euSleight of Mind
https://euro-math-soc.eu/review/sleight-mind
<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 Cook is an economist, a composer, a storyteller (as the author of thrillers), and he performs as a magician. Several magical tricks rely on creating an intuitive expectation and then come up with a totally different result. This creates amazement and unbelief in the audience. This is also very much the effect of a paradox. Given that Cook is not a professional mathematician himself, it comes as a surprise to find rather much abstraction and mathematics in this book.</p>
<p>Logical paradoxes are often found in popular science books discussing mathematics, games, and puzzles. Many of these "popular" paradoxes you can also find in this book but there are many more. Although this book is written for a general public, it is not leisure reading, since the discussion of the paradoxes goes in depth and that requires precise definitions and sometimes it touches upon the foundations of logic, mathematics, probability, or whatever topic the paradox is about.</p>
<p>The different topics are arranged in different chapters and the format is always similar. There is a general introduction to the subject, and that involves the definition of the concepts that are required for the discussion of the paradoxes to follow. These are precise but the selected terms and the associated technicalities are restricted to a minimum. Only what is essential is defined and only as precise as needed. For example in the chapter on probability it is defined what a probability space is, and that involves a sample space, a sigma-algebra, and a probability function, which are described by words, rather than formulas. Of course it is also explained how a random variable and its density function are defined and the Bayes theorem is introduced (this inevitably results in a formula). So, there are some formulas, but they are suppressed as much as possible, describing the definitions mostly in words and by using examples. I guess this is intended not to shy away the non-mathematician, but if you are a mathematician, then, given the intended rigour, it feels a bit awkward and verbose. Of course some formulas cannot be avoided, for example to illustrate what is in the Principia Mathematica of Whitehead and Russell a formula here and there is unavoidable.</p>
<p>When Cook comes to the many examples of paradoxes, it assumes an attentive reader because the lack of formulas requires sometimes complicated sentences that are often almost philosophical. Also here, a returning format is used. First the paradox is formulated, wherever possible, mentioning its origin. Cook usually tells a story to make the paradox concrete for the reader, rather than formulating it in its mathematical or abstract form. Then the opposing explanations (often there are only two) are formulated. The main discussion then explains why one is wrong and the other is correct. Sometimes there are more possibilities and more than one explanation is possible depending on how some components are defined or interpreted, which happens when the problem is ill-posed or under-defined.</p>
<p>Let me give some examples that illustrate the types of paradoxes and the depth of the discussion. A first chapter is dealing with infinity, which is not the simplest one to start with, but it is also the underlying concept in some subsequent chapters. It is clearly a concept that has caused a lot of confusion throughout the history of mathematics and logic. First we are instructed about bijections and countable sets, Cantor's diagonalization process, the cardinals $\aleph_k$, and the continuity hypothesis. Then the paradoxes can be explained: Hilbert's Hotel, Stewart's HyperWebster Dictionary, and many more. After introducing some additional group theory also the Banach-Tarsky theorem is explained in some detail. Not really a proof, but still the reader is given some idea of why this seemingly impossible result holds. Zeno's paradoxes of motion are of course somewhat related to the concept infinity, and so these are discussed making use of what was obtained in the previous chapter. Thomson's lamp is also related. If a lamp is alternately switched on and off at time instances $1−2^{−n}$, then deciding whether at time $t=1$ the lamp will be on or off is impossible.</p>
<p>With chapter four, probability is introduced. The Simpson paradox and the Monty Hall problem are probably the best known but there are others that allow much more variations and require much more discussion. In the chapter on voting systems we are introduced to social choice theory and Arrow's impossibility theorem. This is not completely unrelated to the topic of game theory which plays a role in, for example, price setting in a economic system. The Braess paradox is the unexpected result that by adding an extra road to a traffic system, the traffic may be slowed down.</p>
<p>With self-reference we are back to the foundations of mathematics with axiomatic set theory, and, among others, the paradoxes of Russell (the set of al sets that are not a member of themselves) and the liar (I am always lying). Inevitably this leads to Gödel's incompleteness theorems, a theory of types, the ZFC axiomatic system, etc. Also the unexpected hanging is a tough paradox discussed here. Somewhat in the same style is the chapter on induction, where some elements of formal logic are introduced.</p>
<p>A chapter involving geometry has curves, areas, and volumes with fractal dimension. There is not really a paradox here, but the fact that a dimension can be a fraction and need not be integer is considered to be paradoxical. But there are other simpler geometric examples. In many calculus books, we find the hard-to-believe fact that we can create an infinitely large overhang by stacking bricks if brick $k$ (numbered from top to bottom) overhangs the underlying one by $1/(k+1)$. This is an example where the mathematical fact that $\sum_{k=}^\infty 1/k$ diverges is replaced by a "story" of stacking overhanging bricks. Some typical mathematical beginners errors can also give some unexpected results, dividing by zero for example, or summing divergent series.</p>
