European Mathematical Society - world scientific
https://euro-math-soc.eu/publisher/world-scientific
en Linear Algebra and Optimization with Applications to Machine Learning. Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning
https://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The "with applications" in the title of the book should be read as "applicable" because it provides the fundamental mathematics, but it does not explicitly treat the applications that are mentioned. In this volume I, these fundamentals are basically linear algebra, and in volume II it is promised to cover optimization. I can imagine though that there is some interaction like for example the important subject of linear programming. Except for a few illustrative examples, the applications themselves are supposed to be covered in other courses or textbooks. There is a bit of statistics, which I think is also important for the applications mentioned, but probability and statistics as well as calculus is assumed known, but anyway, whatever is needed from these is recalled briefly.</p>
<p>Since this volume introduces the fundamentals with applications in mind, it is in some sense similar to a first course in linear algebra that I have been teaching to engineering students for many years. This volume has also some numerical procedures and even matlab codes. Those I taught in a separate numerical course. The procedures discussed include Gaussian elimination, Cholesky factorization, QR decomposition, eigenvalue and SVD algorithms, Krylov and Lanczos methods, but it skips the basic numerics of rounding errors, error propagation, numerical stability, and the more analytic problems such as numerical quadrature, differential equations, zero finding, etc. The latter also have a definite link with linear algebra, but clearly including all applications of linear algebra is an interminable task.</p>
<p>I wrote my own lecture notes, not satisfied with the existing books that were not providing the desired abstraction and that spent too many glossy pages on the introductory level with many examples and applications. In many ways my notes were very similar to the material covered here, which is why I like this book so much. But since I was trying to cover as much as possible in the most efficient way restricted by the amount of credits that were assigned to the course, my notes were much more concise. This is quite different in this book since, at its introductory level, it is almost encyclopedic and it is like the text I would have liked to write if I were not restricted by a time limit to cover all the material. For example, I spent some time explaining that finite dimensional real vector spaces and linear maps can be treated in an isomorphic way by discussing $\mathbb{R}^n$ and matrix algebra, and then I could just do matrix algebra. Not so in this book. The abstract vector spaces remain present throughout the book. Infinite dimensional vector spaces are a problem because there you need infinite sums and convergence, which require topology to define convergence, and the maps are operators. The authors here maintain some elements of function spaces but certain analysis aspects are not really covered in detail. But otherwise almost all the proofs are fully written out.</p>
<p>Thanks to LaTeX it is nowadays no problem to produce a professionally looking text. The illustrations however require different tools and producing good quality graphics is a challenge. Clearly the authors of this book had the same problem with graphics that I also had. The text is excellent, but the graphics are definitely of lesser quality. Unfortunately it is not only a problem of how they are generated and reproduced, also they do not always make very clear what they are supposed to illustrate.</p>
<p>What would one expect in a basic linear algebra course for engineering-type students? I think that should include vector spaces and linear maps and how they relate to matrices, the rank of a matrix with range and null space, determinants (in my opinion as little as possible), linear systems with Gaussian elimination, normed and Euclidean spaces, orthogonalization and QR, eigenvalue and singular values with generalized inverses and least squares, and geometric interpretation of all these concepts. All this is extensively discussed in this book. The numerical and matlab algorithms and certainly the iterative methods, I would rather expect in a more specialized numerical course, but it is not completely unexpected that they are found here in this book. More unexpected are the following: A discussion about the Haar wavelet with some signal and image processing; the chapter on linear systems is introduced with a discussion about the computation of interpolating Bézier curves; and the computation of the matrix exponential, important for the solution of differential equations, is discussed to some extent. There is also an extensive discussion of groups (SU(2), SO(3), and quaternions. This is important for robotics. Finally, there is much material related to graphs: graph Laplacians, clusters, and graph drawing. The final chapter about polynomial factorization and the Jordan form is less elementary and not always found with the same detail in a basic course. So there is a lot of material, and I can imagine that one wants to make a selection. No advise is given by the authors and it would be difficult anyway since successive chapters rely on previous ones. The authors have earmarked only some sections that they considered to be more advanced and these can be skipped initially.</p>
<p>To conclude, I can definitely recommend this extensive book on linear algebra that is both self contained and thorough and mostly remaining at a basic level. I have always considered linear algebra a basic tool in many applications. The applications mentioned in the title are computer vision, robotics and machine learning but they are not really discussed. However the Haar wavelet is a hint to image and signal processing, robotics are related to the study of the quaternions and the rotation groups, the linear algebra and the graphs are useful for a lot of applications, thus also for machine learning. To come to the actual applications though will need extra material, continuing the basics given here, but also some extra material for example statistics, and optimization (the latter is promised in volume II). Every chapter has a list of exercises (no answers are provided). There is a bibliography which lists mostly books and an index (12 pages). With a book of this size, the index can never be extensive enough. I tried to look up some topics that were not listed. On the other hand, I noted separate entries for "Jordan block" and "Jordan blocks", which makes no sense of course. I know from experience that collecting an index in an artisanal way is, just like generating good graphics, a time consuming task that needs patience and many iterations.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a extensive and thorough introduction to linear algebra that includes some extras like wavelets, Bézier curves, groups of rotations, quaternions, and applications in graph theory, that are of particular interest for applications in computer vision, robotics, and machine learning.</p>
</div></div></div></div><span class="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/jean-gallier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean Gallier</a></li><li class="vocabulary-links field-item odd"><a href="/author/jocelyn-quaintance" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jocelyn Quaintance</a></li></ul></span><span class="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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9789811206399 (hbk), 9789811207716 (pbk), 9789811206412 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 175.00 (hbk), GBP 85.00 (pbk), GBP 70.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">824</div></div></div><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/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li><li class="vocabulary-links field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.worldscientific.com/worldscibooks/10.1142/11446" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/11446</a></div></div></div><span class="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/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span><span class="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/15axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15Axx</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/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65d19" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D19</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65t05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65T05</a></li></ul></span>Fri, 10 Apr 2020 06:07:24 +0000Adhemar Bultheel50668 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra#commentsThe Soma Puzzle Book
https://euro-math-soc.eu/review/soma-puzzle-book
