Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenErnest Irving Freese's Geometric Transformations
http://euro-math-soc.eu/review/ernest-irving-freeses-geometric-transformations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>In its simplest form, a geometric dissection refers to subdividing a polygon into a finite number of pieces and to reassemble them to form another polygon. It is proved that this is always possible, but the challenge is then to obtain it with a minimal number of pieces. Proving the minimality for an orthogonal polygon is shown to be NP-hard. If every piece can pivot around a vertex that stays connected to a neighbouring piece during the transformation, it is called a hinged dissection. When unfolded, it forms a string of connected pieces. Piano-hinged means that the connection is not at the vertices, but that edges are connected so that this corresponds to folding a paper version, but these are not considered in the present book.</p>
<p>Frederickson starts this book with a short history of dissections. They were first studied in the 19th century by some mathematicians a.o. Farkas Bolyai, but they became popular in the 20th century when they featured in newspaper puzzle columns by Sam Lloyd and by H. Dudeney, (who collected them also in their collected puzzle books) and later M. Gardner. Harry Lindgren provided a way to derive dissections by overlapping tessellations of the plane around the middle of the 1900s. That is also the time that Freese was preparing his work on the topic.</p>
<p>In his book Dissections: Plane & Fancy (1997) Frederickson mentions some 'over 200 plates' prepared by Freese of which a few loose copies were circulation, but the whole manuscript was not located. In his second book Hinged Dissections: Swinging & Twisting (2002), Freese has disappeared from the reference list, but in Piano-hinged Dissections: Time to Fold! (2006), Freese's work is very prominently present in the form of appendices to the chapters written by Fredericson. Some of the plates are reproduced and Frederickson provides comments. What has happened? Frederickson gives an explanation, which is actually a very remarkable story. That story is retold in this book. Freese, who obviously knew about these dissection problems from the puzzlers columns and from puzzle books that he got as a present from his wife. He had collected his ideas on dissections in the form of 200 plates that he finished a couple of months before he died. He wrote in 1957 to a friend that he had been intensively busy preparing them, that a blueprint could be obtained for $28.00, but that probably nobody would be interested in publishing his drawings. After his death several people tried to obtain the manuscript from his wife Winifred but she had written a letter to Ginsburg (a friend of Freese and editor of a journal) asking him to take care of the manuscript. However Ginsburg had died three weeks before her husband, so he never answered and the manuscript was forgotten. Frederickson obtained Winifred's address only after she passed away, but her son Bill was living in the house now. Frederickson wrote him a letter but the manuscript was not found. After Bill died, a cousin found the letter and the manuscript so that Frederickson finally could make a copy of Freese's plates in 2003. That explains why Freese features in the appendices of his 2006 book.</p>
<p>Because Frederickson also got hold of some other material, he can add a biography of Ernest Irving Freese (1886-1957) as the second chapter of this book. A very wild adventurous life this Los Angeles architect has lived. Mostly self-taught, he first worked for an architectural firm, later as an independent architect, but he also travels the country as a tramp, and later goes on a bicycle world tour, working when he needs money or as a crew member on a ship to pay for his fare. He published articles in cycling magazines and in architectural and construction journals. After an earthquake in 1933 he started a campaign to construct safe schools (he was by then father of three). He was an assertive man with strong opinions.</p>
<p>The main purpose of this book is to finally publish the manuscript by Freese. It was originally not conceived as a commercial product, so it is a notebook that consists of loose hand-made geometric constructions with little text in Freese's elegant slanted handwriting. Frederickson has kept the order of the numbering of the places, subdivided them into chapters and provided an introductory text and explaining notes, references, and new results per chapter (Freese has no references) and this text is followed by the relevant plates. So, while in his 2006 book Frederickson used Freese's results as an appendix to his own work, here it is the other way around, it is Freese's work with Frederickson commenting. The plates are beautifully reproduced after being digitally processed to remove stains. Their original size is 8.5 x 11 inch (21.6 x 27.9 cm) which is also the somewhat unusual format of this book. Moreover to keep the originals intact and in the state that Freese has created them, these pages get no headers or page numbers, they only have the original encircled plate numbers.</p>
<p>Freese had divided the plates into sections corresponding to the geometry of the objects. The subdivision into smaller chapters is a decision of Frederickson. The idea is that chapters will group transformations that are somewhat similar so that a common introduction is possible, although Frederickson is also commenting on all the separate plates within each chapter. The chapters are then about transforming isoscele or equilateral triangles, followed by squares, crosses, rectangles, and n-polygons and n-polygrams up to n = 12 and they conclude with some unclassified miscellaneous figures. All the dissections are two-dimensional, so no 3D generalizations. It is made clear with references that some of the dissections were found later by others, when Freese's work was unavailable. Everyone interested in geometric dissections, and this kind of puzzles, either mathematically or recreationally will embrace this publication. But also the readers interested in the history and certainly those who became curious about this mystery man and his manuscript, after reading Frederickson's 2006 book, will be fully satisfied with this respectful reproduction eventually made available for a general public.</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 main purpose of this book is to publish 200 plates illustrating geometric dissections that were produced by E.I. Freese, a Los Angeles architect, shortly before his death in 1957. Due to circumstances, the plates got lost and was only recovered by G. Frederickson in 2003. This book contains a short history of geometric dissections, and a biography of Freese, followed by the reproduction of the plates subdivided into chapters and introduced and commented by Frederickson.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/greg-n-frederickson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Greg N. Frederickson</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/world-scientific-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific publishing</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-981-3220-46-1 (hbk), 978-981-3220-47-8 (pbk), 978-981-3220-49-2 (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">GBP81.00 (hbk), GBP32.00 (pbk), GBP26.00 (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">432</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.worldscientific.com/worldscibooks/10.1142/10460" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10460</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/52-convex-and-discrete-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52 Convex and discrete geometry</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/52b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52B45</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/52-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-01</a></li>
<li class="field-item odd"><a href="/msc-full/05b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05B45</a></li>
<li class="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 06:34:22 +0000adhemar48456 at http://euro-math-soc.euThe Paper Puzzle Book
http://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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</a></li>
<li class="field-item odd"><a href="/author/david-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li>
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Thu, 10 May 2018 06:28:44 +0000adhemar48455 at http://euro-math-soc.euMandelbrot the Magnificent
http://euro-math-soc.eu/review/mandelbrot-magnificent
<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>Benoit Mandelbrot finished his autobiography shortly before he died in 2010. After some editing, it was published as a book entitled <a target="_blank" href="/review/fractalist-memoir-scientific-maverick"><em>The Fractalist, Memoir of a Scientific Maverick</em></a> in 2012 with a foreword by his wife Aliette. In that book he describes in three parts (1) his youth in prewar Warsaw and Paris and later in Tulle (France) during WW2 (2) his education and scientific life in the period 1944-1858 and (3) his life after recognition, i.e. the period 1958-2004.</p>
<p>Liz Ziemska is a Polish born literary agent who came to the US when she was seven. She earned a Bachelor of Science in Biology and a Master of Fine Arts in creative writing and she has written several stories that appeared in fantasy and science fiction collections. What do you think will come out as a story if she first read Mandelbrot's <em>The Fractalist</em> and then, impersonating a juvenile Mandelbrot, she would re-tell the first five chapters using the facts provided by Mandelbrot and mix these with her own fantasy? You do not have to guess because that story has been written and it is called <em>Mandelbrot the Magnificent</em>.</p>
