European Mathematical Society - princeton university press, princeton
https://euro-math-soc.eu/publisher/princeton-university-press-princeton
enTrigonometric Delights
https://euro-math-soc.eu/review/trigonometric-delights
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>In the first two sentences of the preface, Maor writes</p>
<blockquote><p>
This book is neither a textbook of trigonometry —of which there are many— nor a comprehensive history of the subject, of which there are almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences.
</p></blockquote>
<p>I could not think of a better characterisation of the book than this. All I can add to this description is to give an idea of which kind of topics were selected and what kind of applications have benefited from these developments.</p>
<p>The successive chapters are organised more or less chronologically, starting with a prologue about the Egyptian Rhind papyrus from around 16-17th century B.C. and ending with Fourier series (18th century). There is of course much attention for the history, but what strikes me in particular, is how much attention is given to the etymology of the mathematical terminology. The origin of the words algorithm and algebra is described in several publications as originating from the Arab author al-Khwarizmi and from al-jabr, which is part of the title of his book, but what is the origin of words such as sine, secant, and many other common mathematical words? Maor carefully pays attention to this. He also shows how trigonometry, which originally was about angles like in pyramid building problems in Egypt, were somewhat made more abstract, in a geometric context of triangles by the Greek, but later, it became more and more part of analysis. The sine and cosine were not only tabulated for computational purposes, but they became functions so that now we see x in sin x as a real or even a complex number, not necessarily corresponding to a physical or geometric angle. The original idea of an angle in degrees or radians in the goniometric unit circle has become somewhat obsolete.</p>
<p>But of course it all starts with angles and chords in planar circles for the Greek, and even earlier in astronomy, which is essentially a three dimensional spherical discipline as practised by Babylonians and almost any civilisation of antiquity. This is the subject of the first two chapters. Then appeared tables of goniometric values of what became our basic goniometric functions. This opens the possibility to introduce algebra (goniometric identities) and gradually also analysis (involving series) into the discipline. This helped considerably to discover (actually re-discover) the heliocentric interpretation of our solar system and to measure our own planet by triangulation and those practical problems in turn stimulated the development of associated trigonometric identities in triangles. But before the heliocentric model, the trajectories of the planets required also more general curves than ther circle like epi- and epo-circles which allow an easy description in terms of trigonometric formulas. Then Maor ventures into a period of proper analysis with the Sine integral and many other relations and series expansions, not in the least for the fascinating number π. These were obtained by master minds such as Gauss and Euler. As complex numbers entered the picture, with Euler's fabulous formula, we are fully involved in complex analysis, conformal maps and ultimately Fourier analysis.</p>
<p>This marvelous survey by Maor of some episodes in the historical evolution of mathematics also allows to sketch some biographies of important mathematicians. There are the "usual suspects" from Greek antiquity (including Zeno whose paradoxes are discussed when infinitesimals from analysis are introduced). Also Regiomontanus (15th C.), François Viète (16th C.), De Moivre (17th C.), Maria Agnesi and her "witch" (18th C.), Jules Lissajous (19th C.), Edmund Landau (20th C.) are discussed in somewhat more detail. aot only the history and mathematicians, also the applications are well documented: astronomy, cartography, spirographs, periodic oscillation, music; and there are detailed mathematical derivations of several trigonometric and other mathematical identities, conformal maps, series converging to π, the solution of the Basel problem by Euler, how Gauss showed that any trigonometric summation formula can be represented geometrically, etc.</p>
<p>All these items are treated requiring only some elementary trigonometric formulas. Some of the standard identities are collected in appendices. In another appendix we find Maor's plea to re-introduce the unit circle and the geometric definitions of the trigonometric functions like cos and sin being projections of the circular point on x- and y-axis, etc. instead of the "New Math" approach. Also Barrow's integration of sec x is moved to an appendix. All the chapters are completed with a section containing notes and references to the sources used. There are many useful mathematical graphs and some grayscale images.</p>
<p>This is an interesting mixture of mathematical history, illustrating the evolution and the usefulness of trigonometry throughout the centuries, and on top of that, it gives some mathematical training by deriving formulas and identities that are easily accessible with only some elementary mathematics knowledge. The book appeared originally in 1998 and is here reprinted in its original form as a volume in the Princeton Science Library. So this is a fortunate occasion to bring this great book back under the attention of a broad audience.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This a reprint in the Princeton Science Library of the original book from 1998. Maor sketches several episodes on the history of mathematics where especially trigonometry was involved from the Rhind Papyrus to Fourier analysis. The history, the mathematicians, the applications, as well as the derivation of mathematical identities are discussed.</p>