<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 classical <a href="https://en.wikipedia.org/wiki/Soma_cube" target="_blank">Soma cube</a> is a 3D puzzle invented by the Danish polymath Piet Hein in 1933. A 3 x 3 x 3 cube is partitioned into seven building blocks. Each of these blocks consists of three or four atoms (that are 1 x 1 cubes) glued together on matching faces and they have at least one inside corner. One block has three atoms (this is called the V block and consists of a corner atom with two atoms glued to two adjacent faces). All the others have four atoms, and are obtained by adding a fourth atom to the V. Three of the these 4-blocks are "flat" (the L the S and the T where the fourth atom is added in the same plane as the V) and three where the fourth atom is added on top of the V outside the V-plane: The P (the fourth atom is on top the corner atom of the V) and the remaining ones (A and B) are on the other blocks of the V (these are left and right chiral). There exist commercial versions of 4 x 4 x 4 or 5 x 5 x 5 cubes for the diehards, but these will not be considered here.</p>
<p>
There are 240 different ways to put the seven pieces together to form the 3 x 3 x 3 cube. The mathematical background has been fully analysed by John Horton Conway in the <em>Mathematical Games</em> column of <em>Scientific American</em> way back in 1958. So this cannot be the subject of this book. What is presented are problems (and solutions) of what other kind of challenges can be posed using these same building blocks. Even with one block there are problems to solve like which block can give a hexagonal shadow or how small can a hole in a plane be that allows to get all (or some of) the pieces through, or what is the largest hole that can be filled with every piece.</p>
<p>
And then the book continues chapter by chapter posing problems with 2, 3,..., 7 pieces. One or two shapes have to be constructed using a selection of building blocks (possibly with duplicates). Also the chapter involving all the seven blocks adds to the classic problem by asking to construct the cube with constraints or to find non cubic volumes. A nice proposal is to construct fractions where a fraction p/q means that there is a bottom layer of q atoms and a second layer of p atoms resting on the blocks of the bottom layer. One can then construct for example fractions 3/9 and 6/9 (whose sum is 1) and there are other such fractions that sum to 1.</p>
<p>
Note that the 3-block with 3 atoms put in one line (that is block I) is excluded and also two 4-blocks (4 in a row which does not fit in the cube and a 2 x 2 square called O) are excluded. If we add the I and O to the set of seven, then new problems can be added to the already extensive list of problems that will now involve 8 or 9 blocks.</p>
<p>
There is no mathematical analysis in this book. A challenge is just graphically presenting the blocks that can be used and the required result. A colourful graphical language is defined that is used in the solution sections to explain how to generate the solution in several steps. This it is a book purely for the fun of solving puzzles. It is of course possible to solve the puzzles with pen and paper if one has a well developed 3D imagination, but it is of course the intention that you have a set (sometimes two sets) of seven blocks physically available, which can be bought in most toy shops. These often already come along with non-cube shapes that have to be built. The current book will add new problems to the existing ones. If you happen to have already such a set, then this book will provide new challenges 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>
The book presents new challenges using the seven pieces of the classical Soma cube. The problems and the solutions are presented using colourful graphics. No mathematical knowledge is assumed and no mathematics are explained.</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-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li><li class="vocabulary-links field-item odd"><a href="/author/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-981-3275-31-7 (hbk), 978-981-3275-94-2 (pbk), 978-981-3275-33-1 (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">£ 40.00 (hbk), £ 25.00 (pbk), £ 19.95 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">180</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://worldscientific.com/worldscibooks/10.1142/11130" title="Link to web page">https://worldscientific.com/worldscibooks/10.1142/11130</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</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/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li></ul></span>Mon, 05 Aug 2019 10:17:32 +0000Adhemar Bultheel49606 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/soma-puzzle-book#commentsElegant Fractals
https://euro-math-soc.eu/review/elegant-fractals
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is much like the previous one <em>Elegant chaos. Algebraically simple chaotic flows</em> (2010) by this author. There Sprott investigates all sorts of dynamical systems and their chaotic behaviour. By elegant chaos he means the simplicity, symmetry,... of the formulas describing the system. In the current book with subtitle <em>Automated Generation of Computer Art</em>, it is all about the graphical representation, and elegant now refers to what one could define as an aesthetic pleasing graphic. Moreover, this elegance is defined by investigating certain properties of the dynamics that can automatically be optimized by the software to produce the probably most pleasing results.</p>
<p>
So the book starts with some technical definitions of chaos, iteration, basin of attraction, etc. It is explained how to deal with resolution by manipulating the pixels. Artificial variables can be used to add shadow (giving a 3D effect), and to add colour while the original dynamics of the system are maintained. Thus, even for dynamics in the plane, one ends up with six variables (three for space, and three for colour). Next several properties can be evaluated: the probability of generating chaotic behaviour for varying values of the parameters, the probability of clusters, and characteristic indices like Lyapunov exponent, Kaplan-Yorke dimension, correlation exponent, etc. An artificial neural network can then be taught to automatically select the most interesting graphics.</p>
<p>
In subsequent chapters several ways of generating the dynamics that will produce fractals are considered with variations on how to influence the graphical result by changing formulas or parameters, or the way of colouring, how to introduce symmetry, clustering etc. These chapters are dealing with iterated maps, chaotic flows (in fact continuous systems discretized using a simple Euler rule), iterated function systems, escape-time plots (think of Julia sets), spatio-temporal systems (e.g. cellular automata, or differential equations), random fractals (the graphics of random number generation and stochastic systems), and fractal tessellations.</p>
<p>
An epilogue, describes briefly the software that is downloadable from the author's web page both in a PowerBASIC source format or as a Microsoft Windows executable. There you can also find all the graphics of the book in png format. The graphics can be named uniquely using one letter for the type of the generating process followed by 12 letters that indicate what value is taken for the parameters (there are at most 12 parameters that can take no more than 26 different values).</p>
<p>
The focus of the book is purely on the graphics, and <em>not</em> the mathematical analysis of the dynamics, nor the physical interpretation, or the numerical solution. For example that a continuous system is approximated by a simple Euler rule is of no importance since the purpose is to generate interesting graphics and not an accurate solution. Nevertheless, links to the literature are often provided where these systems do get their interpretation, or where they are studied from a more mathematical or numerical side. So this book should be considered as an effort of bridging the gap between mathematics and art, although it is still very technical and automated. Whether true aesthetics criteria are used is disputable. Thus it may not be well appreciated by the artist and it may not be taken seriously by the mathematician. However, it may force the art critique to be more precise about why he or she thinks one picture is more beautiful than another, and it may invite mathematicians to investigate on a more mathematical basis what is observed here experimentally driven by graphical arguments.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is describing the results of generating fractal graphics by a computer program in several different ways and discusses how the variation of parameters can lead to more or less "elegant" (the word beautiful is avoided) pictures. By defining elegant in terms of computable indices, the software can automatically search for the most interesting regions in parameter space to be explored.</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/julien-clinton-sprott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Julien Clinton Sprott</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-3237-13-1 (hbk); 978-981-3237-15-5 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 105.00 (hbk), GBP 29.95 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">268</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://www.worldscientific.com/worldscibooks/10.1142/10906" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10906</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/28-measure-and-integration" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">28 Measure and integration</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/28a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">28A80</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/37f99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37F99</a></li></ul></span>Mon, 05 Aug 2019 10:13:20 +0000Adhemar Bultheel49605 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/elegant-fractals#commentsThe Paper Puzzle Book