<p>The facts: Benoit Mandelbrot is born in Warsaw (1924) in a family with a long Russian-Jewish tradition. His mother was a dentist and his father was a business man who became also a tailor, forced by the circumstances. His father's younger brother, uncle Szolem, was mathematically gifted and introduced Benoit to mathematics at a young age. When Szolem got a position in France in 1936, Benoit together with his younger brother Léon and his parents, moved first to Paris and three years later, when Szolem was appointed at the university of Clermont-Ferrand, they moved to the nearby village of Tulle. After the Germans invaded France, Tulle fell under the "Free" France of the Vichy Régime headed by Marshal Pétain. Szolem escaped the war because he got an appointment in Princeton and migrated with his wife to the US. When also the Vichy France was invaded by the Germans, life became pretty dangerous for Jews and Benoit and his brother narrowly escaped from being arrested. Tulle is infamously remembered for the Tulle massacre in 1944, three days after D-Day, when civilians were executed and many taken captive by an SS tank division as revenge for a successful action of the French Résistance. That is a summary of the first five chapters of <em>The Fractalist</em> and these are also the facts that Ziemska uses in her story.</p>
<p>If she were only repeating these facts, then there would be no point in writing her novella since also the original text is well told, and it is first hand. So, Ziemska adds some fractal imaginative detail-adornments and some more large-scale fantasies. Examples of the latter are certainly the Mandelbrot family hiding from the Germans in a fractal structure invented by Benoit. She also introduces the <em>sefirot</em> as an essential element in Benoit's life. It is a densely but well structured esoteric graph from the Kabbalah with ten nodes that represent all manifestations of an infinite God (or of "G-d, the Mathematician" as Ziemska writes). In Ziemska's view it gave Mandelbrot the insight of iterated function systems, an essential tool for the generation of fractals. To increase the narrative tension, she also added the character of Emile Vallat, a student in Benoit's class in the Tulle period who is another bright student, competing with Benoit for the best grades in mathematics, but Emile's family (his mother is the local librarian where Benoit finds his Book of Monsters, a ficticious book on mathematical objects) is sympathizing with the Germans and so, he is constantly teasing and humiliating Benoit and his brother who try hard to be discrete and hide their Jewish background.</p>
<p>Of course fractals and more generally mathematics are well represented announcing Mandelbrot's future career as the inventor of fractals and their omnipresence in nature. So several names of mathematicians are mentioned: Kepler, Poincaré, Gaston Julia,...; and mathematical terms, not really mathematics, just some name dropping without explanation: Zeno's paradox, Fibonacci numbers, Hausdorff dimension, the volume of a sphere as a multiple integral, the golden section, and of course the Mandelbrot set; and there are the illustrations from The Book of Monsters: the Sierpinski triangle, Koch's snowflake curve, Peano's curve,.... These are all very curious elements to embed in an imaginative novella, making it literally "extraordinary". The beginning and the end of this story is Mandelbrot finishing his memoirs while his wife Aliette is serving him cauliflower for his eightieth birthday, his favourite dish in which he admires the fractal structure.</p>
<p>This is a well told story, with many Jewish elements like for example the role of the <em>sefirot</em> and the dreadful situation of Jews during the war, and the atrocities that in fact any war does to a society. The latter is unfortunately very real, but, although based on facts, it is not a biography of Benoit Mandelbrot's youth. For that component it should be taken for what it is intended to be: a mixture of facts and fiction. Somewhat disappointing from a mathematician's point of view is that geometry is illustrated by "monsters" (and not by "gems") but Ziemska blames Poincaré for that. She uses one of his well known quotes where he claims, referring to an example produced by Weierstrass of an everywhere continuous function that is nowhere differentiable (a typical property of fractals) that "logic can sometimes make monsters that would have to be set grappling with this teratologic museum". But it is certainly Ziemska's fantasy that makes mathematics seem to be some Kabbalistic pseudo-science placing it in the same category as a magician's magic, with the magician anxiously hiding the secrets of his tricks from his public, so that he can perform disappearing tricks in some Hausdorff dimension, impenetrable for ordinary people. But of course this works out very nicely when used in a fantastic story.</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 novella in which Ziemska impersonates the adolescent Benoit Mandelbrot as the narrator telling the story of his youth first in Warsaw and mainly in Tulle in France during the second World War. Ziemska has added several of her own imaginative components to the story.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/liz-ziemska" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Liz Ziemska</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/torcom" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tor.com</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">978-0-7653-9805-5 (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">10.04 USD (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">128</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.tor.com/2017/10/24/excerpts-liz-ziemska-mandelbrot-the-magnificent/" title="Link to web page">https://www.tor.com/2017/10/24/excerpts-liz-ziemska-mandelbrot-the-magnificent/</a></div></div></div>
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<li class="field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
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Tue, 10 Apr 2018 09:17:45 +0000adhemar48379 at http://euro-math-soc.euInfinity: A Very Short Introduction
http://euro-math-soc.eu/review/infinity-very-short-introduction
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is one of the small booklets (literally pocket books: 174 x 111 mm) appearing in the Oxford series <em>Very Short Introductions</em> that treat diverse subjects from accounting to zionism. Infinity, is a concept mainly of importance and practically useful in mathematics, but it has also philosophical and even religious aspects. Stewart is as broad as "a very short introduction" allows and adds a lot of history to his discussion. So much is to be told on only 143 small pages. Although there is obviously a lot of overlap, Stewart's treatment is wider than Marcus Du Sautoy's <a href="/review/how-count-infinity" target="_blank">How to Count to Infinity</a> and Eugenia Chen's <a href="/review/beyond-infinity-expedition-outer-limits-mathematics" target="_blank">Beyond Infinity</a> who stay more on a mathematical playground.</p>
<p>
Infinity, and certainly the infinitely large, has long been something fuzzy that was discussed on a philosophical basis. The Greek were arguing over a distinction between an actual (existing) infinity and a potential version, i.e. that "something" that is beyond all natural numbers, which is never reached by enumeration. They got away with the infinitely small by their concept of commensurability in what was mainly a geometric approach to mathematics. The infinitely small was beyond any possible subdivision of a finite length. Their fundamental common measure was thus finite and that led Zeno to his paradoxes. The infinitely small was somehow tackled when calculus was developed by Newton and Leibniz in the eighteenth century introducing infinitesimals. They represented something almost zero but not quite. When used in calculations one could divide by them, since they were not zero, but at some point, when suitable for the result, they were assumed to be zero. Not very rigorous mathematics that was. It was not until towards the end of the nineteenth century that Georg Cantor brought more insight into the nature of the infinitely large. Stewart guides us through this history and illustrates how the concept of infinity has played a role in several disciplines that all have somehow contributed to how we think of the concept today.</p>
<p>
With a first chapter, Stewart puts forward some puzzles or paradoxes that involve infinity to illustrate that it is not sufficient to say that infinity is that "something" that is beyond all numbers. More precise definitions are needed for the infinitely large as well as for the infinitely small. Examples are the processes that hide irrational numbers like a staircase approximating the diagonal of a square converging to a straight line when its steps become finer and finer, and the regular polygon converging to the circle as it gets more edges. These demonstrate the problem of evaluating $0\times\infty$ in a sensible way. Hilbert's hotel is illustrating that a more precise definition of the infinitely large is required, and Stewart gives some other examples. These puzzles and paradoxes are first raised as questions for the reader to think about. Stewart's explanations of all these confusing statements are given afterwards.</p>
<p>
The second chapter illustrates that infinity is not hidden away in higher mathematics but that it is also embedded into elementary calculus. Gabriel's horn is obtained by revolving $1/x$ for $x>1$ around the $x$-axis. This has the surprising property that its volume is finite even though the surface is infinite. Of course infinity is also hidden in 0.9999... being equal to 1, a fact that astonishes many an undergraduate student, and of course infinity resonates in the decimal representation of irrational numbers. Distinguishing discrete from continuous would not be possible without infinity. Here as in the other chapters Stewart gives quite some attention to history: Dedekind defining the real numbers as sections which are essentially infinite objects, Lambert who proved the irrationality of $\pi$. In the Jain religion of India (600 BCE), people distinguished infinity from enormously large numbers, etc.</p>
<p>
Chapter three is further exploring the historical views of infinity. Space and time were traditionally assumed to be infinite, but when looking at the infinitely small, the situation is different. People had difficulty in dealing properly with infinitely small things. Zeno's paradoxes are examples that illustrate that a sum of infinitely many nonzero numbers can be finite. Since the ancient Greek there has been a distinction between an actual infinity and a potential infinity, a discussion that has continued throughout the centuries among philosophers. Even some theologians claimed that God was the only existing impersonation of something infinite. Some proofs for the existence of God were based on this belief. For mathematics, this distinction is not essential. Mathematical existence is abstract and does not coincide with physical or actual existence.</p>
<p>