</div></div></div></div><span class="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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691202198 (pbk), 9780691202204 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 17.95 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">256</div></div></div><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/paperback/9780691202198/trigonometric-delights" title="Link to web page">https://press.princeton.edu/books/paperback/9780691202198/trigonometric-delights</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li></ul></span>Sat, 16 May 2020 14:47:19 +0000Adhemar Bultheel50783 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/trigonometric-delights#commentsMathematics in Ancient Egypt-A Contextual History
https://euro-math-soc.eu/review/mathematics-ancient-egypt-contextual-history
<div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even"> </div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/MathInAncientEgypt.pdf" type="application/pdf; length=41130">MathInAncientEgypt.pdf</a></span></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/annette-imhausen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ANNETTE IMHAUSEN</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-11713-3</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">234</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span>Tue, 18 Jul 2017 13:10:22 +0000Raquel Díaz47773 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-ancient-egypt-contextual-history#commentse: The Story of a Number
https://euro-math-soc.eu/review/e-story-number
<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 paperback edition in the <em>New Princeton Science Library</em> of a 1994 classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by J. Havil, P. Nahin, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another recent reprint is for example Nahin's <a href="/review/imaginary-tale-story-√-1"><em>An imaginary tale: the story of v-1</em></a> that is also reviewed here.</p>
<p>
This book with the shortest possible title: e, needs a subtitle to make clear what it is about. This is of course the e in the exponential function $e^x$ and it is the number that is the basis of the natural logarithm. However the logarithm function came earlier and the exponential was just a way of inverting the logarithm and it was only accepted as a full-bred function later in history. The idea of multiplying numbers by adding the exponents when they are represented by powers was proposed by John Napier (1550-1617). To make the steps of the successive powers as small as possible, he decided to take as a base a number close to 1, and he defined $L$ to be the logarithm of a number $N$ when they were related by $N=10^7(1-10^{-7})^L$. In a time when all computations (for example in astronomy) were done by hand, the idea and the use of the first logarithm tables (1614) caught on very quickly. After a meeting in 1615 with Briggs (1561-1630), it was decided that a base 10 was a better choice, which corresponds to what is now called the common (or Briggs) logarithm. And that introduced the logarithm function much appreciated by mathematicians, astronomers and engineers of the 17th century.</p>
<p>
So far, no number e is involved, but its roots involve another "power story" about compound interest. With an interest rate of $r$, that is reinvested after the <em>n</em>-th part of the period (e.g. every month or even every day instead of at the end of a year) will give a return $(1+r/n)^n$ per unit. What happens if <em>n</em> tends to infinity? The result, as we know is <em>e</em><em>r</em>, but it took a very long time to come to this result. It needs binomial powers, the concept of infinity and of a limit. This came only after calculus was introduced by Newton and Leibniz, Jacob Bernoulli linked compound interest and the exponential, and it became only fully explored by Euler, the master of them all. Of course there are many precursors before one arrived at calculus. There are Zeno's paradoxes that relate to the concept of infinity. And the Greek approximated the circle by an <em>n</em>-gon with <em>n</em> indefinitely increasing, which lead to this other magic number we know as <em>p</em>. This polygon approximation allowed to approximate the circumference but also the area of the circle, and similar techniques applied to other conic sections. The integral under the rectangular hyperbola $y=1/x$ investigated by Fermat and Descartes is another road to the logarithm. But the proper machinery to compute this integral as an anti-derivative was only provided by Newton and Leibniz. With calculus and infinite sequences available, it became possible to define $y(x)=a^x$ for a number $x$ that was the limit of a sequence of rational numbers, and one could obtain its derivative which has the form $ky$. A natural question is when $k=1$ so that $y$ becomes its own derivative and the answer is $y(x)=e^x$. And there appears the number e like magic.</p>
<p>
Once the exponential and logarithmic functions are known, they show up in all kinds of applications like solutions of differential equations, music scales, spirals, catenary and other curves, hyperbolic functions, and of course the most magic formula showing a family picture of the most famous actors of our number system: $e^{i\pi}+1=0$. Of course the latter directly relates to complex numbers too. The spreading of Leibniz's ideas throughout Europe was mainly due to the Bernoulli family. Leibniz himself died at the age of seventy almost completely forgotten. Jacob was particularly fond of the logarithmic spiral (<em>spira mirabilis</em>) and he had it carved on his tomb. The formula $e^{i\pi}+1=0$ is due to Euler, who, besides the notation for e, is responsible for other standard notations too like $\pi$, $i$ and $f(x)$. Euler was the first to considered the exponential function in its own right, next to the logarithm, and not as just as an inverse by-product. He also derived a continued fraction expansion for e and a series expansion for $e^x$ and so arrived at $e^{ix}=\cos(x)+i\sin(x)$ and expressed the cosine and sine functions with complex exponentials.</p>
<p>
