European Mathematical Society - princeton university press
https://euro-math-soc.eu/publisher/princeton-university-press
enHot 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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links 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><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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/35k05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35s30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35S30</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/35l05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35L05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35k57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K57</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel50806 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/hot-molecules-cold-electrons#commentsDark 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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-j-hand" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David J. Hand</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/62d99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62D99</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/68p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P99</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel50375 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/dark-data#commentsEuler'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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/55-algebraic-topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55 Algebraic topology</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/55-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55-03</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/54-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">54-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel50363 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/eulers-gem-0#commentsThe 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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/mircea-pitici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Pitici</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00b15</a></li></ul></span>Fri, 31 Jan 2020 10:37:23 +0000Adhemar Bultheel50361 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/best-writing-mathematics-2019#commentsCalculus Reordered. A History of the Big Ideas
https://euro-math-soc.eu/review/calculus-reordered-history-big-ideas
<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"></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">Ángeles Prieto</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="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/BressoudCalculusReordered.pdf" type="application/pdf; length=66685">BressoudCalculusReordered.pdf</a></span></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 very-well written book can be strongly recommended as a resource for instructors. Also students will benefit greatly from Bressoud’s journey through centuries to explain how calculus evolved into its current structure.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-m-bressoud" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David M. Bressoud</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-18131-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">$29.95</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">242</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691181318/calculus-reordered" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691181318/calculus-reordered</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/26-real-functions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26 Real functions</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/26a-functions-one-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26A Functions of one variable</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span>Mon, 27 Jan 2020 17:04:58 +0000Ángeles Prieto50335 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-reordered-history-big-ideas#commentsPerspective and Projective Geometry
https://euro-math-soc.eu/review/perspective-and-projective-geometry
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a workbook to learn the techniques of perspective drawing and the theory of perspective and projective geometry. The exercises range from very practical to proving theorems, but it is essentially based on experiments and discovering the theory by practicing on the examples.</p>
<p>
The book starts with an example of a very practical experiment. One person (the director) has to stand immobile with one eye closed in front of a big window at a few meters distance. He or she is looking at the landscape or the buildings outside. The view should have preferably many straight lines. The window is used as a canvas, on which the outside world has to be projected and the director has to instruct other students (the artists) to fix waste tape on the window where the director sees the projected straight lines of the outside world. This is a way to detect how a 3D world is represented on a 2D window-canvas. The first module has several questions that can be answered in the blank space that is left open for that purpose or one has to check true/false answers or choose from multiple choice possibilities. All the pages of the book are perforated so that they can be torn out and handed in for correction or feedback. There is also a section for homework with more questions to answer (marked with circled E: Ⓔ) and art assignments like make drawings or take pictures (marked with a triangulated A) or more theoretical exercises related to theorems and proofs (marked with a squared P). In an appendix to this first module some explanation is given that should lead to a definition of sketches in n-point perspective, which is what the subsequent modules will work to on a more theoretical basis.</p>
<p>