https://euro-math-soc.eu/review/paper-puzzle-book
<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 subtitle of the book : All you need is paper (and scissors and sometimes adhesive tape if you want to be picky), might be tricking you into an imaginary situation of a kindergarten with children producing some artwork for mom, dad, or one of their grand parents. This is a completely different kind of book. You need a well trained set of brains and a strong puzzler's attitude to solve the puzzles that are collected by some of the best.</p>
<p>Ilan Garibi is an Israeli origami specialist, David Goodman is a designer of (mechanical) puzzles, and Yossi Elran is a mathematician, head of the Davidson Institute Science Education Accelerator of the Weizmann Institute in Rehovot, and a big puzzle fan. When they met at a meeting of recreational mathematics and games, the idea for this book was born.</p>
<p>In the best of Martin Gardner's tradition 99 puzzles are collected. Some are classics, some are found in the literature, and others are new. The authors are kind enough to give the origin of the puzzles when appropriate. The number of 99 is just a rough indication because there may be 99 problems formulated, but their solutions, which are given at the end of the chapters, sometimes propose variations or end with an extra challenge left open for the reader.</p>
<p>It may seem not very easy to represent with a static image (or images) in a book, all the necessary operations of folding an cutting that have to be performed in 3D and that sometimes even result in a 3D object. However the different steps are represented using some pictoral vocabulary that is explained in the beginning and that is remarkably clear and easy to read.</p>
<p>The puzzles are grouped according to techniques and topics in ten chapters. Sometimes puzzles are sequential, i.e., you first need to solve puzzle x before you solve puzzle x+1 because solving x is a subproblem of x+1. The puzzles are also rated with one up to four stars. Sometimes the shape of the paper is important for the technique to work: it need to be square or A4, but in other cases it can be just rectangular, or it has to be a long strip. Here is a list of the chapters with some simple illustrative example:<br />
1. Just folding. For example fold a square paper into an equilateral triangle with a follow-up problem to fold the largest possible equilateral triangle that is contained in the square.<br />
2. Origami puzzles. These need so called Kami paper whose sides have different colours, for example black and white. A first exercise is to fold the paper such that the visible areas of black and white are equal. This chapter is rather extensive.<br />
3. 3D folding puzzles. Given a strip of size 1 by 7, fold it into a cube with side 1.<br />
4. Sequence folding. Here one is given for example a square paper with a 2x2 grid defining 4 squares that are marked with the numbers 1 to 4 in lexicographical order. The problem is to fold the paper until it has size 1x1, but such that the squares on the folded stack have the natural order 1,2,3,4. Many variations are possible, starting from different configurations, or allowing a few cuts, etc.<br />
5. Strips of paper. Here of course the Möbius band plays a prominent role, but there are other puzzles to formulate with strips.<br />
6. Flexagons. This is an invention of Artur Stone of 1939 and popularized by Martin Gardner and later picked up by several others. Paper is folded into a polygonal form in such a way that that it has a front and a back side, but it allows for an simple flipping operation such that it is so to speak turned inside-out, showing different faces. One could define it as a flat folded configuration that has more than two faces. As a simple example one could start from a particular configuration of 6 connected squares (neighbouring squares have exactly one edge in common). Both sides have two squares marked 1, two marked 2 and two marked 3. Counting both sides, there are thus four 1's, four 2's and four 3's. This has to be folded into a 2x2 square and the 'first' and 'last' square are taped together so that one gets a sort of Möbius ring object that will allow only a limited number of hinged flips. The 2x2 square has to show the four 1's on the front and the four 2's on the back. By 'flipping' it, one gets all 3's on one side and all 2's on the other. There are three faces that can be shown in turn by flipping.<br />
7. Fold and cut. For example, you have to fold a piece of paper in a certain way and cut it with one straight cut to obtain a prescribed shape like a cross or a star.<br />
8. Just cutting. A classic is to cut a hole in an A4 size paper, such that a person can step through the hole without tearing the paper.<br />
9. Overlapping paper puzzles. It is clear that, given three paper squares, one may arrange them in a partially overlapping way such that all three are only partially visible. This is impossible with four squares. Problems based on this principle can be formulated putting restrictions of the number or size of papers you start with, or restrictions on the shape of the outer boundary of the stacked papers.<br />
10. More fun with paper. This is the miscellaneous section with many diverse fun constructs like putting together a rotator or an helicopter, performing magic tricks, solve (seemingly) impossible bets, etc.</p>
<p>The examples I gave above are just to illustrate the idea of what kind of puzzles are possible. They are usually the first kick start puzzles for the chapter rated with one or two stars. Sometimes these innocent looking problems can be be surprisingly difficult to solve even if they get the lowest difficulty rating. Although the solution methods for the puzzles are reminiscent to geometry, no mathematics is required. It reminds me of the ancient Greek idea of constructions using only compass and straightedge, but this is definitely different and even more basic: there is no compass, and there is no ruler. It is for example difficult to divide an edge of a square in three (or in n if n is odd) equal parts. That is only possible using an iterative pinching procedure. Such basic techniques are explained in an appendix. There is also a (limited) list of books, papers, and websites for further reading.</p>
<p>This is a marvellous book. The diversity of possible puzzles that can be given with these very limited resources, which are basically some paper and scissors, is overwhelming, and the challenges are sometimes very tough. Even the two-star problems may be hard for an untrained puzzler. This is medicine against boredom on long rainy days, but be careful not to get addicted or it may suck up your less empty and sunny days as well.</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 marvellous set of about a hundred puzzles that have to be solved by only folding and/or cutting paper. They were collected by three experts: an origami specialist, a puzzle designer, and a mathematician. Many of these innocent looking problems are really hard to solve, and others seem to be impossible at first sight. It requires geometrical thinking, but no mathematical knowledge is needed. As with many of these mathematical puzzles you need to be able to think outside the box, and sometimes to visualize things in 3D.</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/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</a></li><li class="vocabulary-links field-item odd"><a href="/author/david-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li><li class="vocabulary-links field-item even"><a href="/author/yossi-elran" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yossi Elran</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-3202-40-5 (hbk), 978-981-3202-41-2 (pbk), 978-981-3202-43-6 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP42.00 (hbk), GBP25.00 (pbk), GBP20.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">264</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://www.worldscientific.com/worldscibooks/10.1142/10324" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10324</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Thu, 10 May 2018 06:28:44 +0000Adhemar Bultheel48455 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/paper-puzzle-book#commentsA Friendly Approach To Functional Analysis