The acceptance of negative and complex numbers is another interesting story that Maor takes the opportunity to tell. Mathematics had developed well with only positive numbers (basically only rationals). Negative numbers were known to Hindus, but were neglected by Europeans. It was only when Bombelli used a number line to represent numbers that a meaning could be given to negative numbers. Similarly, it was the geometric interpretation of complex numbers as points in the complex plane that finally began to make sense. That was the merit of several mathematicians among which Gauss. Only later Hamilton gave a formal definition as couples of real numbers with appropriate addition and multiplication, which he later generalized to quaternions. Maor even discusses complex functions and complex calculus of course including the complex exponential and logarithm. Much more on complex numbers and complex functions, although much more mathematically advanced, can be found in Nahin's book telling the story of $\sqrt{-1}$ mentioned above. Another topic that could not have been missed is in the trailing chapter about the transcendence of e (proved by Hermite in 1873). Hermite's proof inspired Lindemann for his proof of the transcendence of $\pi$ published in 1882.</p>
<p>
Maor has written with this book the first (his)story of e as a counterpart for the well documented history of $\pi$. Since the original publication in 1994, J. Havil has compiled a biography on Napier: <a href="/review/john-napier-life-logarithms-and-legacy"><em>John Napier: Life, Logarithms, and Legacy</em></a> (Princeton University Press, 2014) and there were of course also several books that had chapters on the logarithm or the exponential, which emerged in a period where mathematics experienced a boost in Europe. This story has e as the central star, but e has many strings attached to it Thus many other issues of mathematics are also wonderfully told by the author, much to the liking of the public who made it a bestseller. It reads smoothly, is well illustrated, with some more technical material moved to appendices although the main text is not avoiding formulas, it remains quite accessible for a general interested reader. Also the brief interleaving sections on several topics (e.g. how to work with logarithm tables, a list of numbers related to or derived from e, a fictitious meeting between J.S. Bach and Johann Bernoulli, the logarithmic spiral in nature and art,...) are most interesting. So this reprint in this series of classics is most appropriate. It is of course the original publication and this means that no updates are done, no additional comments or recent references are added. For example the MacTuror biography of Briggs (and other sources too) tells us that he died 26 January 1630, while Maor mentions 1631, which could have been corrected. Another example is that it mentions that the largest Mersenne prime mentioned is $M_{2976221}$. That was in 1997 (this is a reprint of the first paperback edition of 1998), but of course there were several more found since then.</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 paperback reprint in the <em>New Princeton Science Library</em> of the bestselling original from 1994. It tells the story of the number e, the less famous sibling of <em>π</em>. It is of course directly connected to the logarithm and the exponential function and to many other topics in mathematics too. </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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691168487 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">US$ 11.95</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">248</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="http://press.princeton.edu/titles/5342.html" title="Link to web page">http://press.princeton.edu/titles/5342.html</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/01a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A99</a></li></ul></span>Wed, 27 Apr 2016 09:15:24 +0000Adhemar Bultheel46899 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/e-story-number#commentsThurston's work on surfaces
https://euro-math-soc.eu/review/thurstons-work-surfaces-0
<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">Raquel Diaz</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">UCM, Madrid</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/Thurston%27sWorkOnSurfaces.pdf" type="application/pdf; length=43970">Thurston'sWorkOnSurfaces.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 is an English translation of the now-classic book "Travaux de Thurston sur les surfaces", by A. Fathi, F. Laudenbach and V. Poénaru, 1979, concerning a classification of surface diffeormorphisms.</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/fathi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">A. Fathi</a></li><li class="vocabulary-links field-item odd"><a href="/author/f-laudenbach" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">F. Laudenbach</a></li><li class="vocabulary-links field-item even"><a href="/author/v-po%C3%A9naru" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">V. Poénaru</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</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">978-0-691-14735-2</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><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/57m99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">57M99</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/32g15-58f99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">32G15 58F99</a></li></ul></span>Fri, 26 Feb 2016 21:22:16 +0000Raquel Díaz46756 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/thurstons-work-surfaces-0#commentsFundamental Papers in Wavelet Theory