The next chapter is a discussion of the infinitely small and how this has triggered the development of calculus. The original historical concept of infinitesimals is now replaced by the concept of a limit. The infinitesimals where revived when in the 1960's Abraham Robinson developed non-standard analysis.</p>
<p>
In geometry, infinity is where the horizon is. It led to the development of perspective in the Renaissance. This is extensively discussed in chapter six, explaining why a ship seems to become smaller as it approaches the horizon, and how this has led to the concept of a point or a line at infinity. The Euclidean plane can be modelled as a disk where infinity is represented by its boundary. More concretely, the line at infinity makes it easy to produce perspective drawings. Eventually this discussion ends in ideas of projective geometry and the mapping of the plane to a sphere and vice versa by stereographic projection, the point at infinity corresponding to the North Pole on the sphere.</p>
<p>
Infinity is a useful concept in mathematics, but how does it appear in a physical world? That is what the next chapter is about. In physical sciences, infinity often leads to a nasty singularity. Stewart discusses three examples. The analysis of the rainbow phenomenon is an optical example. If light is incident at a certain angle, then the intensity of the rainbow would be infinite according to ray optics. This singularity entailed that light had to be reconsidered as a wave. In Newton's gravitation theory a singularity occurs when the distance between particles becomes zero and the potential becomes infinite. For example Zhihong Xia proved in 1988 that by solving equations in a five-body problem, dramatically non-physical solutions are obtained after a singularity. Black holes are singularities in general relativity theory and in cosmology the Big Bang is obviously a singularity. Stewart also explains here why cosmologists are wrong when they use curvature as a parameter that determines whether our universe is finite or not.</p>
<p>
Te last chapter is the discussion of how Cantor came to his proof that the real number are not countable and how this has led to set theory and his transfinite numbers, and how this resulted in a revision of the foundations of mathematics. This story is best known by mathematicians or anyone who is a bit familiar with this kind of mathematical background literature. But again here Stewart follows the historical evolution of who did what and why in brewing up the eventual result.</p>
<p>
This is a lot of information and because of the compact presentation, it will not always be casual reading for a general reader. There are a few references provided per chapter, which might be of interest if the reader wants to look up more details. Some aspects are elaborated more than what is needed for explaining the impact of infinity (e.g. the computation of the angle of the rainbow, the geometry of perspective) but these topics are of course interesting in their on right, and they are usually not found in other treatments of infinity. If you are interested in only the strict mathematical concept of infinity, then Du Sautoy's or Chen's treatises that were mentioned above might be simpler alternatives. But in this booklet, even the experienced reader may have more occasion to learn something new. Some of these non-essential but nevertheless flashes of a that's-interesting-I-didn't-know-that experience will make it worthwhile reading.</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 booklet wants to introduce a general reader to the concept of infinity. With a lot of historical, philosophical, and occasionally theological background Stewart shows how the concepts of the infinitely small and the infinitely large were eventually settled in a mathematical setting towards the end of the nineteenth and early twentieth century when the current foundations of mathematics were established.<br />
</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-1987-5523-4 (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">£7.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">154</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/infinity-a-very-short-introduction-9780198755234" title="Link to web page">https://global.oup.com/academic/product/infinity-a-very-short-introduction-9780198755234</a></div></div></div>
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<ul class="field-items">
<li class="field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
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<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
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Tue, 03 Apr 2018 06:35:02 +0000adhemar48365 at http://euro-math-soc.euMulti-shell Polyhedral Clusters
http://euro-math-soc.eu/review/multi-shell-polyhedral-clusters
<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>When studying materials at a nanoscale, some well structured lattices can be observed. The hexagonal structure of graphene is a well known two-dimensional carbon structure that can live in a three-dimensional world in the form of a nanocone or a nanotube. The Buckminsterfullerene or C${}_{60}$ which is a dodecahedron is a simple example of a closed surface. Also three- or higher-dimensional structures with strict topological geometry have practical applications. Think of a cube which has 8 atoms on its vertices and add atoms at the body center and at the center of the 6 faces and the midpoints of the 8 edges and this will give a hyper-structure with 27 atoms. The cube is divided into 8 sub-cubes but this is easily generalized to a structure consisting on $n^3$ sub-cubes. Hence in mathematical chemistry, an extensive literature emerged that investigated the topological properties of these nanostructures. The simplest ones are the Platonic solids: tetrahedron (T), cube (C), octahedron (O), dodecahedron (D), and icosahedron (I). They are represented by undirected three-dimensional graphs, assuming atoms at the place of the vertices and the edges representing chemical bonds. Plato identifying earth, air, water, fire, and ether with cube, octahedron, icosahedron, tetrahedron, and dodecahedron respectively. Kepler in his <em>Mysterium Cosmographicum</em> used a nesting of the Platonic solids to model the position of the planets in the solar system. These five polyhedra are now revived as basic building blocks in these nanostructures. All kinds of maps can be applied to them to form more complex blocks from which much more complicated constellations can be composed. This book wants to describe some of the structures that can be obtained and tabulate their topological properties. It provides some kind of atlas for particular sets of these structures.</p>
<p>To describe all the complex structures, some introductory chapters are needed to give the necessary definitions from graph theory and of the topological indices of these graphs. The second chapter introduces operations on the elementary structures which will be the main tools to construct the more complicated ones. Some examples of these transforms: the <em>dual</em> of a graph exchanges the role of faces and vertices; the <em>median</em> of a graph takes as vertices the midpoints of the original edges and connects a pair if they belong to originally adjacent edges; and a <em>truncation</em> cuts off the vertices of the polytope by a plane that intersects all its incident edges; and there are more complicated operations possible like stellation, snub, leapfrog, etc. . The result is an atlas of single shell structures.<br />
The third chapter defines how more complex constellations can be formed by adding more shells to the structure. For example, one could add a vertex at the body center of a polyhedron and connect it to the surrounding vertices (the P-centered clusters). Other examples are the cell-in-cell clusters that place a polyhedron inside another polyhedron and connect the nearby vertices of inner and outer cell. Or there can be abstract structures like the 24-cell (a four-dimensional generalization of a Platonic solid).<br />
Depending on what property one is interested in, different notations exist in the literature (a Schläfli symbol like {<em>p,q</em>} or {<em>p,q,r</em>}, Conway's notation, Coxeter diagrams etc.), this can already be confusing, but unfortunately the author has to add another one for the more complex structures. The author is of course not a mathematician, but it is somewhat regrettable that there is not a strict formal definition of the notation in its most general form. One should try to grasp the meaning from the many examples. Another unfortunate fact is that the operations and the more complicated structures get different names in the literature, although median, snub, and stellation are pretty standard. When these are used as abbreviations in the formal notation this can be confusing and so some familiarity with the different nomenclatures is advisable.<br />
Chapter 4 is the last of the "introductory" chapters and introduces symmetry and (structural) complexity, which can be measured by several indices like Euler characteristic, centrality and ring signature. Also for translational and spongy structures and other structure generating techniques such parameters can be computed.</p>
<p>Chapters 5-11 form the main part of the book and describe collections (they form an atlas) of several clusters that are based on the icosahedron, octahedron, tetrahedron, dodecahedron, or constellations like multi-tori and spongy hypercubes. The last chapter 12 requires a bit more (carbon based) chemistry and considers structures with C${}_{20}$ (dodecahedron), or C${}_{60}$ (truncated icosahedron) or D${}_5$ configurations.<br />
Each chapter starts with a short introduction, with some hints on the notation and tables that contain all the so-called figure counts (number of vertices, edges, and faces of successive order, the rank of the structure and the Euler characteristic. Then enlarged pictures of the graphs, one per page, visualize the structure, but they become quickly hopelessly complicated when the structure is a bit more complex, even when they are multi-coloured, it is often hard to distinguish the nested layers of edges inside the cage.</p>
<p>Each chapter has a long list of references, many of which are by the author. Some chapters correspond for a large amount with one of his papers. For example the discussion of the ring structure index in chapter 4 is largely overlapping with the paper C.L. Nagy, M.V. Diudea, Ring Signature Index, in <em>MATCH Commun. Math. Comput. Chem.</em> <b>77</b> (2017) 479-492. It may in fact help to look up some of these papers because the text is really telegraphic, and is clearly a compilation of previous results, and thus not always explaining all the details. Therefore, I consider it is a reference text for the specialist, but I would not recommend it as a first reading on the subject of nanoclusters.</p>