https://euro-math-soc.eu/review/friendly-approach-functional-analysis
<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>Many books were written on this topic already. The current text can be called friendly because when the author gives a definition, he usually connects it to what the reader is supposed to know. These prerequisites are basically not much more than finite dimensional vector spaces. Since this is what functional analysis is about: calculus lifted to the level of infinite dimensional vector spaces. A lot of effort goes into pointing out the similarities and the differences. Since also Lebesgue integrals are used, an appendix introduces this concept in the one dimensional case.</p>
<p>But friendly does not mean that the reader is spared from effort. The text is peppered with many (197) exercises. The formulation of the exercises and the solutions provided at the end form an essential part of the book and they actually fill the larger part of its pages. Thus the reader who is new to the topic is supposed to work hard to properly grasp all the ideas. By including the solutions, the book is suitable for self study, although the text grew out of lecture notes used by the author while teaching at the London School of Economics.</p>
<p>On the other hand, the subject is vast, and since this is only an introduction not all the most difficult proofs are provided in full extent and not all the most complicated issues are discussed in all detail. It is important to mentioned that, even though the subject is abstract, attention is also paid to the application aspect. In fact the introduction starts with an optimal control problem as a motivation to embark on functional analysis. In a later chapter when differentiation and its application to optimality conditions is discussed, the Euler-Lagrange equation is derived and it is applied to classical Hamiltonian and Poissonian mechanics. This forms also the basis for a discussion of quantum mechanics in the chapter on Hilbert spaces. Compact operators are a reason to go into the subject of finite dimensional Galerkin approximations of the operators, which is important for numerical computations. Weak solutions of differential equations are obtained using distributions, discussed in the last chapter. This means that also physics and engineering students will appreciate this approach.</p>
<p>The text is rather dense and to the point and covers an enormous variety of topics, even though the main text has only six chapters packed on 258 pages. Here is a superficial sketch of the contents. The first chapter makes the step from vector spaces to normed spaces and to Banach spaces. The next obvious things to tackle in any calculus course are continuity and differentiation, which are also here the subjects of chapters 2 and 3. These include some operator theory with the open mapping theorem, some spectral theory and even a proof of the Hahn-Banach theorem. The application of the Fréchet derivative in optimization and the application in mechanics was mentioned above. The chapter on inner product and Hilbert spaces may be the easiest ones to deal with since these spaces behave mostly like finite dimensional vector spaces when separable, and they are frequently used in all kinds of applications: approximation, Gram-Schmidt orthogonalization, generalized Fourier series, problems involving self-adjoint operators, etc. The next chapter shows that to study operators, things become easier when we restrict ourselves to compact operators. The final chapter on distributions (to stress that it is only an introduction to the subject, its title is A glimpse of distribution theory) has as one of its applications the weak solutions of differential equations but also allows to extend Fourier analysis. A few of the topics that are considered to be more advanced (like the open mapping theorem, the dual space and the Hahn-Banach theorem, and the spectral theory of compact self-adjoint operators) are marked by a star, so that they can be skipped it needed.</p>
<p>This is a nice and modern introduction to the topic. The impatient or more advanced reader can just read the text, skipping the exercises (perhaps skip only their solutions, not all their statements since they often are formulations of additional properties to be used in further proofs), and so get a quick idea. Where possible, examples clarify some basics and some subtleties of the definition or the property. If, on the other hand, the reader is a student who wants to become proficient in the subject, then solving the exercises, or an many as possible, is an excellent way to acquire the necessary skills. The flexible possibilities provided —the text can be used as lecture notes for a course, or as a tool for self study, and even as a handbook to look up some definitions or theorems— is another of its great advantages.</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 modern introduction to functional analysis for mathematics or engineering students who are familiar with finite dimensional vector spaces. There are many exercises that form an integral part of the text. Topics discussed are Banach spaces, continuity and differentiation, Hilbert spaces, compact operators, and distributions. Applications include classical and quantum mechanics and optimization problems. The book can be used for self-study, as a guide for a course, and even as a reference work.</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/amol-sasane" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amol Sasane</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-78634-333-8 (hbk), 978-1-78634-334-5 (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">GBP 98.00 (hbk), GBP 56.00 (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">396</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></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://www.worldscientific.com/worldscibooks/10.1142/Q0096" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/Q0096</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/46-functional-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46 Functional analysis</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/46-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46-01</a></li></ul></span>Tue, 20 Feb 2018 14:08:08 +0000Adhemar Bultheel48277 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/friendly-approach-functional-analysis#commentsThe Power of Computational Thinking
https://euro-math-soc.eu/review/power-computational-thinking
<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>
Computers play an increasingly important role in mathematics and the converse is also true, old and new branches of mathematics are increasingly important in computer science. Traditionally, languages, science, and mathematics were the corner stones and the main tools in education to develop skills needed to understand the world. Obviously those shape our scientific thinking in a broad sense. The authors believe that more and more also a way of computational thinking is another pillar that should support our educational system. It relies on logic and mathematics and results in an algorithmic approach to problem solving. That involves not only a sequential description of successive steps, but it also includes abstraction and generalization (not solving a particular problem, but solve a whole class of similar problems), it requires checking all the details (no loose ends, and including also all the most unlikely situations), and it needs analysis, and it forces to look for optimal (shortest, fastest, ...) algorithms. Note that all these properties also apply to solving a mathematical problem. And yet there is a difference the authors claim. And I tend to agree, since indeed, I also have known students who were good in mathematics, and yet had difficulties to understand or assimilate an algorithm and others were marvelous computer programmers, but were very reluctant when mathematical theorems and proofs were involved.</p>
<p>
The two authors of this book are mostly involved in education and with this book they want to illustrate what this computational thinking means outside the context of a lesson or a school environment. So they chose the subtitle ``Games, magic and puzzles to help become a computational thinker''. They obviously love magic and tricks and much of what they illustrate relies on these. Part of the text is based on their contributions to the online magazine <em>Computer Science for Fun</em> located at <a href="http://www.cs4fn.org/" target="_blank">www.cs4fn.org</a>.</p>
<p>