This first module described above is an example of how all the 13 modules are organized, although some are more theoretical, and none of the others have an appendix. Some of the questions are incomplete and the student has to guess what the question is. That has of course been prepared in previous questions, but still it may be a problem to correctly complete the sentence, which makes it impossible to continue with the next questions. Therefore, I believe this is not a workbook for self-study, a teacher should be guiding the process but it remains a challenge for the student whose responsibility is to discover the proper way to go or to detect the concepts and the theorems that support the constructions.</p>
<p>
To illustrate how one moves from the window taping experiment to the theory, we note that in section 2, one has to analyse and complete the graphic representation of a 3-dimensional construction of two thick tiles on top of each other in the form of the letter T, drawn in (a 1-point) perspective. And then module three is stuffing up the theory with definitions and properties of points, lines, segments, planes, and module four is introducing geometry in $\mathbb{R}^2$ and $\mathbb{R}^3$ with the announcement of Ceva's and Menelaus's theorems. Module 5 extends the Euclidean space in (2 and 3 dimensions) by introducing ideal points, lines and planes, that are essentially the points at infinity. This gives the extended spaces $\mathbb{E}^2$ and $\mathbb{E}^3$. This seems to complicate things, but it actually simplifies life since no exception has to be made for these ideal objects which is the whole idea of projective geometry. To formalize the perspective drawings, meshes and maps are defined in $\mathbb{E}^3$</p>
<p>
. The latter allow to project a 3D scene onto a 2D plane and it can be used for example to correctly project equispaced segments (say vertical poles of a fence or a square tiling of a floor) onto non-equispaced distances on the canvas or how to correctly draw a poster on a wall that is represented on a canvas in perspective.</p>
<p>
The theory continues as above, but always in connection with practical problems related to the plotting of 3D scenes on a 2D canvas. The next issue is Desargues theorem (two triangles are in line perspective if and only if they are in point perspective). It is formulated in terms of meshes in $\mathbb{E}^3$ and a proof is to be derived. On the other hand, this module also introduces exercises in GeoGebra (a free interactive software package for geometry, algebra, and other computations). An elementary introduction to using GeoGebra is added in an appendix. Now the student should know enough about projective geometry to move objects around with concepts like (perspective) collineations, homologies and how these connect with harmonic sets. One may now experiment by moving points in a GeoGebra plot.</p>
<p>
Herewith the modules move somewhat towards numerics. For example the position of the designer in the taped window example could be found by someone who moves herself into a position where the scene outside aligns with the tapes on the window. But now, at this stage of the book, it is possible to compute the distance of the viewer's eye to the canvas from the perspective drawing. Or it should be possible to derive from the 1-point perspective projection of a box whether it is a cube or not. Module 10 allows to draw boxes in 2-point perspective, respecting the actual distances and module 11 catches the cross ratio of four points as a numerical invariant allowing to draw lines in perspective that are equidistant in reality, a problem that was also previously considered in module 6. Another invariant is a more complex $h$-expression relating distances between 8 points in a rectangular configuration. This is named after and proved by Howard Eves (1911-2004 — the authors give wrong dates 1913-2000). Also the Casey angle of 4 collinear points is another invariant as proved by M. Frantz and named after John Casey (1820-1891) (the theorem here is not to be confused with what is generally known as Casey's theorem).</p>
<p>
The last two modules introduce the Cartesian coordinate system, projective coordinates and linear algebra and even some topological concepts like the Möbius band and the possible shape of $\mathbb{E}^2$ (the four colour theorem does not hold for this space since six colours are necessary in general).</p>
<p>
Every module starts with a one-page graphic, which is to be worked on as one progresses in the module, and thus it is somehow the target and motivation for the module that is coming up. For reference, the main definitions and theorems are summarized in an appendix. A last appendix deals with writing mathematical prose, and that includes style, punctuation, use of formulas and words, etc. This is of course important for any student who has to learn to write mathematics, but even more so it will be important for students who follow the course with an artistic background, and who have not been exposed very often to mathematical texts. I assume however that they may find the abstract mathematics of this text a bit hard to do.</p>
<p>
The book is a nice mixture of mathematics (with rather abstract concepts and proofs), but with practical introductions of these concepts and interesting applications in art as well as in practical situations. Moreover, it gives an introduction to the computer system GeoGebra, and a hint towards linear algebra, analytic geometry and topology. There are excursions into historical aspects, music, photography, and many other tracks. But foremost it is a workbook with an extensive list of many different assignments ("cool" problems to use the language of the authors). More material related to these modules is available at the <a href="http://www.fumikofutamura.com/mathart" target="_blank">Futamura website</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 a workbook to instruct the techniques and the mathematics of perspective and projective geometry. It has many blank spaces where the student can write the answers to the questions and of course there are many graphics, partially set up, that he or she has to complete. The pages are perforated and can be torn out to be handed in for correction and feedback.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/annalisa-crannell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Annalisa Crannell</a></li><li class="vocabulary-links field-item odd"><a href="/author/marc-frantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marc Frantz</a></li><li class="vocabulary-links field-item even"><a href="/author/and-fumiko-futamura" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">and Fumiko Futamura</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">9780691196558 (hbk), 9780691196565 (pbk), 9780691197388 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 97.00 (hbk), £ 40.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">296</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691196558/perspective-and-projective-geometry" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691196558/perspective-and-projective-geometry</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/51n15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51N15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51A05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span>Fri, 20 Dec 2019 14:54:13 +0000Adhemar Bultheel50114 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/perspective-and-projective-geometry#commentsCurves for the Mathematically Curious