<p>The book is number 10 in the Springer series <em>Carbon Materials: Chemistry and Physics</em>. Diudea is a very prolific writer in this area. He was co-editor of two other books in the series: <em>Diamond and Related Nanostructures</em> (vol. 6, 2013) and <em>Distance, Symmetry, and Topology in Carbon Nanomaterials</em> (vol. 9, 2016). Perhaps because of the pressure to publish on the very quickly evolving subject, the quality of English and mathematical editing of this book could have been much better. For example at several places articles are missing (the structure of entire polytope, p,45) or in excess (however at the Plato's time, p.125) or typos produce words that are just wrong (inconsistences, p.39, convex hall, p.42); names are misspelled (Platon p, 77; Hässe, p.45); references are wrong (reference to graph 1.2.4 should be 1.2.2, p.7, Fig. 4.4 refers to top and bottom figures but there is no bottom figure, p.67). For a chemist, this might be nitpicking but it will irritate many a mathematician that variable names are inconsistent ($s_1$ and $S_1$ for the same operation on one line, p.29, and on p.34: $p_4T,P_4(C),p_4(D)$ are three different notations for the same operation $p_4$ on one line); roman and math font are mixed (header of two successive tables 4.3 and 4.4: once in roman, once in italic, p.66); symbols and terms are not defined or used before they are defined (I do not find the meaning of $f_n$ defined, but it probably means an $n$-gon, if so, then the headers $f_5$ and $f_6$ in table 3.2 should be $f_4$ and $f_5$, p.47, in the same table the meaning of $c_n$ and $M$ are not explained, chapter 1 uses RS and CS for row-sum and column-sum, without telling, while in chapter 4, RS is ring signature chapter 2 uses P for polytope or platonic solid, but in the atlas (p.32) it is a prism, in the atlas symbols for rhombic (Rh), antiprism (A), pyramid (Py) were not explained before. Notations like $(3.4)^2$ (fig.2.3), $5.6^2$ (Thm. 2.1) and similar ones on page 33 were used without explanation, since the definition of ring signature follows only in chapter 4); sentences like "Computations at a higher level of theory: Hartree-Fock and DFT have been performed with the HF/6-31G(d,p), B3LYP/6-31G(d), B3LYP/6-31G(d,p) and LDA/3-21G(d) sets, on Gaussian 09 (Frisch et al. 2009). PM6 computations were done with the VSTO-6G(5D;7F) set." (p.400) are very cryptic when the abbreviations are not explained; and most unfortunately, the figures, a prominent feature of this book, are not always helpful (the atlas is like a picture book of all these clusters, and while the graphics are very useful for simple structures, they soon have too many edges in the more complex ones to make anything clear); another typesetting glitch: the text explaining the figure at the bottom of page 31 is on top of page 32.</p>
<p>Even though this may not be the best place to start, I think the subject is a very interesting one where there is work for mathematicians. It is the resurrection of a subtopic of crystallography 2.0 and graph theory requiring somewhat more geometrical (and chemical) insight than just studying the symmetry groups, but it is simpler in a sense because only topological properties are needed, which means that the structure is completely characterized by the 0-1 adjacency matrix.</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 provides an atlas describing graphs and their topological properties representing several atomic nanoclusters in complex constellations. There is a brief introduction about different concepts from graph theory, about mappings of the Platonic solids, and about topological figure counts (the number of faces of successive rank: vertices, edges, faces,...) and topological indices. The book is richly illustrated with various pictures of the different hyperstructures.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/mircea-vasile-diudea" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Vasile Diudea</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-nature" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature</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">978-3-319-64121-8 (hbk); 978-3-319-64123-2 (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">158,99 € (hbk); 118,99 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">457</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319641218" title="Link to web page">http://www.springer.com/gp/book/9783319641218</a></div></div></div>
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<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
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<li class="field-item even"><a href="/msc/52-convex-and-discrete-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52 Convex and discrete geometry</a></li>
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Mon, 19 Mar 2018 09:09:21 +0000adhemar48344 at http://euro-math-soc.euIslamic Geometric Patterns
http://euro-math-soc.eu/review/islamic-geometric-patterns
<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>What are the mathematics behind Islamic geometric decorations? What is the essence that makes it so recognizable? One possible characterization is by pointing to the symmetry, hence group theory is what is needed to describe it. However, that may catch some of the local symmetry, which of course is part of the beauty of these designs, but it does not completely explain the overall structure as well as the finer geometric aspects of doubling and interweaving lines that define the patterns. Thus the description of the 17 wallpaper groups is not the end of the story.</p>
<p>Jay Bonner, who is a creative designer of these patterns gives here a detailed description of the underlying polygonal techniques that can be combined to form a myriad of possible designs. He comes to his conclusion by comparing the many designs that were used throughout the Islamic cultural history and by distilling from these the techniques that were possibly used. Some of the designs, and hence the assumed underlying techniques, were more popular than others or were particular for certain regions or periods. The possibilities of the more complicated ones were not always fully explored and they give rise to new original designs. After the decline of the craftsmanship of these Islamic designs, some renewed interest in the subject arose in the second half of the twentieth century. Some books were written on the mathematics of the symmetry groups used, and it became a popular subject for documentaries and picture books, but Bonner now supersedes the latter less mathematical approaches with this monumental encyclopedia. It is not only a nice picture book with over a hundred photographs of decorative art on monuments (in chapter 1), but there are also the 540 other illustrations, many of which consist of several parts that illustrate the construction and the results of the designs.</p>
<p>The first chapter starts with a quick survey of design techniques with pointers to many illustrations in subsequent chapters where a more technical discussion is given. The main objective of the chapter however is to illustrate by a chronological summary how the different techniques were used throughout the centuries of Islamic culture from the Umayyad Caliphate (7-8th century) till the Mamluk Sultanate in Egypt (13-16th century), how they evolved in Eastern Islamic countries as well as in North Africa and the Western Al-Andalus, and how the techniques were adopted in non-Muslim cultures.</p>
<p>The second chapter is a bit more technical and summarizes different classification methods. One can for example look at an underlying regular tessellation (isometric, triangular, rectangular, hexagonal), or the known plane symmetry groups can be used to classify the designs, but the method proposed by Bonner is by design methodology, and he gives arguments why the polygonal technique is probably the one that was historically most commonly used, and hence the proper way to classify. Other authors have proposed that historically different methodologies were used but there is less evidence for those proposals or they are only useful for simpler designs. The polygonal technique starts from a polygonal tessellation of the plane. Pattern lines in these polygons will define the eventual design. These pattern lines emerge at points on the edge under particular incidence angles and intersect the pattern lines from the other edges. Once the polygons are put together to form a tessellation of the plane, the global design will protrude and the underlying polygonal stratagem can be forgotten.</p>
<p>The incidence angle of the pattern lines at the midpoints of the edges can be acute median or obtuse, and there is a fourth possibility in which pattern lines start from two symmetric points on the edges. Depending on the incidence angles and the underlying polygonal pattern rotational symmetry will occur. The most common are fourfold, (with squares and 8-pointed stars), sixfold (with 3-,6-,12-, and even 24-pointed stars), or fivefold (5- and 10-pointed stars), but occasionally also sevenfold symmetry was used, and in the more complex designs we also find 11, 13-pointed stars. Usually the stars appear at the vertices of some regular polygonal grid and/or its dual.</p>
<p>The longest chapter by far is chapter three which is a thorough discussion of the polygonal technique. One possibility is to start from a tessellation of the plane that consists of one or several types of regular polygons (triangles, squares, hexagons, octagons). Sometimes one needs the systematic inclusion of an irregular polygon, which is then called a semi-regular grid. The pattern lines can be narrow or invisible like when they just delimit coloured mosaic tiles, or they can be widened or doubled. Moreover they usually do not just intersect but they form an ingeniously interweaving pattern.</p>