<p>Finally Matt Cook has invited some colleagues to discuss paradoxes from physics. With statistical mechanics, the reader learns about entropy, Maxwell's Demon, and other classics such as the Brownian Ratchet driven by Brownian motion, and the Feynman's sprinkler problem. The unexpected results of special relativity are well known, and quantum physics is still difficult to understand in all its consequences and different interpretations are still discussed today.</p>
<p>In the final chapter the age-old question whether mathematics is discovered or invented is tackled. As one might expect, the answer is not exclusive for one or the other.</p>
<p>Mind, the paradoxes that are mentioned in this survey, are only few and exemplary for the many examples that can be found in this book (there are over 75). I can imagine that for readers who are totally mathematically illiterate, some steps may be hard, if these use terminology or arguments that are taken for granted. Nevertheless also those are considered potential readers because there is a short addendum introducing some very elementary mathematical notation. Cook also added a rather extensive bibliography, but many of the references are papers where the paradoxes were originally formulated, or papers discussing the solution. Thus not really the popularizing kind of literature for further reading. The index though is well stuffed and useful, since there is sometimes cross referencing across the chapters.</p>
<p>I could spot a typo in the discussion of the Banach-Tarsky theorem. When discussing successions of irrational spherical rotations left, right, up, down, denoted as L,R,U,D, strings of these letters are formed to denote points on a sphere. Uniqueness requires eliminating the succession of opposite rotations (free group). Thus UD, DU, LR, or RL are not allowed in a string. However in the table page 25 appears the string DUL which is not allowed.</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>Several paradoxes are analysed in depth. Some are well known others are less familiar: Zeno's paradoxes, Monty Hall problem, Banach-Tarsky theorem, paradoxes related to voting systems, self reference, but also statistical mechanics, special relativity and quantum physics and many more pass the review. The finale is a discussion of the ultimate question: Is mathematics invented or discovered?</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-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Cook</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</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">9780262043465 (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 34.95 (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">368</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/sleight-mind" title="Link to web page">https://mitpress.mit.edu/books/sleight-mind</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/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</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/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</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/01a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A15</a></li>
<li class="field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
<li class="field-item even"><a href="/msc-full/81p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P05</a></li>
<li class="field-item odd"><a href="/msc-full/63a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">63A10</a></li>
<li class="field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li>
<li class="field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li>
</ul>
</span>
Wed, 01 Apr 2020 11:54:56 +0000adhemar50644 at https://euro-math-soc.euBeautiful Symmetry
https://euro-math-soc.eu/review/beautiful-symmetry
<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>Adult colouring books with complicated mandalas and other fine patterns are currently popular. Long before this hype, the work or M.C. Escher and the Islamic patterns such as in the Alhambra in Spain, display explicit and complicated symmetric patterns that have attracted the interest and admiration of people of previous centuries whether they are mathematicians or not. Alex Berke is a computer scientist who is currently graduate student at the MIT Media Lab. He must have thought that the colouring hype of symmetric patterns could be easily applied to teach some elements and concepts of mathematical group theory. The result is this paper colouring book and, in parallel, an on-line version where you see all the rotations, reflections, and translations dynamically illustrated.</p>
<p>The book has no colours, and shows only patterns in black and white. So, the colouring experience is certainly the most blatant fun aspect of this book. However the colouring should not be inspired by personal artistic creativity. The book is challenging in the sense that the reader has to colour the graphics in such a way that certain restrictive conditions are satisfied. For example in a given black-and-white D4 pattern (4-fold rotational symmetry and 4 mirror symmetry axes) the colours should only leave 2-fold rotational symmetries, or looses all its mirror symmetries. Thus the reader is introduced to some mathematical aspects of symmetry to eventually learn the concepts of a cyclic group Cn (containing only n-fold rotations) or the dihedral group Dn (having the symmetry of a regular n-gon with n-fold rotations and n different symmetry axes). In a playful way the reader-colourist learns the concept of a (finite) group and its order. He/she realizes quickly that there must be something like a subgroup, and that the order of a subgroup must divide the order of the group, and that a cyclic group has just one generator, and similar elementary properties.</p>