They start with Jean-Dominique Bauby, editor in chief of the magazine <em>Elle</em> who, after a stroke, was completely paralyzed and could only blink with an eye. Solve the problem `What is the most efficient way for him to communicate?'. This leads to a binary search algorithm and efficient encoding of an alphabet based on a frequency analysis. Similar cutting of the problem in half at each step is illustrated with a card trick to predict the value of a card from a stack. It turns out that this also applies to mechanically selecting punched cards earmarked with a binary system of slits and holes.</p>
<p>
Cut hive puzzles consist of a hexagonal grid partitioned in subsets each containing a number of neighboring cells. If such a subset had <em>n</em> cells, then the numbers 1,2,...,<em>n</em>need to be placed in these cells, but no cells with the same number can touch each other. This involves not only designing the successive logical steps and rules of the type if... then... for a solution method but it also introduces another important element: patterns. Pattern matching and recognition is important for computational thinking.</p>
<p>
Other puzzles are more like traveling salesman problems: visit a number of squares of a grid with knight moves, or visit a number of interesting tourist attractions in a city, or the classical Königsberg bridge problem. These require a clear representation of the data. In this case clarifying the problem structure via graphs.</p>
<p>
Then a big leap is taken to artificial intelligence. How can a robot learn? Although they don't name it, they touch upon the basics of genetic algorithms, and discuss one of the earliest chatbot ELIZA, and the Turing test. A logical next step is to illustrate the idea of a simple neural network. Since it is a binary example, it also connects with logic binary circuits and simple Boole algebra. Again it is a big leap to then discuss the question if ever artificial intelligent robots or computers can take over our world. The authors reassure their readers that this will certainly never happen because the internet of things consists of things that are too different and moreover they do not have a reason to dominate the human race. So it would only be possible if we let them do it.</p>
<p>
Here we are halfway through the book and here the authors take a step back and discuss some grid games. There is not only the grid of pixels on a screen, but more important is Conway's game of life, or the spit-not-so game. The latter is a game in which two players have to select words from a list until they have 3 words with the same letter in them.</p>
<p>
The remaining chapters involve more tricks and magic using some simple mathematics and most of all pattern matching. For example patterns implied by prime numbers or detecting patterns in an image using simple digital filters, etc. But also the basic principles of CAT scans and MRI are explained. Psychological misleading or make-belief can help in tricks (a variant of the goat-rabbit-cabbage that have to cross the river, or visual illusions, or Weber's law). This is fun, but it seems not to be related to computational thinking, unless one tries to model the human brain and how it experiences its environment.</p>
<p>
The authors conclude by summarizing what they find to be important for computational thinking. It is not only the algorithmic idea, but also modeling the problem and the situation based on scientific arguments, sometimes it requires heuristics, but logic and pattern matching are always important. The representation of the problem can be important to make abstractions and generalizations and it can help to decompose the problem into smaller subproblems. Of course computational problems arise by certain needs people have, so this interaction is also important, understanding what they want and evaluation of the resulting algorithmic product by collecting feedback from users are definitely of practical importance.</p>
<p>
So, I believe it is clear what the authors want to convey to the reader: what is computational thinking and what is covered by that concept. It is certainly not thinking the way a computer thinks but it is linked with how humans function in everyday life, and how this can be transferred to a computational machine. It is however not so clear to me who they want to convey the message to. Are this the children who have to learn these skills? Are it the policy makers who have to define the education programs? Are it the generally interested readers? I believe they address all of them but do not really bring the message to any of them. There is too much magic, tricks, and puzzles to really bring the concepts to children, and they try to cover a really broad spectrum and they touch only superfluously on some topics. There are key words that keep appearing repeatedly typeset in bold in the text like: logic thinking, decomposition, algorithmic thinking, pattern matching, etc. but there is no systematic treatment. After reading the book it gives the impression that magicians must be the best computational thinkers, and you better start practicing card tricks before engaging in programming a computer. But I believe this is not what they had in mind.</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 explain what they understand by computational thinking. It is much broader than algorithmic thinking and involves logic, analysis, pattern matching, and even understanding the human-computer interaction. They bring their message illustrating it with with low-level examples, often involving puzzles, card tricks and magic.</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-curzon" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paul Curzon</a></li><li class="vocabulary-links field-item odd"><a href="/author/peter-w-mcowan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Peter W McOwan</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-78634-183-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 48.00 (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">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></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://www.worldscientific.com/worldscibooks/10.1142/q0054" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/q0054</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97q99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97Q99</a></li></ul></span>Sun, 19 Feb 2017 09:49:10 +0000Adhemar Bultheel47468 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/power-computational-thinking#commentsPrime Numbers, Friends Who Give Problems: A Trialogue with Papa Paulo
https://euro-math-soc.eu/review/prime-numbers-friends-who-give-problems-trialogue-papa-paulo
<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>
Paulo Ribenboim is a number theorist, born in Brazil in 1928, who is living in Canada since 1962 where he was professor at Queen's University. In the tradition of the Socratic dialogues, he wrote this trialogue on prime numbers in which he is Papa Paulo and his opponents are Eric and Paulo. These two are interested in prime numbers. In fact, it is Eric who starts asking elementary questions about primes to which Papa Paulo answers. Soon Eric is joined by his friend Paulo (the other Paulo), and Papa Paulo is renamed to be P.P. (for obvious reasons he is not really happy with that alias). I believe Ribenboim wants to picture Eric and Paulo as young adults, and this is how I first imagined them, also since P.P. is addressing them as such, but then they seem to be very knowledgeable about many things (for example they point out to P.P. that Édouard Lucas was French or Eric who is said "to have travelled a lot"), things you would not expect from teenagers asking elementary questions about prime numbers. Whatever they are, it is just a minor glitch in the story, which does not affect the mathematics.</p>
<p>
The discussion thus starts at a very elementary level but after a while it gradually turns into a course on prime numbers with formulas, computations, theorems and proofs. There are some intermissions in italic like <em>`Eric paused for a while, then continued'</em>. Here is another one: P.P. tells that Fermat after his death meets Saint Peter who has to decide on whether he should be sent to Heaven, Hell, or Purgatory. Fermat is confronted with his little lie about the short proof that he had for his last theorem but that the margin was too small to contain it. In that chapter P.P. is discussing the primality of Fermat numbers and states at the end that it is not known whether there are infinitely many Fermat numbers that are prime or that are composite. Then that chapter ends with the funny remark: `<em>The effect of this strong statement of ignorance caused this reaction on Paulo and Eric: Poor Fermat, he may stay in purgatory forever.</em>'</p>
<p>