<p>But regular or semi-regular tilings are relatively simple and soon Bonner moves to tessellations composed of regular and irregular polygons decorated with suitable pattern lines that fit nicely together obtained by one of the four design possibilities (acute, median, obtuse, 2-point). Bonner systematically discusses the different possible symmetries that can be obtained in this way. There are two variants of the fourfold symmetry. The A version has a large and a smaller octagon and seven other polygons to tessellate. The B version has only one octagon and five other polygons, but still that leaves many possible tessellations. The fivefold system obviously involves decagons and pentagons but can also include many other convex and concave polygons. This fivefold system was very popular and Bonner discusses several variations depending on the shapes of repeat units, that are rhombi, rectangles, or hexagons, These repeat units will fill up the plane by translation. It's not a coincidence that the golden ratio appears in these designs. Sevenfold symmetry occurs is more complicated to deal with and therefore probably less frequently used. The starting point is a tetradecagon and a heptagon and pattern lines can be constructed by connecting the midpoints of edges that are <em>k</em> = 1,...,6 positions apart.</p>
<p>A second group of design methods are called non-systematic patterns by Bonner. This technique allows the construction of more enigmatic stars with 9,11,13, or 15 points. While in the previous group, a tessellation was formed using a limited set of polygons, in this group, just one characteristic polygon is used (rhombus, triangle, square, rectangle, hexagon) that tessellates the plane. The generation goes as follows. Take one of the polygons and generate at each of its vertices, equispaced radii are generated such that the incident edges of the polygon are two of them. The intersection points of the radii are used to generate a design pattern consisting of smaller polygons, and the whole design is then translated to cover the plane. Bonner describes many examples using this kind of technique, some are historical, but there are also possibilities for original designs.</p>
<p>The most complex design technique is called dual-level design. Basically one starts from a coarse level that generates a set of lines that are widened. These wide strips are decorated with a fine gain design, which is then extended to the whole plane. This gives highly complex structures of which historical examples exist. Although there are only two levels used, it has the characteristics of self-similarity and it creates possibilities for new multilevel designs. In a short final section, some ideas are given about how to apply such techniques to decorate a dome or a sphere.</p>
<p>I do realize that my previous attempt to capture the main points of the design methodologies is totally inadequate since one needs the graphics to understand them properly. You may want to look up the author's Facebook page or the website of his company, but none will match the abundance and clarity of pictures in this book.</p>
<p>In a short chapter 4 Craig Kaplan describes the software building blocks that will be needed to generate the pictures on a computer: tilings, fitting polygons together, generating patterns lines, producing rosettes, how to join widened lines or generate the weaving effects etc. There is very little mathematics here and it remains a high level description so that it will need additional computer and mathematical skills to actually produce the graphics, but it gives at least some useful guidelines.</p>
<p>The book is very carefully edited, especially the graphics are extremely nice and very informative. The only strange typo I could spot was that σ is called "delta" on page 361. The book is not written by a mathematician, nor is it written for mathematicians. It is an artistic designers (hand)book for Islamic(-like) geometric patterns. There is very little mathematics, but I am sure all mathematicians will love the beauty of the designs non the less. While reading the text, it takes a while to get used to the terminology. There is a glossary with a set of terms that are briefly explained at the end of the book, but these are necessarily short and their meaning will become only gradually more clear. When chapter one starts with a brief survey of the techniques, one is pointed to pictures in later chapters to get an idea of what is meant, but the proper explanation comes only in chapters 2 and 3, and if you are really interested how the graphics can be produced on a computed, one has to read chapter 4. Mathematicians may be used to books that are arranged in the opposite order: start with the definitions and tools and end with the applications. The (often forward) references to pictures in this book are however carefully and consistently done, so that with a lot of paging back and forth one becomes gradually familiar with the content and the ideas proposed. The book has the looks of a coffee table book, but it requires more than just casual reading to understand the design methods.</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 marvelously illustrated book about the Islamic decorative art that is immediately recognized by its geometric patterns. The possibilities of combining designs for basic patches on diverse polygonal tiling strategies leads to a wealth of different patterns, for which some classification is proposed. The first approach is mainly historical with many pictures of the actual decorations, but there are many more graphics generated by computer to illustrate the patterns and how they are generated and repeated to fill the plane.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/jay-bonner" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jay Bonner</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-verlag-new-york" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Verlag New York</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">978-1-4419-0216-0 (hbk); 978-1-4419-0217-7 (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">116,59 € (hbk); 91,62 € (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">620</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9781441902160" title="Link to web page">http://www.springer.com/gp/book/9781441902160</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/05b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05B45</a></li>
<li class="field-item odd"><a href="/msc-full/01a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A30</a></li>
<li class="field-item even"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li>
<li class="field-item odd"><a href="/msc-full/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li>
</ul>
</span>
Mon, 19 Mar 2018 08:44:30 +0000adhemar48343 at http://euro-math-soc.euIndefinite Inner Product Spaces, Schur Analysis, and Differential Equations
http://euro-math-soc.eu/review/indefinite-inner-product-spaces-schur-analysis-and-differential-equations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is volume 263 of the Birkhäuser series on <em>Operator Theory Advances and Applications</em>. It is devoted to Heinz Langer on the occasion of his eightieth birthday. Two other volumes in this series were celebrating Langer: <em>Contributions to operator theory in spaces with an indefinite metric</em> (OT106, 1995) on the occasion of his sixtieth birthday and <em>Operator theory and indefinite inner product spaces</em> (OT163, 2004) on the occasion of his retirement at the University of Vienna.</p>
<p>
The titles of these three volumes already illustrate that operator theory in indefinite inner product spaces form the focus of Langer's research. Langer was born in Dresden in 1935. After his his PhD and his habilitation at the TH Dresden he became a professor leading the institute of probability and mathematical statistics. His stay in Odessa in 1968-69 where he met M.G. Krein strongly influenced his career and his research interests. He also spent research stays at several Western universities as well, which was not obvious in the time of the DDR. In 1969 he left East Germany permanently to become a professor in Dortmund, later in Regensburg, and finally, in 1991, he accepted a position in Vienna where he stayed until his retirement.</p>
<p>
This is just a very brief summary, but Bernd Kirstein has a much longer, and richly illustrated contribution in this book. It is the ceremonial address on the occasion of the honorary doctorate awarded to Heinz Langer by the TU Dresden in 2016. He received many other prizes among which another Dr. h.c. from Stockholm University in 2015. Kirstein sketches in detail the people that were influential on Langer's career. Many of them became colleagues and friends. Among them are the most important names in the domain: Krein, Nudelman, Iokvidov, Potapov, Sakhnovich, Gohberg (who founded the OT series in 1979), Adamyan, Arov, Potapov, and many more. Kirstein describes this from his own perspective, hence the paper describes also the history of the Schur analysis group in Leipzig that he is leading together with his mathematical twin brother Bernd Fritzsche. Kirstein also illustrates the difficulties in maintaining relationships among mathematicians in an East block country and their colleagues who had left for Israel or another Western country before the fall of the Iron Curtain in 1989.</p>
<p>
A list of the publications of Heinz Langer (op to January 2017) is also included in the biographical part I of this book. A similar list in OT163 in 2006 had 171 entries, while the current one has 203 (the last one from 2017) which illustrates that Heinz Langer at his age is still an active researcher and collaborator. And the latter is what the main content of this book really is: an illustration of the influence that Langer had on other people who worked on topics related to the subjects that are close to the heart of his own research, always prepared to listen and collaborate. These topics include nonlinear eigenvalue problems, indefinite inner product spaces such as Krein and Pontryagin spaces and applications in mathematical physics.</p>
<p>
A collection of sixteen research papers, (some are longer surveys, others are short communications, all together over 420 pages) form the main part II of this volume. The titles of the papers and their authors are available on the publisher's website (see this book's meta-data elsewhere on this page) so that I do not need to repeat them here. The papers are listed in alphabetical order of the first author, but in their introduction, the editors subdivide them into five (overlapping) classes. The largest group falls under the broad title <em>Schur analysis, linear systems and related topics</em>. These papers are about Carthéodory and Weyl functions, Nevanlinna-Pick interpolation, scattering theory, L-systems and an inverse monodromy problem. In the group about <em>Differential operators, inverse problems and related topics</em> which is broad as well, we find papers related to the pantograph delay equation, and spectral and other properties for a selection of other operators. Two papers are explicitly dealing with <em>Pontryagin spaces</em> and one paper is about probability and is classified as <em>Non-commutative analysis</em>. <em>Positivity</em> is a keyword that can be assigned to almost all the papers in the volume, but it groups the remaining three texts where positivity has a key role.</p>