<p>In a first part only rotations and cyclic groups are discussed. Then reflections are introduced, which, in combination with rotations, generates the dihedral groups of regular polygons. Next translations are added and things then become a little bit more challenging when these are combined with reflections and rotations to generate (infinite) frieze groups. Colouring the patterns becomes more difficult since one has to deal with horizontal and vertical mirror reflections, and glide reflections as well as rotations. The seven frieze groups are obtained, but to illustrate that the mathematics stay in the background, there is for example no formal theorem or proof that these seven friezes are the only ones possible. The reader is just asked to try and use a combination of the known transformations to find a pattern that is different from the seven friezes obtained, but of course none can be found.</p>
<p>While frieze groups are generated using translations in only one direction, wallpaper groups are based on translations in two directions. This gives many more possibilities, and yet there are only 17 different wallpaper groups. Berke goes through all 17, basically just showing the patterns and asking the reader to find the rotations, symmetry axes, and other transformations that are involved.</p>
<p>Berke has chosen for his book the orbifold notation of Conway to indicate the frieze and wallpaper patterns. This is not so easy to remember and interpret for a lay person. So, for reference, this is recalled in an appendix, just like one can find there also some very elementary mathematical definitions. Answers to the challenges are not given there but some can be found on-line at the website of the book, where one should look for <a target="_blank" href="http://www.beautifulsymmetry.onl/solutions">solutions</a>. If you go to the <a target="_blank" href="http://www.beautifulsymmetry.onl/">online version</a> you can also find dynamically generated frieze and wallpaper patterns, and even more circular patterns that will interactively generate random mandala-like <a target="_blank" href="http://www.beautifulsymmetry.onl/circular-pattern">circular patterns</a> using the building blocks used throughout the book. The idea is that these are downloaded and printed so that they can be coloured. The menu of the online version also has a link to fractals (Sierpiński triangle, Sierpiński arrowhead curve, Pythagoras tree), which are not discussed in this book. All this is based on software that is also available via github (it's an open source project).</p>
<p>The book is both fun and challenging. If you love colouring patterns, you will enjoy it, and you will learn some (elementary) concepts of group theory. The fact that you are challenged to colour in such a way that you meet the conditions that are the subject of the exercise, is the learning element. Little children who love to smudge their colour books filled with princesses, gnomes, and flowers, will have no message here. If you want to learn something about groups, you may have lost that childish kind of love for colouring, and then it may look like a lot of colouring to learn only a little. So it may be just right for the youngsters (or adults) who love to puzzle, and solve riddles and sudokus, and that usually form the bulk of the public at math fairs. But they will perhaps not actually be colouring every pattern throughout the book. I can imagine that they will only choose some that they will actually colour and do the colouring for the others only mentally. If you are teaching an introduction to groups it will be a wonderful tool to generate problems to test your students with, and there is a lot of inspiration to be found if you want to organize a hands-on session at a math meeting for youngsters or for a generally interested public. The mathematical interest of eager colourists can be raised and perhaps they will look up some books or websites to learn more on symmetry groups, and detect these patterns also in Islamic decorations or in the work of Escher.</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 this workbook, symmetric patterns are drawn that the reader is supposed to colour to train his/her insight into concepts from elementary group theory such as rotations, reflections, and translations leading to group structures like cyclic and dihedral groups, friezes and wallpaper patterns.</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/alex-berke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alex Berke</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</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">9780262538923 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 19.95 (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">168</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/beautiful-symmetry" title="Link to web page">https://mitpress.mit.edu/books/beautiful-symmetry</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/20-group-theory-and-generalizations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20 Group theory and generalizations</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/20-01-12-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20-01, 12-01</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
<li class="field-item odd"><a href="/msc-full/20f65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20F65</a></li>
<li class="field-item even"><a href="/msc-full/20f50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20F50</a></li>
<li class="field-item odd"><a href="/msc-full/52b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52B45</a></li>
<li class="field-item even"><a href="/msc-full/52c20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52C20</a></li>
</ul>
</span>
Sat, 07 Mar 2020 06:35:30 +0000adhemar50550 at https://euro-math-soc.euDark Data
https://euro-math-soc.eu/review/dark-data