The latter illustrates that the conversation that has mostly a serious mathematical aspect, also has instances with funny components. Besides these few italic parentheses and some notes at the end of the chapters in which some biographical notes are added about a person that was mentioned (Euclid, Euler, Mersenne, Legendre, Fermat, and many many more), the whole book is just reproducing the conversation among the three protagonists. There is another bit of an unrealistic aspect to this trialogue when it comes to all the computations and formulas or formulations of theorems with their proofs. The latter formulation include the titles `Theorem' and `Proof' in bold and end with an q.e.d. message. This is something you only find in a printed mathematics book, not in a conversation, unless the discussion is taking place while the characters are writing down what they are saying as it is printed. This is indeed how we should read it because at some point P.P. <em>says</em>: `You are sharp-eyed, but what I <em>wrote</em> is correct.' (my emphasis). So he <em>wrote</em> it, not <em>said</em> it. Although P.P. is in fact to be identified with the author (Ribenboim is the meta-P.P.), he basically only reproduces the conversation and does not tell us much of the meta-story about the who, how, what, and where of the actors outside what is in the conversation. So there is only a very thin sketch of their personality, and only few circumstantial remarks in the conversation go besides the mathematical discussion. Ribenboim is just following the literary genre of the Platonic dialogues seasoned with contemporaneity and humour.</p>
<p>
Although the reading is light, the book is not easy for a truly unskilled reader since, as the book advances, the mathematics get more and more involved. In the beginning it is about the Euclidean algorithm, gcd, lcm, modular arithmetic, the Wilson theorem, Fermat numbers, and Mersenne primes, up to primality testing and public key encoding. But when it comes to the prime number theorem, it requires real numbers, the log and exp functions and the logarithmic integral and for the formulation of the Riemann hypothesis, complex numbers and complex functions, series, analytic continuation and much more advanced mathematics need to be introduced. Nevertheless, the `technical stuff' is left out as much as possible. At some point, one of the intermissions read: <em>Papa Paulo was visibly happy with the presentation of the important theorem of Dirichlet on primes in arithmetic progressions. He was particularly elated to have been able to hide all the technical innovation needed to prove the theorem in its general form...'</em></p>
<p>
Towards the end of the book many more curious facts and conjectures about prime numbers are formulated (twin primes and the likes, conjectures by Goldbach, Sophie Germain, Bunyakovskii, Schinzel and Sierpinski, and many others). There are even conjectures by Papa Paulo and by Eric.</p>
<p>
Paulo Ribenboim has written some dozen books almost all published by Springer. This one is published by World Scientific, so I do not know how much is fiction and how much is truth, but the last chapter is about publishing the notes of the trialogue. P.P.'s usual (fictional) publisher Marcel Spank at Gold Springs Publishing Company New York does not like the original title <em>Prime Experiments Explained to Boys and Girls</em> and proposes <em>The Story of Two Boys in Love with Prime Numbers</em> (from this it should be understood that Eric and Paulo are indeed boys and not adults). Eventually Spank turns down the manuscript. This at the time of P.P. writing this chapter it is still uncertain whether the notes will be published or not. It is also in this chapter that P.P. gives his unconventional idea about why there are so few women in mathematics: He, being a man, gets his best ideas while shaving, and women don't shave, hence....</p>
<p>
As a conclusion, I liked the book and at some stages it is absolutely funny. The reader should however be prepared to swallow all the mathematics, the theorems, the proofs, the formulas and all the computations. As long as only integers are involved, in principle anybody motivated enough can understand what is going on. When it becomes more involved around the formulation of the prime number theorem, it may become a bit more difficult to hang on, but then it becomes interesting again when all these mysterious properties about prime numbers are conjectured. I can imagine that it will get smart young people interested in starting a career involving number 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>
Two boys, Eric and Paulo, start asking questions about prime numbers to Papa Paulo. The conversation between those three grows into an introduction to number theory, in particular to the properties of primes and all the interesting questions and conjectures that can be formulated about them. The whole story is told in the form of a trialogue, but it involves theorems and proofs as well. It is intended for a broad audience, and yet it gives an introduction to what the Riemann conjecture is all about.</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/paulo-ribenboim" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paulo Ribenboim</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-981-4725-80-4 (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">£36.00 (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">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/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</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://www.worldscientific.com/worldscibooks/10.1142/9836" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9836</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/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</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/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</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/97f60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F60</a></li></ul></span>Fri, 02 Dec 2016 11:45:40 +0000Adhemar Bultheel47308 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/prime-numbers-friends-who-give-problems-trialogue-papa-paulo#commentsMethods of Differential Geometry in Classical Field Theories. k-symplectic and k-cosymplectic approaches
https://euro-math-soc.eu/review/methods-differential-geometry-classical-field-theories-k-symplectic-and-k-cosymplectic
<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">José Luis Guijarro Regalado</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/Recension.pdf" type="application/pdf; length=94382" title="Recension.pdf">Brief recension with the main formulas inside the book</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>The book generalizes the hamiltonian and lagrangian mechanics to classical field theories of first order.</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/manuel-de-le%C3%B3n" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Manuel de León</a></li><li class="vocabulary-links field-item odd"><a href="/author/modesto-salgado" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Modesto Salgado</a></li><li class="vocabulary-links field-item even"><a href="/author/silvia-vilari%C3%B1o" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Silvia Vilariño</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9789814699754</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">87,58€</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">126</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/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</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><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/53-differential-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53 Differential 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/70h" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70H</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/70h03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70H03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/70h05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70H05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/53b50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53B50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/53c80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53C80</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/53d42" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53D42</a></li></ul></span>Thu, 10 Nov 2016 13:46:14 +0000José Luis Guijarro Regalado47265 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/methods-differential-geometry-classical-field-theories-k-symplectic-and-k-cosymplectic#commentsThe “Golden” Non-Euclidean Geometry
https://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-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>
Fibonacci numbers and their relation to the golden ratio are among the few mathematical items that gained some publicity among non-mathematicians. The golden ratio ($\phi=1.68033...$) is well known since antiquity and it played an important role in Euclid's <em>Elements</em> and in the work of many other mathematicians. It also shows up in phylotaxis and spirals that appear in nature. And it relates to harmony, another term that has been studied in a mathematical sense since the Greek. The golden ratio has therefore gained some mythical and even mystical status, the latter often has to be understood in a (pejorative) numerological sense.</p>