<p>
This volume will of course be of interest to anyone who knows or collaborated with Heinz Langer, but more generally for anyone working in one of the topics that he was, and still is, interested in, and this is a broad field as illustrated by the papers in this volume. So it may be that not all the papers are interesting for a particular reader, but in that case there is of course also the possibility to download an electronic version of a particular paper from the publisher's website, like one would do for a journal paper.</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 set of papers collected to celebrate the eightieth birthday of Heinz Langer. The broad research field of Langer can be described by keywords as enumerated in the title of this book. Besides the set of selected papers that fall under these topics, there are also some biographical data like a list of publications of H. Langer and a long and richly illustrated paper by B. Kirstein sketching the career of Langer and his influence on the Schur analysis group in Leipzig.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/daniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">daniel alpay</a></li>
<li class="field-item odd"><a href="/author/bernd-kirstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Kirstein</a></li>
<li class="field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-nature-birkh%C3%A4user" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature/ Birkhäuser</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">978-3-319-68848-0 (hbk), 978-3-319-68849-7 (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">116.59 € (hbk); 91.62 € (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">522</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319688480" title="Link to web page">http://www.springer.com/gp/book/9783319688480</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
<li class="field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
<li class="field-item even"><a href="/imu/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial Differential Equations</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/46-functional-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46 Functional analysis</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/46n99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46N99</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li>
<li class="field-item odd"><a href="/msc-full/47a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A40</a></li>
<li class="field-item even"><a href="/msc-full/93c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">93C05</a></li>
</ul>
</span>
Tue, 13 Mar 2018 08:03:28 +0000adhemar48324 at http://euro-math-soc.euThe moment problem
http://euro-math-soc.eu/review/moment-problem
<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 moment problem, or one should say moment problems (plural) because there are several different classical moment problems. Some ideas can be found in work of Chebyshev and Markov, but Stieltjes at the end of the nineteenth century was one of the first to formally consider the moment problem named after him. Given a sequence of numbers ($m_k$), is there a positive measure $\mu$ such that $m_k=\int x^k \mu(dx), k=0,1,2,\ldots$? In the case of Stieltjes, the measure was supposed to have a support on the positive real line. First of all one wants to find out under what conditions such a measure exists, then when the solution is unique, and when it is not unique to characterize all possible solutions. Soon (around 1920) other versions were formulated by Hamburger (when the support of the measure is the whole real line) and Hausdorff (when the support is a finite interval) and some ten years later the trigonometric moment problem was tackled by Verblunsky, Akhiezer and Krein where the support is the complex unit circle. There is a basic difference between the trigonometric moment problem and the other classical moment problems on (parts of) the real line. In the latter situation, the existence of a solution is guaranteed by requiring the positivity of Hankel matrices whose entries are the moments. In the trigonometric case, the Hankel matrices are replaced by Toeplitz matrices. The latter involve also the moments $m_{-k}=\overline{m}_k, k=1,2,\ldots$ which are automatically matched as well. Not so for the other moment problems. When also imposing moments with a negative index in those cases, this is called a <em>strong</em> moment problem. When only a finite number of moments are prescribed, this is called a <em>truncated</em> moment problem.</p>
<p>
The importance of the moment problem is a consequence of the fact that it is at the crossroads of several branches and applications of mathematics. It relates to linear algebra, functional analysis and operator theory, stochastic processes, approximation theory, optimization, orthogonal polynomials, systems theory, scattering theory, signal processing, probability, and many more. No wonder that the greatest names in mathematics have contributed to the problem with papers and monographs. Because of the many connections to different fields also many approaches and many generalizations have been considered. The previously described moments are called power moments because of the $x^k$, but one could also prescribe moments based on a set of other functions $M_k(x)$. Traditionally, the Hausdorff moment problem is formulated for the interval [0,1], but one may consider any finite interval $[a,b]$ just like the Stieltjes moment problem could be formulated for any half line $[\alpha,\infty)$. Other generalizations lifts these problems to a block version, by assuming that the moments are matrices and the measure is matrix-valued, or the variable $x$ can have several components, resulting in a multivariate moment problem.</p>
<p>
The fact that today, 100 years after Hamburger and Hausdorff, this is still an active research field is another proof of the importance of moment problems. Many books did appear already that were devoted to moment problems or where moment problems played an essential role. Some classics are Shohat and Tamarkin <em>The Problem of Moments</em> (1943), Akhiezer <em>The classical moment problem and some related questions in analysis</em> (1965), Krein and Nudelman <em>The Markov moment problem and extremal problems</em> (1977). The present book is a modern update of the situation. It gives an operator theoretic approach to moment problems, leaving aside the applications. The univariate classical problems of Hamburger ($\mathbb{R}$), Stieltjes ($[0,\infty)$) and Hausdorff ($[a,b]$), appear both in their full and their truncated version. Also the trigonometric moment problem is represented but by only one chapter.<br />
The introduction to these problems is quite general. It is showing how integral representations for linear functionals can be obtained, and in particular how this works for finite dimensional spaces, and for truncated moment problems. Another essential tool is giving some examples of how moment problems can be defined on a commutative *-semigroup. Indeed, all what is needed is a structure with an involution (which could be the identity) and it should allow the definition of a positive definite linear functional so that it can give rise to an inner product on the space of polynomials (and its completion). With gross oversimplification one could say that a sequence is a moment sequence if the associated linear functional is positive and the solution corresponds to the measure that appears in an integral representation of the functional. For real problems, the involution is the identity: $x^*=x$, for complex problems, the involution $x^*=1/\overline{x}$ allows to treat the trigonometric moment problem at the same level as the real moment problems.<br />
This general approach is not really needed for the classical one dimensional moment problems that are treated in part I and the truncated version in part II, but the generality of the introduction allows more easy generalizations to the multivariate case and its truncated version that are discussed in parts III and IV respectively. What is treated in the first two parts are the classical results: the representation of positive polynomials, conditions for the existence of a solution of the moment problem, Hankel matrices, orthogonal polynomials and the Jacobi operator, determinacy (i.e. uniqueness) of the solution, the characterization of all solutions in the indeterminate case, and the relation with complex interpolation problems for Pick functions. For truncated moment problems one may look for some special, so called N-extremal, solutions which lie on the boundary of the solution set, or a canonical solution or solutions that maximize the mass in a particular point of an atomic solution.</p>
<p>
For the multivariate case, it takes some more work and we do not have the classical cases where the measure should be supported and generalizations can go in many different directions. Nevertheless, the corresponding chapters in parts III and IV go through the same steps as in the univariate case as much as possible. What are representing measures and when are polynomials positive? By defining the moment problem for a finitely generated abelian unital algebra, and using a fiber theorem that characterizes moment functionals, some generalizations of the one-dimensional case can be obtained (like for example a rational moment problem) or moment problems on some cubics. Determinacy of the multivariate moment problem is given in the form of a generalized Carleman condition, moments for the Gaussian measure on the unit sphere, and complex one- and two-sided moment problems are all discussed. Characterizing a canonical or extreme solution(s) is not as simple as in the one-dimensional case. Only for the truncated multivariate problems Hankel matrices are introduced and atomic solutions with maximization of a point mass can be characterized.</p>
<p>