https://euro-math-soc.eu/review/curves-mathematically-curious
<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>
Julian Havil has already produced several popular math books. Some of them have been reviewed here: <a href="/review/impossible-surprising-solutions-counterintuitive-conundrums" target="_blank">Impossible?</a> (2008), <a href="/review/irrationals-story-numbers-you-cant-count" target="_blank">The Irrationals</a> (2012) and <a href="/review/john-napier-life-logarithms-and-legacy" target="_blank">John Napier</a> (2014). In this book, containing an anthology of ten iconic curves, he takes another angle of approach to tell more stories about mathematics. Havil's popular math books are more of the recreational kind. I mean that while telling his story, he is not hiding away the mathematics. There can be many formulas and derivations that are however easy to follow with some background in basic calculus.</p>
<p>
The curves selected have names. That is because they are in some sense important. If it is the name of a mathematician, it is, like often in mathematics, not always the name of the one who first studied the object. This is again illustrated by Havil in this book when he explores the history underlying the origin of the curve. There is one chapter for every curve. Sometimes it is just one particular curve described by a unique formula like the catenary, but often these curves have parameters or it is just a whole collection of curves with a special property like space filling curves. "Why these ten?" is an obvious question to ask, and Havil has anticipated this because he opens every chapter with a section that explains why he has chosen this curve. Whatever reason he gives, what is important for the reader is that there is always a story or stories worth telling that can be connected to that curve and in some cases these also have a very long history. Let me illustrate the yeast of the book by a telegraphic survey of the ten chapters.</p>
<p>
1. <em>The Euler spiral</em>. Its parametrizations are analyzed and the connection with elastic curves and Fresnel integrals. It is also known under other names (e.g. Coru spiral and clothoid), and Havil also explains the history of how and why this has happened.</p>
<p>
2. <em>The Weierstrass curve</em>. This is defined as an infinite sum and it is probably the first fractal ever described: a continuous function that is nowhere differentiable. The proof of Weierstrass for these properties is included.</p>
<p>
3. <em>Bézier curves</em>. This is an introduction to the characterization of these curves and how they are constructed by the Casteljau algorithm. There are two fun stories connected to these curves. One is about a Bézier curve called Lump which is the name of a dachshund as it was sketched by Picasso caught in one smooth Bézier curve. Havil provides its control points. Another story on the side is about how these curves are used to design letter fonts.</p>
<p>
4. <em>The rectangular hyperbola</em>. This is an excellent occasion to tell the history of how logarithms were invented. This is of course described in much more detail in Havil's book about John Napier.</p>
<p>
5. <em>The quadratrix of Hippia</em>. The history of this curve is connected to the classic Greek problem of trisecting an angle using only compass and straight edge, but the story would not be complete if one did not recall also the other "impossible" problems of squaring the circle and doubling the cube. The quadratrix is formed by the intersection of two moving lines one translating and another rotating at constant speed. If one could construct that curve, then trisecting an angle and squaring the circle became possible as well as constructing segments whose length is a unit fraction or a square root. The latter are examples of how Havil manages to add some extra mathematics of his own to a well known story.</p>
<p>
6. <em>Two space-filling curves</em>. Cantor, Hilbert, and Peano, are three names connected with these curves. The construction of these curves is of course related to the study of cardinality. The Peano curve is a continuous map from a unit interval to a unit square but it is not surjective.</p>
<p>
7. <em>Curves of constant width</em>. These are curves like the Reuleaux triangle that looks like a triangle that is slightly inflated, and yet shares many properties with a circle. If it is used as a drill, it will produce square holes (with slightly rounded corners). But there are several generalizations to study. Again, the latter are typical examples of Havil's mathematical extras.</p>
<p>
8. <em>The normal curve</em>. This bell shaped curve is probably best known since it represents the normal probability distribution and it is related to the accumulation of rounding errors in long computations. No introduction to probability or statistics is possible without it. There are a few less known names of mathematicians that show up in the birth history of this curve.</p>
<p>
9. <em>The catenary</em>. This is the curve formed by a chain loosely hanging from its fixed extremes. It looks deceptively like a parabola, but it isn't and that has fooled some mathematicians of the past. It is of course a place to discuss also the other hyperbolic functions. This is one of the curves that has been used to shape bridges and arches. It is also the shape of the road on which one can smoothly drive with square wheels.</p>
<p>