<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>Dark matter and dark energy in cosmology is the matter and energy that cannot be directly observed with the techniques currently available, but we know that it must be there since it is the only explanation for the data that we observe and that cannot be explained by all the measurable matter and energy. This may be our first insight into the existence of dark data. Currently we live in a period of big data thanks to computers and the World Wide Web (part of if is called dark as well) and there are many different ways to collect and process these data. However there are also many different ways in which our analysis may lead to the wrong conclusions because part of the data are missing or wrong, or "dark" as Hand calls it. These dark data can exist for many different reasons. This book is taking a closer look at the phenomenon. Countless examples are described in the book (mostly for UK data). What type of obfuscations can darken our data? Why are some datasets dark? What are the consequences? What could possibly be done to remedy the situation?</p>
<p>Hand starts with a kind of taxonomy of dark data. In what kind of situations are we dealing with dark data? He describes, with many examples, fifteen different phenomena that can lead to dark data. Some are quite obvious like missing data that we know are missing (known unknowns), but there might also be data missing that we are not aware of (unknown unknowns). Some data are intentionally wrong (falsification) or unintentionally (over-simplification or rounding). Conclusions may be obtained for a whole population or over a larger period of time based on data that were only collected for part of the population or were only valid at a certain moment in time (extrapolation), etc. Quite often, the data are wrong or misused for more than one reason.</p>
<p>There is not a formal definition in this book, but nevertheless, using his examples, Hand explains what the different types of dark data are and how they come about, and identifies some of the concepts that he uses throughout the book. For example dark data caused by "self-selection" refers to the fact that data are corrupted because some participants, invited for and online poll, decide not to participate, or prefer not to answer some of the questions. There are problems of designing the sample (even a sample can be big data, in any case the data collected should represent the whole population for which the conclusion is supposed to hold), one has to be careful not to miss what really matters (like causality between data used and the conclusion derived), data can be corrupted by human errors, by summarising or simplifying or rounding the data. People can manipulate data in a creative way (like tax evasion) or corrupt data by deliberately feeding false data (criminal activity, insurance fraud).</p>
<p>Hand also has a chapter on science and dark data, not only were scientists in the course of history tricked in their conclusions by dark data, some also contributed by falsifying published data intentionally, or they may have been biassed by a general belief or intuition. John Ioannidis threw the cat among the pigeons with his 2005 paper <a target="_blank" href="https://doi.org/10.1371/journal.pmed.0020124"><em>Why Most Published Research Findings Are False</em></a>. Reproducibility was recognised as a major problem and research institutions are demanding data management plans in funding applications. Data management has grown into big business interfering with problems of privacy and GDPR. Distinguishing truth from reality, has become increasingly difficult in our digital world. Encryption, verification, identification, authentication, etc. can hardly keep up chasing creative fictionalisation. Artificial Intelligence algorithms based on machine-learning try to analyse the data that are too massive for humans to deal with, but even these machines can be led astray by dark data.</p>
<p>Thus it has become a major problem to recognise dark data and to know how to deal with it and avoid wrong conclusions. This is what Hand is discussing in the first two chapters of part II of the book (part II has a third chapter that is also the last chapter of the book which is summarising the taxonomy that was described in the early chapters). First we need identify why some data are missing. Here Hand considers three different types: it can be a random phenomenon but related to the missing data (UDD = Unseen Data Dependent) like some may be reluctant to give their BMI when it is high. Or missing data may depends on the data previously observed (SDD = Seen Data Dependent) like a BMI not given because it has increased since the last registered observation. Finally, data are missing but that does not depend in any way on the data observed (NDD = Not Data Dependent). Recognising the mechanism behind the missing data is important because it defines how one should deal with the data, for example on whether and how to complete the missing data or not. Effects of NDD and SDD can be be cured, but UDD is more difficult to deal with.</p>
<p>Dark data can also be beneficial if it is detected and if that leads to a reformulation of the question that we want to answer, or it may lead to strategic elimination of some data that would bias the result. To avoid dark data, it helps to randomise the sample and even hide data from the researcher (like not revealing who did and who did not get the placebo). An obvious way to fill up missing data is to use averages, but a somewhat strange advise is to fill up these data by simulation. It is a valid way to generate data in case of a simple model with known probability distribution, but when it involves a complicated model, then these models are simplifications of reality built upon observations that may involve dark data. Similarly machine-learning techniques involve massive analysis of data that may be corrupted. In these cases it can only be hoped that over the repeated simulations or over the whole learning process the wrong data are not systematic and are averaged out over the iteration. Or one may apply techniques such as boosting and bootstrapping to reduce bias. Bayesian statistics helps to test hypotheses and thus confirm or refute intuitive assumptions. Cryptography can help to make data anonymous so that people are more willing to provide correct data or to prevent the introduction of false data by fake persons. It may even help to make some data deliberately dark, not making them available to users but still using them in computations.</p>