The book appears in the series <em>Graduate Texts in Mathematics</em> which means that it is conceived as a as a text that could be used for lecturing with proofs fully included and extra exercises after every chapter as well as notes the refer to the history and the related literature. It is however marvellously capturing the present state of the art of the topic. So it will be also a reference work for researchers. It captures a survey of the univariate case and indicates research directions for the multivariate problem. The list of references at the end of the book has both historical as recent publications, but it is restricted to what has been discussed in the present book. Schmüdgen has published two books before on operator theory, so he knows how to write a book on a difficult subject and still keep it accessible for the audience that he is addressing (graduate students and researchers). Lists of symbols are really helpful to remember notation. The fact that on page 4 Chebyshev and Markov are situated in 1974 and 1984 respectively is just a glitch in an otherwise carefully edited text.</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 operator approach surveying classical one-dimensional moment problems, but the setting is general by formulating the problem on an abelian *-semigroups. This allows to also capture an introduction to multivariate moment problems which is much more recent and a subject that is still in evolution. The characterization of moment sequences, associated linear moment functionals, and determinate as well as indeterminate problems for the full or the truncated problems are discussed. Particular canonical and N-extremal solutions or solutions with a maximal mass point are discussed.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/konrad-schm%C3%BCdgen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Konrad Schmüdgen</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-internationa" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Internationa</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">978-3-319-64545-2 (hbk); 978-3-319-64546-9 (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">84,79 € (hbk); 67,82 € (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">535</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319645452" title="Link to web page">http://www.springer.com/gp/book/9783319645452</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/42a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A70</a></li>
<li class="field-item odd"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li>
<li class="field-item even"><a href="/msc-full/44a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A60</a></li>
</ul>
</span>
Tue, 13 Mar 2018 07:38:36 +0000adhemar48323 at http://euro-math-soc.euThe Turing Guide
http://euro-math-soc.eu/review/turing-guide
<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>
Jack Copeland is a professor at the University of Canterbury, NZ, director of the <a href="http://www.alanturing.net/">Turing Archive for the History of Computing</a>, co-director of the <a href="http://www.turing.ethz.ch/" target="_blank">Turing Center of the ETH Zürich</a>, and he has written or edited several books about Turing and his work. So he seems to be also the driving force behind this new collection of papers devoted to the life and the legacy of Alan Turing. Only four authors are explicitly mentioned on the cover of this book, but the collection contains 42 papers authored by 33 persons with very diverse backgrounds. Fifteen of the 42 papers were (co)authored by Copeland. Four of the papers by older authors (three of them have known or collaborated with Turing) are published posthumously.</p>
<p>
Alan Turing (1912-1954) hardly needs any introduction. Most people will know him as a codebreaker of the German Enigma at Bletchley Park during the second World War. They probably also have heard of his tragic death covered by a veil of uncertainty: was it an accident or suicide. He was convicted in 1952 to chemical castration for having a gay relationship. Only in 2013 he was rehabilitated by a royal pardon. Some may also have an idea of what a Turing Test is. A mathematician or a computer scientist will almost certainly also know that he proved independently but almost simultaneously with Alonso Church that Hilbert's <em>Entscheidungsproblem</em> was unsolvable. Turing proved it by reducing it to a halting problem which is undecidable on a universal Turing Machine. Many books and even films tell the story of Turing and of all the activities at Bletchley Park. The Turing Centenary Year 2012 which triggered the publication of many more and the recent (loosely biographical) film <em>The Imitation Game</em> (2014) have spread the knowledge about Turing in a broader audience. Bletchley Park may now be a major tourist attraction park, but the confidentiality that was kept by the British authorities about what was developed there during the war concerning cryptanalysis and the early digital computers has delayed the historical disclosure of the role played by Turing and other scientists in that period. Somewhat less known, but very familiar to biologists is Turing's work on morphogenesis which he developed during a later stage in his life. The book has eight parts that cluster papers about eight different aspects of Turing's life and legacy.</p>
<p>
Thus Turing was much more than just a codebreaker. His universal machine was an essential theoretical model in proving results about the foundations of mathematics, logic, and computer science. Because of his work at Bletchley Park while the first digital computing machines were being assembled during and just after the war, he was intensively involved in writing original software, a user's manual, and he has even contributed to the design of circuits and hardware. The introduction of machines that could be instructed to perform less trivial tasks raised concern about the future of Artificial Intelligence and Turing contributed with several variants of his Turing test in an attempt to define what intelligence really meant. He called his ultimate version of 1950 the 'imitation game'.</p>
<p>
It should not be forgotten, that, even though his scientific interest and contributions are broad, Turing was fundamentally a mathematician. It is less known that his Kings College Fellow Dissertation (1935) involved a proof of the Central Limit Theorem. It was little known that this was proved already in 1920 by Jarl Lindeberg and so Turing's result was never published. He also worked on group theory, in particular the word problem, on number theory (the Riemann hypothesis and normal numbers) and of course the code breaking involved statistical analysis and hypothesis testing. Turing exploited these statistics in his algorithms Banburismus and later Turingery. After the war he was also doing numerical analysis (LU decomposition, error analysis,...). His work on morphogenesis was also mathematical and involved diffusion equations that model the random behaviour of the morphogenes.</p>
<p>
This collection of papers is produced for an interested but general audience. Formulas are kept to a minimum and technical discussion is maintained at an accessible level. It may not be the best choice to read as a first introduction to Turing and his work. Better introductions that are less chopped up in different papers are available. On the other hand, if you have read already several books about Turing and his work, I am sure you will find here some anecdotes and historical facts that you did not know yet in each of the eight parts of the book.</p>
<p>
A first part is biographical. The timeline by Copeland is useful to place everything in a proper historical sequence. There is a testimony of Sir John Dermot Turing, Alan's nephew, and another by the late Peter Hilton an Oxford professor who worked with Turing at Bletchley Park.<br />
Part two is more history in which Copeland explains about the Universal Turing Machine conceived by Turing to solve the Entscheidungsproblem. It has also a noteworthy contribution by Stephen Wolfram, the creator Mathematica and Wolfram-alpha, who praises Turing for initiating computer science.<br />
The third part is the most extensive one and puts the codebreaking and Bletchley Park in the spotlight. Some of the texts are by people who worked there and who give an account of how everyday life was during the war, other papers are explaining how the Enigma machine worked and how it could be broken.<br />
In part four the first computers as they developed after the war are in the focus. The Colossus machines were computers that were used since 1943 for codebreaking, These facts were only declassified in 2000 so that one got the impression that the original ideas and prototypes came from von Neumann at Princeton who developed the ENIAC and the EDVAC. However, the University of Manchester had a small scale computer <em>Baby</em> (1948) that was running a few months before the ENIAC and Turing at the National Physical Laboratory developed the Automatic Computing Engine (ACE) that was operational in 1950. Turing even wrote a manual on how to program the machine to play musical notes.<br />
The fifth part is about computers and the mind: chess computers, neural computing, and the working of the human brain. It also has a remarkable text by novelist David Leavitt about Turing and the paranormal.<br />
The next two parts are about Turing's biological (morphogenesis) and mathematical (cf. supra) contributions. The final part has two papers contemplating the Turing thesis (1936) which claims that a Turing machine can do any task a human computer can do. Similar claims were made by Zuse and Church, but whether the whole universe can be seen as a computer, obviously depends on what you call a computer.<br />
In the last chapter about Turing's legacy in different disciplines we find many references to books and other media that can be consulted for further information.</p>
<p>
The remaining pages offer a short biography of the contributors, references to some books about Turing, and a list of published papers by Turing. The many references and notes from the contributions are also gathered at the end. The book ends with a very detailed index, which is of course very welcome and obviously non-trivial with that many different authors.</p>
<p>
In summary, this is a welcome addition to the existing generally accessible literature that gives additional testimony of the brilliant mind of Alan Turing. There is historical as well as technical material that will be appreciated also by specialists whatever their discipline: history, mathematics, biology, computer science, or philosophy.</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 papers about Alan Turing, his life and legacy. It has biographical and historical details and explains the influence of Turing on codebreaking, artificial intelligence, computer science, mathematics, biology, and philosophy.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/jack-copeland" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jack Copeland</a></li>