10. <em>Elliptic curves</em>. These are the most complex curves of the book. They are related to Diophantine equations and they are most famous for their use in cryptography.</p>
<p>
It is clear that the variety of topics is very broad: form constructions with compass and straight edge to cryptography and from the foundations of mathematics to the design on fonts with Bézier curves and the Casteljau algorithm. There are also seven short appendices explaining some preliminaries or expanding on some topics. However the first appendix is a surprise. On one of the very first pages of the book (page ii, before the title page) there are two 13 × 41 blocks of decimal digits or a number <em>N</em> of over 500 digits spread over 13 lines. No reference, no explanation. The explanation comes in the first appendix. It shows a complicated formula whose main ingredient is a modulo 2 formula for an expression depending on <em>x</em> and <em>y</em>. It thus gives a 0 or 1 depending on <em>x</em> and <em>y</em> which are assumed within certain bounds. The bounds for <em>y</em> depend on a number <em>N</em>, It turns out that it describes the pixel values within a rectangle of a page that will reproduce a pixelated image of the formula on a 106 × 17 pixel grid. Thus the <em>N</em> is the decimal representation of the binary number with 106 × 17 = 1802 bits giving the bit pattern of the pixmap one wants to generate. The two blocks at the beginning of the book give the two <em>N</em> values needed to reproduce the title of this book in pixel-form. The idea is from a 2001 paper by computer scientist Jeff Tupper.<br />
A few pages further at the beginning of the book on page vi shortly after the title page, there is a mathematical doodle with nine wild curves symmetrically arranged in a 3 x 3 matrix, and a trigonometric formula. No further explanation, hence leaving it as a puzzle and a challenge to tease the reader.</p>
<p>
There is more serious mathematics to be found in some other relatively long excursions in the chapters. Many of them are following some historical evolution of the problem. For example in the chapter on the normal distribution there is a lot of formula manipulation to move from a binomial distribution, via summing binomial coefficients and Bernoulli numbers, to finally arrive at the exponential expression. The discussion that a bijective map from the unit interval to the unit square cannot be continuous is illustrated by following the steps of the proof of continuity and non-differentiability as given for the Peano curve. The move from an parametrization of the Euler spiral to a simple one, parametrized by arc length, is fully explained and variations in the parametrization can produce very frivolous curves. And there are more not-so-trivial derivations in other chapters that can set the reader on a DIY path for further exploration. The fun items on page ii and vi will certainly trigger the interest of the mathematical puzzlers to find explanations or variations. The conclusion of the book is that $x^2+(\frac{5}{4}y-\sqrt{|x|})^2=1$ is the most important curve of all and it is indeed a lovely one.</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 book Julian Havil selects ten iconic curves to tell entertaining stories about mathematics. The stories are written for a broad audience, but still there is also a lot of juggling with formulas. Some basic background in mathematical calculus should however suffice.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/julian-havil" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">julian havil</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">9780691180052 (hbk), 9780691197784 (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">£ 24.00 (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">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebraic-and-complex-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebraic and Complex Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/53a04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A04</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/14h50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14H50</a></li></ul></span>Mon, 02 Dec 2019 07:18:48 +0000Adhemar Bultheel50003 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/curves-mathematically-curious#commentsOpt Art
https://euro-math-soc.eu/review/opt-art
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Robert Bosch is a mathematician who has produced computer generated art that can be found on his <a href="http://www.dominoartwork.com/" target="_blank">website</a> and that he also presented at the Bridges mathart conferences. The pleasing optical effect of his graphical art is generated by solving some constrained optimization problem. The objective function is often simple, but the challenge is to design and formulate appropriate constraints.</p>
<p>One possible example is the construction of mosaics to represent a given image in a recognizable way where the elementary tiles of the mosaic are used like dots in halftone images or in pointillist paintings. Given is a restricted set of (small) tiles to choose from and some (large) picture. Put a relatively fine square grid over the picture and replace each square of the grid (let's call them pixels) with a tile from your set whose greyscale approximates the original greyscale of that square. The first example in this book is using flexible Truchet tiles to build the picture. A Truchet tile is a square tile with a diagonal dividing the square into a black and a white triangle. By introducing a parameter to move the midpoint of the diagonal along the other diagonal it is possible to generate tiles whose average greyscale ranges from 75 percent black to 75 percent white. Finding the optimal parameter for each tile to match the greyscale of the corresponding square pixel of the image is a large constrained optimization problem. Note that a Truchet tile has four rotationally symmetric siblings. This can be taken into account by defining larger composed tiles in which the unit tiles have a prescribed orientation. Obviously what is done for squares can be extended to any other shape of tile with rotational symmetry that fills the plane. Representing coloured instead of greyscale images can be an extra complication.</p>