https://euro-math-soc.eu/review/fundamental-papers-wavelet-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book traces the development of modern wavelet theory by collecting many of the fundamental papers in signal processing, physics and mathematics stimulating the rise of wavelet theory, together with many important papers from its further development. These papers were published in a variety of journals from different disciplines, making it difficult to obtain a complete view of wavelet theory and its origin. Additionally, some of the most significant papers have been available only in French or German. Heil and Walmut bring together these documents into a book allowing the reader a complete view of the origins and development of wavelet theory. The volume is an excellent book and a first-class reference for the history of wavelets. By collecting all these various papers together in one volume, the editors and the publisher are offering a wonderful gift to graduate students as well as to researchers in engineering and mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">knaj</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/c-heil" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">c. heil</a></li><li class="vocabulary-links field-item odd"><a href="/author/d-f-walnut" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. f. walnut</a></li><li class="vocabulary-links field-item even"><a href="/author/eds" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">eds.</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</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">0-691-12705-0 </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 49,50</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/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier analysis</a></li></ul></span>Sun, 23 Oct 2011 12:07:28 +0000Anonymous40035 at https://euro-math-soc.euMax Plus at Work - Modelling and Analysis of Synchronized Systems - A Course on Max-Plus Algebra and Its Applications
https://euro-math-soc.eu/review/max-plus-work-modelling-and-analysis-synchronized-systems-course-max-plus-algebra-and-its
<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 first textbook on max-plus algebra, which represents a convenient tool for the description and analysis of discrete event systems like traffic systems, computer communication systems, production lines and flows in networks. The book is divided into three main parts and an introductory chapter, which illustrates main ideas in an informal way. The first part provides the foundations of max-plus algebra viewed as a mutation of conventional algebra, where instead of addition and multiplication the central role is played by the operations maximization and addition, respectively. It starts with the definitions of fundamental concepts (max-plus algebra and semiring, vectors and matrices over max-plus algebra) and an investigation of their properties. Then the spectral theory of matrices over a max-plus semiring is built, followed by a study of linear systems in max-plus algebra and their behaviour in terms of throughput, growth rate and periodicity. </p>
<p>The first part of the book ends with two chapters dealing with numerical procedures for the calculation of eigenvalues and eigenmodes. The second part starts with an introduction to Petri nets and their subclass, event graphs that are shown to be a suitable modelling aid for constructions of max-plus linear systems. Real-life applications related to timetable design for railway networks are discussed. It covers construction of large-scale systems, the throughput and periodicity of such systems, delay propagation, stability measures for railway networks and optimal allocation of trains and their ordering. The last part deals with various extensions (stochastic extensions, min-max-plus systems that also contain a minimization operation and thus enable modelling of a larger class of problems, and continuous flows on networks viewed as the continuous counterpart of discrete events on networks). The whole text ends with a bibliography, a list of frequently used symbols and an index. The book can be warmly recommended to final-year undergraduate students of mathematics, as well as to all interested applied mathematicians, operations researchers, econometricians and civil, electrical and mechanical engineers with quantitative backgrounds.</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">mhyk</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/b-heidergott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">b. heidergott</a></li><li class="vocabulary-links field-item odd"><a href="/author/gj-olsder" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">g.j. olsder</a></li><li class="vocabulary-links field-item even"><a href="/author/j-van-der-woude" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. van der woude</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">ISBN 0-691-11763-2 </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 50</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/93-systems-theory-control" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">93 Systems theory, control</a></li></ul></span>Sun, 23 Oct 2011 12:06:33 +0000Anonymous40034 at https://euro-math-soc.euGeneral Theory of Algebraic Equations
https://euro-math-soc.eu/review/general-theory-algebraic-equations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book contains the first English translation of the monograph, “The General Theory of Algebraic Equations”, which was published in French in 1779 by Etienne Bézout (1730-1783) under the name, “Théorie Générale des Équations Algébriques“. The book became very popular after its publication. The monograph was translated by Eric Feron, Professor of Aerospace Engineering at Georgia Institute of Technology. The book presents the Bézout approach to the problem of how to solve systems of polynomial equations in several variables and his new notation for the polynomial multiplier, which simplified the problem of variable elimination by reducing it to a system of linear equations. The book describes the major Bézout's result called “the Bézout theorem“, his deep analysis of systems of algebraic equations, his uses of determinants for finding a solution of systems of linear equations, his approach to integration and differentiation of functions, etc. The English translation can be recommended to everyone who is interested in problems of solving systems of polynomial equations and inequalities because it describes this mathematical problem from a historical perspective.</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">mbec</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/e-b%C3%A9zout" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">e. bézout</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</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">0-691-11432-3</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 32,50</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/12-field-theory-and-polynomials" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">12 Field theory and polynomials</a></li></ul></span>Sat, 22 Oct 2011 17:49:25 +0000Anonymous39983 at https://euro-math-soc.euFearless Symmetry - Exposing the Hidden Patterns of Numbers