<p>As mentioned above, the description is an exploration of a major problem in data analysis with an attempt of classification, analysing causes, mechanisms, and to some extent also suggest mitigations. However most of the book consists of examples and particular cases to clearly explain the ideas. Nowhere, however is there a concrete or general well defined statistical, mathematical, or algorithmic solution given. This is mainly a wake-up call that clearly points at a major problem that any scientist has to be aware of and that he or she should think about how to deal with. Certainly statisticians, applied mathematicians, computer scientists, but in fact anyone dealing with data (big or not) should be well aware of the "darkness" of their data.</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 description of how important it can be that in our treatment of data, some of them are missing, or fake. Conclusions derived from these corrupted data can be biassed or wrong. How do we recognize the dark data? How can we deal with the phenomenon? These are answers that Hand deals with in this book. The approach is mainly descriptive with an abundance of examples, mainly from data related to the UK situation. Suggestions are given, but no concrete precise or detailed mathematical or statistical analysis or algorithms is discussed in detail.</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-j-hand" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David J. Hand</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">2020</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">9780691182377 (hbk), 9780691198859 (ebk), 9780691199184 (abk) </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">£ 26.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">344</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/9780691182377/dark-data" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691182377/dark-data</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/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</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/62d99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62D99</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/68p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P99</a></li>
<li class="field-item odd"><a href="/msc-full/68t09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68T09</a></li>
</ul>
</span>
Mon, 03 Feb 2020 12:29:51 +0000adhemar50375 at https://euro-math-soc.euEuler's Gem
https://euro-math-soc.eu/review/eulers-gem-0
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The fact that the book is reprinted in its original version as a volume of the <em>Princeton Science Library</em> is a quality label as such. For completeness, I should here also mention the subtitle: "The polyhedron formula and the birth of topology". This rules out that by the "gem" is not meant the other famous Euler formula $e^{i\pi}+1=0$, but that it concerns the polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 (the letters stand for Vertices, Edges and Faces).</p>
<p>
A short introduction serves as a teaser for the reader and explains the kind of problems that will be discussed in the sequel. A reasonable way to start the full story is to give a biography of Euler. He is recognized as one of the greatest mathematical minds of all times. Next, the reader is surprised by the fact that it is not so straightforward to define a polyhedron, or at least to identify the ones that one wants to focus on. So the reader is warped back to the Greek origin when the five Platonic solids (discussed in Book XIII of Euclid's Elements) and the thirteen Archimedean solids were studied. We then have to take a leap to the Renaissance of the fifteenth century before the polyhedra and the Greek knowledge was rediscovered. Kepler later built a whole world view and a solar system on polyhedral shapes. And then came Euler, who detected his above mentioned "gem". It is so simple an observation that it comes as a real surprise that, as far as we know, it was missed by everyone so far. Some explanation may be that previously one concentrated on the vertices and the faces (or the solid angles), but Euler also considered the edges as essential components of a polyhedron. Richeson reproduces Euler's proof (1750-51), but the formula does not hold for all (regular) polyhedra. When does it hold and when does it not? The reader is expecting to read the answer in the next chapter, but then Richeson surprises again by revealing that Descartes may have been the first one to have discovered the formula because a similar relation was found posthumously in his notes (notes that were miraculously saved from oblivion). However, a complete rigorous proof was only given by Legendre (in 1794), a proof that Richeson also explains later in the book.</p>
<p>
From the story told so far, there are already many historical mathematicians involved and Richeson gives every time some short biographical sketch to situate him (so far only men) as a person who existed and lived a life of his own. It is not just some abstract name used to identify a result. Note also the way in which Richeson builds up his story. He takes the reader along to think about what polyhedra are, for what polyhedra does the formula hold, and how could it be adopted to hold in more general case? Euler and his proof are some kind of a climax, but then Descartes shows up as an unexpected twist, and Legendre's proof is not based on properties of planar faces but (another surprise) requires geometry on a sphere with geodesics. This shows how well the book is written and how Richeson manages to fascinate the reader, and make her curious about what is coming up next.</p>