<p>As an example of a constrained optimization problem some elementary introduction to the simplex method is given and it is illustrated how such a problem can be solved. It is not necessary to understand all the mathematics since computations are done by optimization software. One only needs to know how to feed the problem to the software. Most applications in the book use the Gurobi optimization software, except for the travelling salesman problems, for which the Concorde TSP package is used. Once the software is available, the previous idea of Truchet tiles matching the greyscale of a picture can be applied by using any dictionary of tiles with different average degrees of darkness. It becomes more challenging when one uses domino tiles, which for this purpose are double nines, that means that the number of dots on half the domino is not between zero and six as usual, but they have between zero and nine dots. Thus there are 55 domino tiles in a complete set ranging from double blank to double nine without repetition. The extra complications with respect to the previous problems are that these tiles are rectangular, and one has to decide how they are oriented to cover two square pixels of the original image. Moreover one can restrict the available dominoes to be only a finite number of complete domino sets that should be used completely. Thus there is only a finite number of copies of each tile. Two methods are explained to solve this problem.</p>
<p>Another goldmine to dig from is the travelling salesman problem (TSP) and all its applications. First it is explained what the optimization problem is and how to solve it approximately and how to avoid disjunct subtours. One should first select random points that are closer together where the image is dark and sparser where the image is light. That is called stippling. Again there are algorithms to solve this stippling problem and they will generate a set of points. This is based on MacQueen's unsupervised learning algorithm to detect clusters. Once the dots are chosen, they need to be connected by one and only one TSP tour, hence producing a piecewise linear Jordan curve that connects all the points in one non-intersecting tour that is of minimal length, at least approximately. Plotting this path with a black line, will show a graph that from a distance will again give a greyscale representation of the original image. It you do not like the piecewise linear curve, it is of course possible to modify the path slightly to turn it into a smooth curve. On the other hand, one could plot a white ribbon on a black background, weaving into a knot like on some Celtic or Arabic graphics. After stippling and finding the TSP contour using extra constraints that prevent points on the ribbon and the contour to "cut" the ribbon where it is not allowed, we are left with the ribbon as blank areas. Depending on symmetry conditions and the imposed constraints it is quite remarkable to detect what is the inside and what is the outside of the Jordan curve that represents the route of the salesman. Some parts of the ribbon are inside while others are outside, which is counter intuitive since from a distance the ribbon looks like in one piece.</p>
<p>Other abstract designs can be obtained by visualizing the knight's tour on a chess board. Another challenging problem is to design a nontrivial maze in such a way that within the outer boundary of the maze, all the squares should be visited exactly once to reach the center. The fine-touch is to design it in such a way that it shows some pleasing pattern. Under the title "Mosaics with side constraints" we find several other variations on the previous techniques that obey some extra restrictions to make the problem more challenging. A very nice idea is based on Conway's game of life. Also this game of life is played on a square grid where every square in the grid represents a cell. The game is a discrete dynamical system in the sense that a cell will live (a dot is present) or die (no dot) depending on some simple rules like the number of its neighbours that are alive. One may collect a number of cells in a larger composite tile that remains stationary under these dynamics and, depending on the number of living cells (dots) it will represent a greyscale tile, which can be used in the previous way to form a mosaic. However, more challenging is to generate composite tiles with cells that alternate between two states, but that do not interact with neighbours. Then one can obtain a dynamic image with a blinking effect.</p>
<p>It is clear that there are many ways to apply the idea of using optimization problems with carefully designed constraints to generate some nice pictures like Robert Bosch illustrates here. I love the book. If you want to start designing yourself, you will find it is far more challenging and probably far more addicting than any game that has been designed for you.</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>It is explained how (constrained) optimization can be used to generate pleasing visual mathematical art with optical effects like mosaics, or nicely designed mazes.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/robert-bosch" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Bosch</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">9780691164069 (hbk), 9780691164069 (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">£ 24.00 (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">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/control-theory-and-optimization" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Control Theory and Optimization</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691164069/opt-art" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691164069/opt-art</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/90c90" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90C90</a></li></ul></span>Mon, 25 Nov 2019 09:46:26 +0000Adhemar Bultheel49950 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/opt-art#commentsTales of Impossibility
https://euro-math-soc.eu/review/tales-impossibility