https://euro-math-soc.eu/review/fearless-symmetry-exposing-hidden-patterns-numbers
<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>Number theory is an old and difficult subject. It is possible to have a problem that is easy to formulate (e.g. Fermat’s last theorem) but very difficult to solve. A variety of different methods have been developed over the centuries. This book is devoted to methods in number theory connected with (hidden) symmetries, their realizations by means of groups and their representations. The book is designed for a wide audience of non-specialists. The authors were willing to make an attempt to explain exciting discoveries in mathematics for a larger public without special training in mathematics. They did a splendid job. They are able to describe an eminent role of symmetries in number theory by carefully explaining what it means to represent something (a group) by something else. The first part of the book reviews basic algebraic notions and introduces the Legendre symbol and the law of quadratic reciprocity. The Galois group and its role in number theory is the main topic of the second part. The last part treats various reciprocity laws. It also indicates the prominent role played by ideas of symmetry in the proof of Fermat’s last theorem. The book describes very nicely and in simple terms key ideas of the field so that they can be appreciated by people with no particular mathematical education. It is also inspiring and useful for general mathematicians who are not specialized in the field.</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">vs</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/ash" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. ash</a></li><li class="vocabulary-links field-item odd"><a href="/author/r-gross" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r. gross</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</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">0-691-12492-2</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</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span>Sat, 22 Oct 2011 17:02:06 +0000Anonymous39977 at https://euro-math-soc.euThe Pythagorean Theorem. A 4,000-Year History
https://euro-math-soc.eu/review/pythagorean-theorem-4000-year-history
<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 interesting book is devoted to the Pythagorean theorem, which is the most frequently used theorem in all branches of mathematics and which is learnt in geometry at school. The author shows the long route the Pythagorean theorem has taken through cultural history and he analyses its role in the development of mathematical thinking, research and teaching.<br />
The book starts with a description of the earliest evidence of knowledge of the theorem, which was known to the Babylonians over 4000 years ago. It continues with an analysis of the work of the Greek mathematician Euclid, which immortalized this theorem as Proposition 47 in the first book of his famous Elements. Then the book describes the role of the theorem in Greek mathematics (Archimedes, Apollonius, etc.) and its positions in pure and applied mathematics, as well as its influence on the arts, poetry, music and culture through the Arabic world, the Middle Ages, the Renaissance and the New Age in Europe up to Einstein's theory of relativity. The most beautiful proofs of the theorem are shown and explained. The book contains an index and many interesting photos and pictures. It can be recommended to readers who want to learn about mathematics and its history, who want to be inspired and who want to understand important mathematical ideas more deeply.</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">mbec</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/e-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">e. maor</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</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">978-0-691-12526-8 </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">USD 24.95</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>Sat, 01 Oct 2011 08:47:15 +0000Anonymous39831 at https://euro-math-soc.euHow Mathematicians Think. Using Ambiguity, Contradiction, and Paradox to Create Mathematics
https://euro-math-soc.eu/review/how-mathematicians-think-using-ambiguity-contradiction-and-paradox-create-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>A lot has been said and written about the philosophy of mathematics, yet not enough. There are too many unanswered questions. For example, is mathematics discovered or created? There still seems to be room for a good book in this field, and this one is wonderful, a must-read for everyone interested in mathematics, philosophy and/or history. One of the most pervasive myths about mathematics is that it is a dull technocratic discipline carried out by grim computer-like minds that have no feelings and who work without intuition, ambiguity or doubt and produce strictly formal algorithms and theorems free of contradictions, conflicts and paradoxes. Such myths, unfortunately still rather common among ‘outsiders’, call for such a book.<br />
The author, a great mathematician and philosopher, and also a practitioner of Zen-Buddhism, shows how essential non-logical qualities are in mathematical research and creativity and that the secret of successful mathematics is not in its logical structure, or at least not only there. Excellent discussions are presented about ambiguity, contradiction, paradox and their central role in the world of mathematical discovery.</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">lp</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/w-byers" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">w. byers</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-princeton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</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">978-0-691-12738-5 </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 35</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>Fri, 30 Sep 2011 17:23:51 +0000Anonymous39355 at https://euro-math-soc.eu