<p>
And next chapter is again some kind of a surprise because it introduces the problem of the Bridges of Köningsburg and how Euler solved the problem which is considered as the origin of graph theory. Not so surprising though if you know that a graph consists of vertices and edges. Cauchy uses this idea by projecting a polyhedron on a plane, giving a plane graph that can be analysed to prove Euler's formula. Now Richeson's story takes off into graph theory and applications: recreational mathematics (the game of sprouts and Brussels sprouts invented by Conway), the four colour problem for planar maps (and other graph colouring problems). Graph based proofs for the polyhedra formula and generalisations concludes the graph theory subject.</p>
<p>
Next Richeson embarks on proper topology as the rubber sheet version of the usual geometry. This requires some new concepts and a classification of all surfaces. Therefore one needs to know when a surface is or is not homeomorphic to another and thus are topologically the same. For example, a torus is a sphere with a handle, a Möbius band is the same as a cross cap, and the projective plane is a sphere with a cross cap. Classification is connected to the definition of the Euler characteristic (or Euler number as Richeson calls it). Make a finite partition of the surface and count the "rubber versions" of vertices, edges and faces, then the Euler formula gives the characteristic <em>χ</em></p>
<p>
which is an invariant for the surface (2−2<em>g</em> for a sphere with <em>g</em> handles, and 2−<em>c</em> for a sphere with <em>c</em></p>
<p>
cross caps). This characteristic and the orientability of the surface allows some classification as started by Riemann but only completed in 1907 by Dehn and Heegaard.</p>
<p>
I have to say that, although now Richeson is still explaining things at an introductory level of topology, (and continues to do so), it will take a more persistent and motivated topological layman to follow in pace and read on. We arrived now at about two thirds of the main text of the book and the mathematical level is not decreasing for the last part. It continues with knots and Seifert surfaces (whose boundary is a knot or link), the hairy ball theorem for vector fields on a sphere and more generally the Poincaré-Hopf theorem on surfaces with boundary, Brouwer fixed point theorem, the angle excess theorem for a surface, the Gauss-Bonnet theorems about the total curvature of an orientable surface, Betti numbers, and Richeson ends with an epilogue about the Poincaré conjecture. All of these are nicely presented in a smooth and logical succession by Richeson, but they are too technical to be discussed at the level of this review. However, for example an undergraduate mathematics student should not have a serious problem to read on.</p>
<p>
Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious. Richeson also adds in an appendix building patterns that can be used to make paper models of polyhedra, of a (square and edgy, yet topologically a perfect) torus and even (the paper realization that looks like) a Klein bottle, or a projective plane. In the text he also gives advise on how to prepare the liquid to make soap-bubble models. These are aids to help visualising the surfaces if the graphics of the text do not suffice. There is a long list of papers referred to in the text, but also an appendix with an annotated survey of recommended literature.</p>
<p>
Except for an additional preface by the author, the book is the unaltered reprint of the original version of 2009. Thus for example the facts of Martin Gardner passing away in 2010 and Perelman refusing the Millennium Prize for proving the Poincaré conjecture were still unknown in 2009. Although the latter was to be expected since he had already declined the Fields Medal in 2006 and an EMS Prize.</p>
<p>
The first half of the book can be considered as a popular science book on a mathematical subject written for everyone. Depending on the motivation or knowledge of the reader this might or might not include the part on graph theory. Once Richeson dives deeper into topology, it becomes more a popular science book for the mathematics student of at least an amateur mathematician. People who are interested in this book may also be interested in a more recent book by Richeson that has also been reviewed here <a href="/review/tales-impossibility" target="_blank"><em>Tales of Impossibility</em></a>.</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 the book of 2009, that is now reprinted in the <em>Princeton Science Library</em>. Richeson gives an account of 2500 year of mathematical history that runs from the Greek's approach to regular polyhedra to the modern problems of topology, all centred around Euler's polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 and its generalisations.</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-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</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">9780691191379 (pbk), 9780691191997 (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">£ 14.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">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://press.princeton.edu/books/paperback/9780691191379/eulers-gem" title="Link to web page">https://press.princeton.edu/books/paperback/9780691191379/eulers-gem</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/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</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/55-algebraic-topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55 Algebraic topology</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/55-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55-03</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/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li>