<p>
Fibonacci numbers (denoted $F_n$) is a term coined by Édouard Lucas in the 19th century, who also introduced the sequence of Lucas numbers (denoted $L_n$). Both sequences are solutions of the difference equation $x_{n}=x_{n-1}+x_{n-2}$. The initial conditions for the Fibonacci sequence are $(x_1,x_2)=(1,1)$, while for the Lucas numbers it is $(x_1,x_2)=(1,3)$. The limit of ${x_n}/{x_{n-1}}$ equals $\phi=({1+\sqrt{5}})/{2}$ in both cases. These numbers are defined for all integer indices by $x_{-n}=(-1)^nx_n$.</p>
<p>
In a first chapter, the authors give a brief historical survey, summarize some properties of the Fibonacci and Lucas numbers and they introduce hyperbolic functions: the symmetric Fibonacci hyperbolic sine $sFs(x)=({\phi^x-\phi^{-x}})/{\sqrt{5}}$ and cosine $cFs(x)=({\phi^x+\phi^{-x}})/{\sqrt{5}}$. Similarly for the Lucas versions $sLs(x)$ and $cLs(x)$ but these do not have the denominator $\sqrt{5}$. Their graphs look very much like the graphs of the standard hyperbolic functions.</p>
<p>
The second chapter is about harmony. The old Greek <em>Music of the spheres</em> was picked up by Pacioli and Kepler. But soon the text comes down to one of Stakhov's pet horses, namely that harmony is a forgotten pillar in mathematics. Counting and classical measure theory are the two other pillars that have resulted in conventional mathematics. However by rejecting Cantor's axiom (a 1-to-1 correspondence between the reals and the points on a line) and a consistent application of the golden ratio and its generalizations, a different measure theory, number system, and geometry can be developed. This is what he calls harmonic mathematics. He considers a delayed version of the above difference equation which leads to the introduction of a new representation of number systems and his $p$-Proportion Codes. However this is soon replaced by another generalized Fibonacci sequence, defined for any real $\lambda>0$ by $F_\lambda(n+1)=\lambda F_\lambda(n)+F_\lambda(n-1)$, with $F_\lambda(0)=0, F_\lambda(1)=1$ and the limiting ratio $\phi_\lambda=(\lambda+\sqrt{4+\lambda^2})/2$ which is a root of the characteristic equation $x^2-\lambda x-1=0$. The above Fibonacci hyperbolic functions can be generalized by replacing $\phi$ by $\phi_\lambda$ and the $\sqrt{5}$ by $\sqrt{4+\lambda^2}$. They are denoted as $sF_\lambda$ and $cF_\lambda$. Note that (up to a factor 2) the classical hyperbolic functions are obtained as a special case of the $\lambda$-Lucas numbers by choosing $\lambda=e-1/e$. For $\lambda=1,2,3,4$ we get the golden, silver, bronze, and copper relations, referred to as the metallic relations.</p>
<p>
The third chapter is about Hilbert's fourth problem, in which it is asked to design new forms of non-Euclidean geometry. The formulation was however rather vague and different proposals were made but it remained unclear whether the problem was (completely) solved or not. So the authors have their own interpretation and solve their form of the fourth problem using the hyperbolic functions introduced above. Lobachevsky's hyperbolic geometry is based on classical hyperbolic functions. Replacing the classical ones by the hyperbolic $\lambda$-Fibonacci functions they get different hyperbolic geometries. To obtain a similar generalization for spherical geometry, yet another type of Fibonacci functions are needed. There are of the form $SF_\lambda(x)=c_\lambda\sin(x\ln\phi_\lambda)$ and $CF_\lambda(x)=c_\lambda\cos(x\ln\phi_\lambda)$ with $c_\lambda=2/\sqrt{4+\lambda^2}$. The $\ln\phi_\lambda$ factor appears here for the sake of harmony. A similar form can be obtained in the hyperbolic case giving a true hyperbolic geometry in harmony mathematics. They consider many more relations and formulas in this context and claim that the Clay Mathematics Institute made a mistake by not putting Hilbert's fourth problem on their list of millennium problems. So the authors claim to have actually solved a self declared millennium problem.</p>
<p>
The next chapter 4 is about the qualitative theory of dynamical systems based on harmony mathematics. Hence the `golden' and also the other metallic proportions show up again. It is a simple observation that a metallic ratio $\phi_\lambda$ (which is an irrational number) can be approximated from above and below by ratios of successive $\lambda$-Fibonacci numbers. This simple fact is exploited in a complicated framework of foliations of a 2D manifold. First foliations of such a manifold are introduced, which is then specialized to the 2D torus $T^2$. These foliations are characterized by a Poincaré rotation number $\omega$. In the particular case that it happens to equal a metallic proportion, then it can be approximated by ratios of Fibonacci numbers and hence the irrational foliation is approximated by rational ones. Since integral curves for flows of a dynamical system are foliations, this may also be applied in a context of dynamical systems. This chapter is much more mathematical with long mathematical proofs which do not seem to be easily accessible for a general public.</p>
<p>
A last chapter is about the fine structure constant in physics. Like the mathematical millennium problems, there is a list of physical millennium problems. The first of these problems is asking whether all dimensionless parameters of the physical universe are calculable. Here the fine structure constant $\alpha$ is declared to be fundamental and hence is the constant to be discussed. The approach taken here is by looking at the Lorentz transform in special relativity theory. It is a transformation of the space-time vector whose matrix can be written as a direct sum of the identity and a hyperbolic rotation over an angle $\theta\in(-\infty,\infty)$. In view of the preceding items it is again a natural thing to replace the classical hyperbolic sine and cosine functions of the rotation angle by the hyperbolic Fibonacci sine and cosine ($\lambda=1$) of an appropriate angle $\psi$ and so obtain a Fibonacci special relativity theory. Here however $\psi\in(-\infty,0)\cup(2,\infty)$ because singularities appear at 0 and 2. Moreover, the speed of light in vacuum has to be made variable. It decreases with the age of the universe. It will be $c^*$ (the classical value) for $\psi\to-\infty$ and it is $c^*/\phi^2$ for $\psi\to-\infty$. The physical meaning is that the Big Bang corresponds to $\psi=0$, the interval (0,2) is the dark age before galaxies were formed (the speed of light is imaginary), and for values larger than 2 this corresponds to the light age, when the stars were formed that created light in the universe. To the left of the origin is the black hole situation with the arrow of time reversed.<br />
In 2000, N.V. Kosinov proposed a formula $\alpha=10^{-43/20}\times\pi^{1/260}\times \phi^{7/130}$. Inspired by this formula, the authors propose to let $\alpha$ depend on $\psi$ by replacing the $\phi$ in this formula by their $\psi$-depending speed of light. The result is an $\alpha(\psi)$ with $\psi=\lambda_0 T$ where $T$ is the age of the universe (in billions of years) and $\lambda_0$ a constant. This $\alpha$ is decreasing with $\psi$ in the black hole range until it becomes 0 at the Big Bang. In the same range the speed of light drops from $1/\phi$ to 0. In the dark age, the derivative is positive and goes from 0 to $\infty$ just like the modulus of the speed of light does, and in the light age it drops from infinity to a little bit below its current value of about $7.29\times 10^{−3}$. Of course as a consequence of the varying $\alpha$, also other values that depend on it will change with the age of the universe. In an appendix allusion to multiverses is made when the $\phi$ in the previous setting is replaced by $\phi_\lambda$ with $\lambda\ne 1$.</p>
<p>
The first author Alexey stakhov is a Ukrainian mathematician with a PhD in computer science, who lives in Canada since 2004. He has published many papers and books in which he has proposed many of his original, sometimes controversial, ideas. Chapter 2 clearly summarizes some of his previous work. The second author is Samuil Aranson who is a Russian mathematician, now living in the USA whose domain is differential equations, geometry and topology. It is therefore clear that he must be the main author for chapters 3 and 4, which also explains the somewhat different and more mathematical style. Scott Olson is a professor of philosophy and religion in the USA, who wrote a book on the golden section and who seems to be helping with the English editing of this book.</p>
<p>