<li class="field-item odd"><a href="/author/jonathan-bowen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan Bowen</a></li>
<li class="field-item even"><a href="/author/mark-sprevak" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mark Sprevak</a></li>
<li class="field-item odd"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li>
<li class="field-item even"><a href="/author/et-al" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">et. al</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-1987-4782-6 (hbk), 978-0-1987-4783-3 (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">£ 75.00 (hbk), £ 19.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">576</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/the-turing-guide-9780198747833" title="Link to web page">https://global.oup.com/academic/product/the-turing-guide-9780198747833</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li>
<li class="field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
<li class="field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li>
<li class="field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li>
<li class="field-item even"><a href="/msc-full/03d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D10</a></li>
<li class="field-item odd"><a href="/msc-full/03b07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03B07</a></li>
<li class="field-item even"><a href="/msc-full/68-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-06</a></li>
<li class="field-item odd"><a href="/msc-full/68q05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Q05</a></li>
<li class="field-item even"><a href="/msc-full/92c15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92C15</a></li>
</ul>
</span>
Tue, 13 Mar 2018 07:33:47 +0000adhemar48322 at http://euro-math-soc.euClosing the Gap
http://euro-math-soc.eu/review/closing-gap
<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>Vicky Neale has a degree in number theory and is now lecturer at the Balliol College, University of Oxford. She has a reputation to be an excellent communicator. This also shows in this marvellous booklet in which she gives a general introduction to the advances made in the period 2013-2014 in the quest for a solution of the twin prime conjecture. But she also explains how mathematicians think and collaborate.</p>
<p>The twin prime conjecture is claiming that there are infinitely many prime numbers whose difference is 2 like 3 and 5 or 11 and 13. It is easy to explain what prime numbers are, and it is even possible for anyone to understand Euclid's proof that there are infinitely many primes. The twin prime conjecture is however still one of the long standing open unsolved problems: easy to formulate and understand but hard to solve. Several attempts and generalizations were formulated. For example it can be claimed there are infinitely many primes whose difference is an even positive integer N. The twin prime conjecture corresponds to N = 2.</p>
<p>And then, in April 2013, Yitang Zhang could prove that the latter generalization holds for N equal to 70.000.000, a major breakthrough. Within a year N was reduced to 246. Neale presents the different steps that were obtained in this reduction almost month by month as a thrilling adventurous quest.</p>
<p>Scott Morrison and Terence Tao, two mathematical bloggers quickly used Zhang's approach to reduce the N to 42.342.946. Tim Gowers, another active blogger proposed a massive collaboration and a Polymath project was set up by Tao. This Polymath platform is a totally new way of collaboration between mathematicians that Gowers had proposed back in 2009. The blog is fully in the open and anyone who wants to take part can dump some guesses or partial ideas on the website. The results are published under the author name D.H.J. Polymath and the website shows who has collaborated in the discussion. Neale spends some pages to discuss this kind of collaboration and comments on its advantages and disadvantages. The project on the twin primes was numbered Polymanth8 and it turned out to be particularly successful. The problem that had been out for so long now progressed quickly because already in June 2013, N was down to 12.006. In July they reached 4.689.</p>
<p>But while in August 2013 Tao is announces to write up the paper with the Polymath8 result, another twist of plot occurs. James Maynard posted a paper on arXiv in November 2013 in which the bound N is brought down to 700. Independently Tao announced on his blog on exactly the same day that he used the same method to obtain a similar reduction. Using the new method the old Polymath8 was renamed as Polymath8a and a new Polymath8b project was started. This resulted in April 2014 in bringing the bound down to 246. The bound can even be 6, but that requires to assume that the Elliott–Halberstam conjecture (1968) holds, which is a claim about the distribution of primes in arithmetic progression.</p>
<p>But Neale in this booklet brings more than just the account of this thrilling quest to close the gap. She also succeeds in explaining parts of the proofs and she also tells about similar related problems from number theory. For example the Goldbach conjecture: "every even number greater than 2 is the sum of the squares of two primes", or its weak version: "every odd number greater than 5 is the sum of three primes", are two famous examples. The generation of Pythagorean triples is another well known example. But there are other, maybe less known ones like Szemerédi's theorem proved in 1976, which proves as a special case a conjecture by Erdős and Turán: "the prime numbers contain arbitrary long arithmetic progressions". The Waring problem: "every integer can be written as a sum of 9 cubes, or more generally, as a sum of s kth powers, (where s depends on k), which triggered Hardy and Littlewood to count the number of ways in which this is possible. They proved the Waring conjecture by showing that there is at least one way of doing that. Neale also explains admissible sets which were used in a theorem proved by Goldston, Pintz and Yıldırım which was essential in proving and improving Zhang's bound on N. And there is some introduction to the prime number theorem and the Riemann hypothesis.</p>
<p>Neale cleverly interlaces these diversions with the progress on the twin prime problem, which has the effect that some tension is built up and new developments pop up as a surprise. Some of the notions and terminology that popped up in the other problems turn out to be related or at least to be useful in the twin prime problem.</p>
<p>Neale realizes that she is writing for a general audience and carefully explains all her concepts. However, I can imagine that some of the mathematics, like for example the formulas for the asymptotics in the Hardy-Littlewoord theorem involving a triple sum, fractional powers, complex numbers, and gamma functions will be hard to swallow for some of her readers. On the other hand, many of her "proofs" rely on visual inspection of coloured tables, and she has witty ways of explaining some concepts. For example admissible sets are presented as punched cards, a strip with a sequence of holes at integer distances, and the idea is that when this is shifted along the line of equispaced integers, then at least one (or more) primes should be visible in the punched holes. Modulo arithmetic she explains using a hexagonal pencil with the numbers 1-6 printed on its sides at the top, then 7-12 next to it etc. If you put the 6 sides of the pencil next to each other, you get a table of numbers modulo 6, and the primes in this table show certain patterns. Some of the graphics are less functional, yet very nice. On page 6 where prime and composite numbers are explained, a prime number p is represented with p dots lying on a circle, while composite numbers are represented by groups of dots arranged in doublets, triangles, squares, etc. which gives a visually pleasing effect. Other graphics are referring to a pond with frogs, grasshoppers, ducks, reed and waterlily leaves. These may be less instructive, but they are still a nice interruption.</p>
<p>Vicky Neale has accomplished a great job, not only in bringing the mathematics and the mathematicians to a broad audience. We meet some of the great mathematicians of our time like Gowers and Tao, both winners of the Fields Medal. We are informed how mathematical progress works, how new ideas are born. This can be through novel communication channels such as the Polymath, but it can still be a loner who works on a completely different approach who comes up with a breakthrough. Sometimes we can gain from results slumbering in mathematical history, but often it relies on coincidences when someone connects two seemingly unrelated results. And when the time for an idea is ripe, then it happens that two mathematicians independently from each other come up with the same result simultaneously.</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 brings an accessible account about the progress that was made in the period 2013-2014 in attempts to solve the twin prime conjecture. It also sketches the way in which mathematicians think and collaborate, for example through a new communication channel such as the Polymath projects which are online blogs promoted by Timothy Gowers and Terence Tao, two prominent mathematicians, both winners of a Fields Medal.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/vicky-neale" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Vicky Neale</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-1987-8828-7 (hbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 19.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">176</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/closing-the-gap-9780198788287" title="Link to web page">https://global.oup.com/academic/product/closing-the-gap-9780198788287</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/11-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li>
<li class="field-item odd"><a href="/msc-full/11b25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B25</a></li>
<li class="field-item even"><a href="/msc-full/11n13" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11N13</a></li>
<li class="field-item odd"><a href="/msc-full/11p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11P05</a></li>
<li class="field-item even"><a href="/msc-full/11p32" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11P32</a></li>
</ul>
</span>
Tue, 20 Feb 2018 18:22:30 +0000adhemar48284 at http://euro-math-soc.eu