<li class="field-item odd"><a href="/msc-full/54-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">54-03</a></li>
<li class="field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li>
<li class="field-item odd"><a href="/msc-full/51m20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51m20</a></li>
</ul>
</span>
Fri, 31 Jan 2020 10:49:42 +0000adhemar50363 at https://euro-math-soc.euThe Best Writing on Mathematics 2019
https://euro-math-soc.eu/review/best-writing-mathematics-2019
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the 10th volume in this series reprinting every year a collection of diverse texts on mathematics (i.e., not necessarily mathematical papers) that are accessible to a broad public. I have been reviewing these books since 2012, and I have repeatedly explained the idea behind the concept and the kind of papers that are selected in my reviews. These ideas have not changed in this anniversary volume, so I will not repeat them here. If you are not familiar with the concept of the series, you can look it up and read all about it in the previous reviews <a href="/review/best-writing-mathematics-2012">2012</a>, <a href="/review/best-writing-mathematics-2013">2013</a>, <a href="/review/best-writing-mathematics-2014">2014</a>, <a href="/review/best-writing-mathematics-2015">2015</a>, <a href="/review/best-writing-mathematics-2016">2016</a>, <a href="/review/best-writing-mathematics-2017">2017</a>, <a href="/review/best-writing-mathematics-2018">2018</a>.</p>
<p>
This volume reprints 18 papers almost all originally published in 2018. The fact that the subjects of the papers are usually crossing the boundary between two or more domains is one of their interesting features. It is remarkable how smoothly the sequence of papers in this book migrates from one subject into the next, due to a careful selection and collation strategy of the editor.</p>
<p>
For example the first paper links geometry to gerrymandering. The latter is a manipulative subdivision of the sets of voters in a the-winner-takes-it-all system to enforce some outcome of the voting. Finding a fair subdivision is a combinatorial problem that can only be solved in a feasible time using Markov Chain Monte Carlo techniques. This smoothly connects to the next paper about a problem from the <em>Scottish Book</em>, a legendary diary from Polish mathematicians meeting in Lviv (Poland) in the 1930's. The problem posed by Hugo Steinhaus in there gave rise to the ham-sandwich theorem, which is also about a problem of fair partitioning. In two dimensions the problem reduces to cutting a pizza and all of its ingredients distributed on top into fair parts.</p>
<p>
Politicians may be interested in gerrymandering and perhaps even in fair distribution, but they may also have something to say on the educational system, and in how to distribute different subjects that children have to learn over a limited education time. In that respect it is important to know if mathematics learns children how to think. Some claim that this can also be learned by studying languages (like Greek and Latin), computer science, or even by solving brain teasers and puzzles. After a careful analysis of this question in relation with different mathematical subjects, the authors of the next paper, conclude with some recommendations on how to teach calculus.</p>
<p>
Speaking of puzzles, the next paper deals with the Rubik's cube and all its generalizations that were realized practically or that were studied on an abstract mathematical basis. Three-dimensional geometry of the cubes brings the reader to the next paper discussing 3D objects that when viewed from different viewpoints create some optical illusions. This optical paradox is geometrically analysed and ingeniously illustrated using a picture of the object simultaneously with its reflection in a mirror representing the alternative viewpoint. The mirror is a perfect link to the detection of mirror symmetry in string theory, which became an important subject in both theoretical physics and algebraic geometry.</p>
<p>
The illustrations in the texts are grey-scale, but when in the original text they were in colour, then sometimes the caption of a grey-scale image refers a line or area of a certain colour. To mitigate this, colour versions of the illustrations of all the papers are collected at this point of the book. This somewhat hides the abrupt switch to more computer related papers that now follow. The first of these more computational type texts is about the application of a so called probabilistic abacus to find the probability that some event will happen. This computational mechanism was invented by A. Engel in 1975. It simulates a finite game played on a graph based on chip-firing. This computational technique is now known as Engel's algorithm.</p>
<p>
Computers play also an increasing role in the analysis and classification of integer sequences. The on-line encyclopedia (<a href="https://oeis.org/" target="_blank">OEIS</a>) started by Neil Sloane in 1996 had 100k entries in 2004. Sloane's paper in this collection is listing some fascinating examples among which an (in 2018) recent entry 250000. At the time of writing this review (Jan 2020) the OEIS has 331811 entries and counting. If anything is related to computers nowadays, then it is certainly big data. That topic made a bliz career in research funding and was promptly turned into a buzz word. The next paper briefly discusses examples of well known big date problems: from search engines to health care to recommender systems to farming, and I am sure we haven't seen the last of it</p>
<p>