<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>Squaring the circle (i.e. finding the side of a square whose area equals the area of a circle using only compass and straightedge) is probably the best known problem that became iconic for representing something that is impossible to do. But the classical geometric problems from Greek antiquity, equally impossible to solve with these instruments are doubling the cube, trisection a general angle, and constructing a general regular polygon. For the latter two problems it could be done in some cases, but not for a general one. These problems have attracted many cranks who claimed to have solved the problem until Pierre Wantzel published his theorem in 1837 proving that these problems were impossible to solve (which by the way did not stop cranks to continue spamming editors with their solutions).</p>
<p>Richeson describes in this book the history of these problems, and it is quite interesting to read that just trying to solve these problems gave rise to a lot of investigation that resulted in proper theorems. Progress towards the solution was only possible when algebra was introduced and linked to geometry and number theory. But of course the story starts with the definition of the problems and it is illustrated that solving them corresponds to the following problems respectively: given a segment of length 1, construct one with a length $\pi, \sqrt[3]{2}, \cos(\theta/3)$ for arbitrary $\theta$, or $\cos(360^\circ/n)$ for arbitrary $n$. There is a lot to tell about the mathematics as it was practised in Greek antiquity. They had natural numbers of course, but they did not consider other numbers as such. In their geometric framework, it were rather ratios of lengths of line segments or of areas of geometric figures. The number $\pi$ as the ratio of the circumference of a circle over its diameter is an universally known example. However this did not work so well with the area of a circle and its radius since the dimensions of an area and a length did not match. What made sense was comparing the area of a circle and the area of a square, which explains the problem of squaring the circle. Sticking to these ratios suggests that they considered all quantities as being commensurable, that is multiples of a fundamental unit, so that their number concept is essentially one of rational numbers. However $\pi$ for the circle and $\sqrt{2}$ in the Pythagoras theorem were known examples of non-rational numbers which they tried to approximate.</p>
<p>Richeson skims the most important mathematicians of antiquity to illustrate how they dealt with $\pi$ and how they attempted to find quadratures, that is to find squares or rectangles that had the same area as the area of another geometric figure. That could be a circle, but also other ones that showed up in their quest to solve the circle problem, like lunes or some parts of a circle or part of another conic. But it was Archimedes who came up with many formulas and with bounds for $\pi$. Since the compass and straightedge were not able to solve the problem, people tried to relax on these restrictions. With neusis constructions or marked straightedge some of the unsolvable problems became solvable. In the centuries that followed, many ingenious instruments were designed to produce all kinds of curves (quadratrix, conchoid, limaçon of Pascal, spirals, carpenter's square curve, ...). On the other hand, one could try to find out what could be done with less than compass and straightedge. For example what if the compass is rusty and has only a fixed angle, or what if we only had a compass. Georg Mohr was the first to prove (1672) that with only a compass one can do everything that can be done with compass and straightedge. He was forgotten and Mascheroni re-discovered this much later. A straightedge alone however can not do the same job.</p>
<p>When algebra was introduced in Europe by the Arabs, mathematicians concentrated on solving equations using formulas, rather than by geometric constructions. When people started to represent curves by algebraic formulas (traditionally attributed to Descartes), the idea of constructible numbers was born and the first impossibility claims emerged. More complicated quadratures and better approximations for $\pi$ were computed, certainly with the newly invented calculus. With complex numbers new construction methods for regular $n$-gons were produced. They also gave formulas for solving polynomial equations and that paved the way for Pierre Wantzel (1814-1848) to eventually come up with his theorem about the degree of the minimal polynomial of a constructable number from which followed a precise statement about what was possible and what was impossible to produce with compass and straightedge.</p>
<p>Richeson is able to bring the story as a popularizing book about mathematical history with a brief characterization or biography of the mathematicians involved and of course the evolution of mathematics from geometrical ideas of antiquity to the algebraic number theory of Wantzel. It is however far from a "hard core" history book. It has many citations (all in English) and there are many notes at the end, but these are all informative and do not disrupt the reading. The impossible problems discussed are easy to understand and have attracted many mathematical hobbyists in the past. So the discussion in this book is easily accessible from a mathematical point of view. Although every chapter is somehow related to the four problems, Richeson takes a broad view and the computation of $\pi$ or the constructions of curves with mechanical instruments can hardly be called diversions from the main theme. Most amusing are the intermezzo's that he calls "tangents" after each chapter. These give some diversions of all sorts. The first one about how to recognize a mathematical crank is particularly amusing. There are others like what geometry can be done using toothpicks (line segments with a fixed length), how to compute $\pi$ at home, what geometric constructions are possible using square origami paper, there is the story of the Indiana $\pi$-bill in which Edwin Goodwin tried to pass his circle squaring by law, etc. The whole book, both informative and amusing, is a highly recommended read.</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 book Richeson explores the history of the classical Greek (geometric) problems that are impossible to solve with compass and straightedge: trisecting a general angle, squaring the circle, doubling the cube, and constructing any regular polygon. The stories of these problems and related ones are traced until they were proved to be impossible by Pierre Wantzel in the 19th century.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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">9780691192963 (hbk), 9780691194233 (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">£ 24.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">456</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li></ul></span>Mon, 25 Nov 2019 09:10:21 +0000Adhemar Bultheel49946 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/tales-impossibility#commentsThe Master Equation and the Convergence Problem in Mean Field Games