The first two chapters are elementary with a lot of history and simple mathematical relations. Who wants to read more on Fibonacci and Lucas numbers and generalizations can read <a href="http://www.euro-math-soc.eu/review/pell-and-pell%E2%80%93lucas-numbers-applications">Pell and Pell-Lucas Numbers with Applications</a> for a good mathematical treatment and there are of course many popular books on the golden ratio. If you are interested in the golden ratio and harmony, you would certainly want to read <a href="http://www.euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries">The Fibonacci Resonance and other new Golden Ratio discoveries</a>. However chapters 3 and 4 of this book are much more mathematical and create a complicated mathematical framework of foliations, not suitable for a general public anymore, while it only illustrates and applies the fact that the ratio of two successive Fibonacci numbers tend to the golden ratio and hence that this irrational number can be approximated by rationals. The fifth chapter is devoted to physics. The core idea is to replace a classic hyperbolic rotation by a more general one. The physical interpretation is certainly not mainstream and is probably susceptible to critique by theoretical physicists, if they do not consider it to be just numerological mysticism. However, since there is no experimental proof of what is exactly happening at this cosmological scale, it may be another explanation that is as good as many other fantasies. It is clear that the book is mainly collecting results that the authors have published as papers and that are here somewhat streamlined into a more consistent survey. Long lists of references are added after each chapter with many papers of the authors but several are only available in Russian. That this harmony mathematics and Fibonacci numbers and generalizations can solve all these problems clearly adds to the myth of the golden ratio. The typesetting in LaTeX is nicely done. I could spot a few typos but not that serious. For example page 121, a $(dv)^2$ is missing in the equation and on page 232 the Black Hole should correspond to $-\infty<\psi<0$ and not $0<\psi<2$. Also the graphics of chapter 5 are a bit rough and not always very precise. Anyway there are some original ideas to be found in this book. Whether the reader will agree with them or not will depend on who's reading it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a book in which the authors give a summary of some of their work. They study Fibonacci and Lucas numbers and show how these give rise to a new kind of mathematics: the mathematics of harmony. Generalizations of these number sequences and their limits the golden and other metallic ratios are applied to derive a new kind of non-Euclidean geometry, to study foliations and dynamical systems and even a golden Fibonacci version of the special relativity theory in which the fine structure constant from cosmology is analyzed.</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/alexey-stakhov" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexey Stakhov</a></li><li class="vocabulary-links field-item odd"><a href="/author/samuil-aranson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Samuil Aranson</a></li><li class="vocabulary-links field-item even"><a href="/author/scott-olsen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Scott Olsen</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-981-4678-29-2 (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">£98.00 (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">308</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/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</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://www.worldscientific.com/worldscibooks/10.1142/9603" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9603</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/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</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/11b39" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B39</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/53a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A35</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/37d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37D40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/83a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83A05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83f05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83f05</a></li></ul></span>Fri, 23 Sep 2016 08:30:58 +0000Adhemar Bultheel47182 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/%E2%80%9Cgolden%E2%80%9D-non-euclidean-geometry#commentsProblems for Metagrobologists
https://euro-math-soc.eu/review/problems-metagrobologists
<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>
David Singmaster is a retired mathematics professor from London South Bank University. He has published several books on Rubik's cube and a collection of mathematical puzzles. He regularly contributes to puzzle sections of several magazines. This is yet another collection of his puzzles with their solutions. They are truly mathematical as the subtitle states: "A collection of puzzles with real mathematical, logical or scientific content". Speaking of the title, for the less geeky readers it may need some explanation what "metagrobologists" really means. Singmaster realizes that his catchy title may raise question marks in the eyes of the potential buyer. So the first thing he does in his introduction is explain the title. It seems that Rabelais was the first to use <em>metagraboulizer</em> in his <em>Gargantua</em> (1534). It is a humorous version derived from the French <em>grabeler</em> which means to sieve fragments out of a medical substance. It has been translated as 'to puzzle' or 'to make a dunce of somebody', meaning 'to confuse somebody'. The English form metagrobolize was used in the translation of the <em>Gargantua</em> by Thomas Urquhart in 1653. Metagrobolize is in the <em>Oxford English Dictionary</em> described as 'to puzzle' or 'mystify'. The word was popularized among puzzlers since Rick Irby used it in 1981 in the <em>Wall Street Journal</em> in a tongue-in-cheek kind of way. Nowadays Singmaster is one of the best known self-declared metagrobologists.</p>
<p>
Some of the puzzles from the book belong to the puzzler's public folklore, so that the origin is not always clear, but the collection here contains 221 puzzles that were at some point proposed by Singmaster, so that most of them can safely be considered to be his invention or his own variant or generalization of a classical one. Where the origin is recalled, the original is acknowledged together with the generalization.</p>
<p>
The puzzles are not simple. Some allow an easy answer provided the proper insight is used. Others do require some longer calculations (the solution part of the book is thicker than the part where the puzzles are formulated). Many of them also have an additional teaser: asking to consider a generalization or when one solution is found, asking for (at least) one other solution or for all possible solutions if there is some freedom left. Some of these additional problems are still open.</p>
<p>
The puzzles are grouped by theme: arithmetic, digits, geometry, geography, sequences, monetary, clocks, calendar, combinatorics, word puzzles etc. Usually the puzzle is formulated as a short story or a dialogue, for example with characters confronted with a problem and the reader is asked to help them out.</p>
<p>
Let me give some distilled bare-bones examples (after evaporating the story) to provide an idea of the level.<br />
From the digits chapter: 14 x 926 = 12964 is special because the 5 digits appearing in the product of the left-hand side are just the digits that appear in the result of the right-hand side. Twelve such examples were found, but Singmaster's contribution is that there are three examples with only three digits are missing from the list. The problem is to find these.<br />
From the physics chapter: why is there no differential gear on a railroad car while it can still go around the bend?<br />
A geographical problem: How high above the earth should one be to see one third of its surface? Assuming of course that the earth is a perfect ball.</p>
<p>
The problems are rather hard and it may need some puzzle experience to find some of the solutions without looking up the answer. If you are patient enough to solve them all by yourself, there is material for many hours of cheap entertainment here.</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 collection of mathematical puzzles from the master metagrobologist David Singmaster who has contributed puzzles to a mathematical journal since he was a master student in 1963. Since 1987 he contributed regularly to more popular magazines and other media.</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-singmaster" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Singmaster</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-981-4663-64-9 (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"> £23.00 (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">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/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://www.worldscientific.com/worldscibooks/10.1142/9531" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/9531</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Fri, 23 Sep 2016 07:53:21 +0000Adhemar Bultheel47181 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/problems-metagrobologists#comments