What can be computed or even what can be decided is a fundamental question to ask in computer science as well as in mathematics (cfr. the halting problem and Gödel's incompleteness theorem). The next paper explains that deciding whether all materials have a spectral gap (i.e. the gap between the energy of the ground state and the first excited state) is proved to be impossible, using Turing machines and ideas from plane tilings. Computer generated proofs and verifying proofs by computers become more and more common practice. That is illustrated with some historical examples in a paper that is wondering how we should proceed for the future.</p>
<p>
Quantum physics and the quest for a theory of everything has divided physics research. The pure mathematical labyrinth in which theoretical physics has evolved as opposed to the classical empirical physics is not completely unrelated to mathematical models that have been designed for other scientific disciplines. The phenomena one wants to study are simplified to models that isolate some interesting characteristics. Given such a model (as a set of equations and constraints), also solving the models analytically or computationally, may require further simplifications to become feasible. Computed results are validated and when not matching with reality, the model may need adaptation. Is not mathematics of modelling here a kind of empirical science. This brings us on the verge of philosophy about mathematics. More philosophy is in a paper asking what it means that 2+3=5 (what is meant is adding of numbers, not counting quantities), Do the numbers 2 and 3 actually exist? We assume they do, since it is so obvious. But why then prove Fermat's last theorem while it is so obvious that it must hold? More on philosophy, in particular about the link to the history of mathematics is illustrated in a paper about Gregory's notion of infinitesimals and continuity as compared to the Weierstrass approach of epsilon-delta definitions. Some purists think infinitesimals are evil, others consider it a blessing to work with. The authors however conclude that eventually, after closer analysis, the two historical approaches are not that different.</p>
<p>
We humans do not like chaos. We try to make sense of things and are constantly looking for patterns. The Kolmogorov complexity corresponds to finding the shortest program that can describe some (mathematical) object like for example a sequence. This links back to the previously discussed problem of computability or decidability. The seemingly complex problem to describe "the smallest number that cannot be described by less that 15 words" is trivial and yet impossible to grasp. Just like an infinitesimal, something smaller than anything finite and yet not zero is difficult to conceive, and still easy to describe and work with.</p>
<p>
What we believe to be true and what actually is true is, with the constant exposure to information, an important issue in an epoch of fake news. Statistics is in this respect a seemingly scientific tool to sustain some fact, but unfortunately, it is easily misused. A paper discussing this ethical issue gives some recommendations about this like: be open about data and methods, be aware of the limitations of statistics, be open for criticism, etc. and I would like to add to that: be careful about causality claims.</p>
<p>
The two remaining contributions are diverse. One is a plea to return to the original idea of Fields when he installed the Fields Medal. Should one recognize brilliant mathematicians who accomplished something big in mathematics and thus are already "established", or should one celebrate a mathematician who is pioneering a new field in mathematics? The original idea was to stimulate (international) collaboration, not competition. Since the Fields Medal got the status of a mathematical Nobel Prize around the 1960's, that original idea is violated and it became the subject of competition. The last paper is about an Eulogy delivered by Melvyn Nathanson for Paul Erdős in 1996 shortly after Erdős passed away, and some considerations Nathanson has to add now (in 2018). The paradox of Erdős is that he was enormously prolific and versatile, even creating new fields and yet he never embraced the new mathematical domains of the twentieth century. How could he publish such important theorems and yet know relatively little?</p>
<p>
I should also mention the list of interesting books that appeared in 2018 and that get some recommendation from Pitici. As in previous volumes there is also a long list of papers that could have been selected as well for this collection (but they were not) and of other writings such as reviews of books and essays, teaching notes, and special journal issues. Thus this book is again a most interesting collection of mathematics related papers of the usual quality.</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 volume 10 of Picici's annual harvest of papers on diverse topics related to mathematics that are collected from different journals and books. The contributions relate mathematics to philosophy, history, education, communication, computer science, games, puzzles, statistics, etc. Most of them were published in 2018 and are written for a generally interested readership.</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/mircea-pitici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Pitici</a></li>
<li class="field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</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">9780691198675 (hbk), 9780691198354 (pbk), 9780691197944 (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"> £ 66.00 (hbk), £ 20.00 (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">287</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/9780691198675/the-best-writing-on-mathematics-2019" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691198675/the-best-writing-on-mathematics-2019</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/00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00b15</a></li>
</ul>
</span>
Fri, 31 Jan 2020 10:37:23 +0000adhemar50361 at https://euro-math-soc.eu