https://euro-math-soc.eu/review/master-equation-and-convergence-problem-mean-field-games
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In Mean Field Theory (MFT) one studies models for decision making in a large population of $N$ interaction agents. The agents are supposed to have only marginal impact so that the behaviour of one (average) agent is determined by its own state and by a distribution that describes the states of all the others while these are varying in time. The states of the agents are thus described by a stochastic differential equation (SDE) according to which the system evolves in time to some steady equilibrium state. The solution of this optimal control problem is described by a continuous Hamilton-Jacobi-Bellman equation expressing necessary and sufficient conditions for optimality of a value function that at every instant of time determines the optimal behaviour of an agent, depending on the state of the system. Thus the mathematical disciplines involved are mainly control theory and (stochastic) differential equations. Mean Field Games (MFGs) thus assume actually a continuous set of infinitesimal agents rather than a large set of discrete players. Since around 2006-2007, where the framework was properly formulated the subject has boomed. These models were originally introduced because of their applications in economics but later also many other applications in engineering and informatics emerged, and it became an applied mathematical topic that deserves further analysis in its own right, independent of the concrete application.</p>
<p>
In a finite Nash differential game (NDG), the state variables are also functions of a continuous time, but there is a discrete set of $N$ players. If $N$ is not too large, then such an $N$-Nash system has been investigated and its Nash equilibrium is well understood. To find an equilibrium state, there is a set of coupled differential equations that describes the time evolution of the individual value functions for the $N$ agents. The value function of an agent evolves in time depending on the time-varying states of all the agents in the system and this defines the instantaneous behaviour of the agent. Given the value functions at some horizon time $T$ for all the agents, the system has to be solved backward in time, and once these value functions for the individual players are known, one can compute their individual trajectories forward in time, where there is some individual noise added for every player as well as some global common noise for the whole system. This is how an $N$-Nash differential game is solved. One would hope that the equilibrium of an <$N$-Nash system tends to an equilibrium for the corresponding MFG with a continuum of players as $N$ tends to infinity. The purpose of this book is to prove that under appropriate conditions, in some sense, this is true for a large class of MFGs,</p>
<p>
Instead of analysing the complicated Nash system with $N$ really large, the authors use as their main approach the <em>Master Equation</em> (ME), which is an MFT concept imported from systems in physics and chemistry. This ME describes the expected asymptotic behaviour, and thus avoid the complexity of the huge Nash game. This reduces the system of infinitely many differential equations to just one stochastic differential equation whose solution is a trajectory for some average value function $U$. This $U$ depends on time $t$ and the state of the system. According to MFT the state is split into the state $x$ (a $d$-dimensional vector) of the individual agent (because of symmetry it does not matter which one) and a distribution $m$ characterizing some average state of all the other players. It is then proved that under appropriate conditions the value function $v$ of an individual player (any player) of an $N$-player differential Nash-system converges as $N\to\infty$ to the equilibrium solution provided by the Master Equation at a rate of $1/N$. Also the state trajectory $X$ of any player convergences in a probabilistic sense like $1/N^{\frac{1}{d+8}}$ to the associated asymptotic trajectory, (solutions of the McKean-Vlasov SDEs using the $U$ previously obtained).</p>
<p>
This project started with the intenton of writing a paper, but by making it somewhat self-contained and because of the generality of the result, it grew out into a book. Here "self-contained" needs to be understood as "for someone familiar with Nash systems" since it requires some knowledge of the subject that is silently assumed. If this knowledge is not present, then some extra reading of the cited references will be required to understand the details. For example, the first chapter is a rather extensive introduction to the problem and the concepts used, it gives a summary of the results to be proved and surveys how this is structured in subsequent chapters with some guidelines on what to read, depending on the knowledge and the interest of the reader. Equation (1.2) and (1.3) on page 4 describe a Nash system as</p>
<p>
\begin{eqnarray*}<br />
&&-\partial_t v^{N,i}(t,\boldsymbol{x})-\sum_{j=1}^N\Delta_{x_j} v^{N,i}(t,\boldsymbol{x})-<br />
\beta \sum_{j,k=1}^N \mathrm{Tr} D^2_{x_j,x_k} v^{N,i}(t,\boldsymbol{x})\\<br />
&&+H(x_i,D_{x_i},D_{x_i}v^{N,i}(t,\boldsymbol{x}))+<br />
\sum_{j\ne i} D_p H(x_j,D_{x_j}v^{N,i}(t,\boldsymbol{x}))\cdot D_{x_j} v^{N,i}(t,\boldsymbol{x})=F^{N,i}(\boldsymbol{x}),~~~~(t,\boldsymbol{x})\in[0,T]\times (\mathbb{R}^d)^N,\\<br />
&&v^{N,i}(T,\boldsymbol{x})=G^{N,i}(\boldsymbol{x}),~~~\boldsymbol{x}\in(\mathbb{R}^d)^N,~~~i\in\{1,\ldots,N\},<br />
\end{eqnarray*}<br />
and the individual trajectories of the agents by<br />
\[<br />
dX_{i,t}=-D_p H(X_{i,t},Dv^{N,i}(t,\boldsymbol{X}_t))dt+\sqrt{2} dB_t^i+\sqrt{2\beta}dW_t,~~t\in[0,T],~~i\in\{1,\ldots,N\}.<br />
\]</p>
<p>
It is explained that the first system is the Nash system considered with $v^{N,i}$ the unknown value functions depending on $\boldsymbol{x}=(x_1,\ldots,x_N)$, $H$ is the Hamiltonian, $\beta\ge0$ is a parameter, and $T\ge0$ is the time horizon. The second describes the optimal trajectories $X_{i,t}$ of the states of the players, with $B_t^i$ individual noise, and $W_t$ some common noise (both are Brownian motions). This is about all the explanation given, so that a reader unfamiliar with the subject will have some problem already on page 4. Although the statement is more fully introduced in chapter 2, and there is an extra appendix with explanation about derivatives with respect to random variables, there is no further explanation about the notation <$D_p$ or $\Delta_{x_i}$ or about an expression for the Hamiltonian (except that it is related to the cost that agent $i$ has to pay). However, if one is familiar with the basics, then the introduction of the main results and the proofs are well explained. For the MFG system, and the Master Equation, similar expressions are introduced, except that the state is split up as $\boldsymbol{x}=(x,m)$ where $x$ is the state of an (infinitesimal) individual and $m$ the distribution of the state of all the others. Since $m=m(t)$ is time-varying, also its evolution over time has to be monitored separately. Thus we get a similar but somewhat different set of equations for the MFG and the ME. Since these require derivatives with respect to the distribution $m$, the appendix is needed to explain this concept in more detail. The ME is an essential element in this book, and although known in other fields, it can be slightly different as it is applied here. So care has been taken to explain it in the present situation also on an intuitive basis in the introductory chapter. This is helpful to assimilate the subsequent chapters.</p>
<p>
The analysis holds under several restrictions that are carefully explained with many links to the literature. For example boundary problems for $t\in[0,T]$ are avoided by assuming periodic (in time) solutions, $F$ and $G$ satisfy some monotonicity condition, $H$, $F$, and $G$ are supposed to be smooth enough. Some of these restrictions can be removed or generalized but they are maintained here mainly for simplicity. Although it is not explained, the current approach is potentially useful for numerical implementation. Also the full proof in chapter 3 of the existence of the equilibrium for the MFG assumes the first order case ($\beta=0$ and thus no common noise). The second order system ($\beta>0$ with common noise) is further explored in chapters 4 and the ME in chapter 5, with the eventual convergence proof in chapter 6.</p>
<p>
This book appears as a volume in the <em>Annals of Mathematics Studies</em> and it is a major contribution to the state of the art in MFGs which is a must read for researchers in the field. It seems like several preliminary versions of this text were previously made available on the Web. There are few typos, which is an achievement for a book with that many formulas. Even with that many formulas and technicalities, the book is still quite readable, because the authors use the book format (and not a more compact paper format) to explain all their steps carefully. Because of its structured approach, it could be used as a textbook for an advanced course on the subject.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The authors give a proof that under certain rather general conditions the equilibrium solution for a Nash differential game with <em>N</em> players converges when <em>N</em> goes to infinity to the equilibrium solution of a mean field game (i.e., with a continuum of players). The approach taken is by introducing and analysing the so called Master Equation for the system. This basically reduces the system of <em>N</em> coupled differential equations to one stochastic differential equation describing the situation for an average player because the state of the system is characterized by the state of that average player while the state of all the others is described by a distribution.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/pierre-cardaliaguet" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Pierre Cardaliaguet</a></li><li class="vocabulary-links field-item odd"><a href="/author/fran%C3%A7ois-delarue" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">François Delarue</a></li><li class="vocabulary-links field-item even"><a href="/author/jean-michel-lasry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean-Michel Lasry</a></li><li class="vocabulary-links field-item odd"><a href="/author/and-pierre-louis-lion" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">and Pierre-Louis Lion</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691190709 (hbk); 9780691190716 (pbk); 9780691193717 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 136.00 (hbk); £ 62.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">224</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/control-theory-and-optimization" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Control Theory and Optimization</a></li><li class="vocabulary-links field-item odd"><a href="/imu/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691190709/the-master-equation-and-the-convergence-problem-in-mean-field-games" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691190709/the-master-equation-and-the-convergence-problem-in-mean-field-games</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/91-game-theory-economics-social-and-behavioral-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91 Game theory, economics, social and behavioral sciences</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/91a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91A80</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/49-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/49n75" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49n75</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/91h15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91H15</a></li></ul></span>Mon, 21 Oct 2019 14:15:00 +0000Adhemar Bultheel49831 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/master-equation-and-convergence-problem-mean-field-games#comments