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Book reviews published on the European Mathematical Society websiteenMulti-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>
<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/52b10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52B10</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/52b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52B15</a></li>
<li class="field-item odd"><a href="/msc-full/92e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92E10</a></li>
<li class="field-item even"><a href="/msc-full/68r10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68R10</a></li>
</ul>
</span>
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>
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<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
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<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>
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<li class="field-item even"><a href="/msc-full/44a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A60</a></li>
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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.euA Concise Guide to Communication in Science and Engineering
http://euro-math-soc.eu/review/concise-guide-communication-science-and-engineering
<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>Contrary to what the title suggests, this book is not so concise since it discusses all the aspects of what is needed for successful communication in STEM disciplines. It is focussing on all kinds of traditional communication (papers, posters, lectures, reports, theses) but no books, blogs or vlogs. A remarkable feature is that practically everything that is discussed is illustrated with examples from real publications. These are inserted in a box with grey background and it is carefully indicated where the problem is or which lines or paragraphs illustrate how it is properly done. The good and the bad examples are easily distinguishable because the bad examples are marked with a fat <b>X</b>. Of course, indicating the source of all the examples requires a long list of references (737), but fortunately the reader does not have to look up all of them. The references about the issues on communication are conveniently separated from the others, but this list still contains 391 references to papers, books and websites, but again these are the sources used and one is not required to look them all up.</p>
<p>There are obviously general guidelines of how to organize any written or oral communication. For example, the use of proper English is one of them, and this chapter takes a substantial number of pages in this book. The author points to common errors, subtle differences in meaning (for example 'which' and 'that'), punctuation, gender neutral formulations (he or she), the use of 'I' or 'we', etc. But there are specific chapters for each of the forms of communication that were listed above, be it written or oral.</p>
<p>Extra chapters are devoted to mathematics, to statistics, and to graphs. The chapter on mathematics is relatively short. The guidelines are to take care of a logical structure, motivate and give insight, do not use a symbol or a notion before it has been defined, and on the technical side, use proper fonts, be consistent in notation, don't start a sentence with a symbol, and take care of punctuation. All of these are very recognizable errors.</p>
<p>Somewhat surprising to me was the chapter on data description and statistical inference. This is really a technical chapter about means, medians, deviation, quartiles, confidence intervals, data fitting, regression, trends, data smoothing, and statistical significance. Even (simple) matlab code is included in the notes at the end of the book to compute some of these. Perhaps this is not such a bad idea because it happens that statistical arguments are misused in some publications, even though it should be expected that authors in science and engineering should be trained in this subject.</p>
<p>Graphics form often a weak point of a publication. A proper caption should explain what is plotted, the axes should be labelled, have tick marks and units should be mentioned. The graphs should not be overloaded with too many different curves or data. If colours are used, a colour scale should be added, and perhaps confidence intervals are appropriate. Of course resolution needs to be high enough for publication, but that is usually required by the published, and it does not get published if the quality is not good enough.</p>
<p>More guidelines are provided to help you through a refereeing an publishing process once a paper is submitted to a journal. The role of impact factors and other ranking systems, and how you can promote your paper.</p>
<p>A separate chapter is devoted to ethical issues. What authors to list and in what order? What sanctions can be the consequence of plagiarism, or even self-plagiarism? Here, like in the rest of the book, the text refers to what is generally considered to be ethical or that the community considers to be the right way to behave. It is not the opinion of the author, nor is the text imperative on all topics.</p>
<p>Each chapter can easily be read separately. It is clearly spelled out in the introduction what is discussed and it always ends with a checklist of items to be verified before the work is finished. A PhD student who gives his first conference lecture or prepare his first poster may want to check the appropriate chapter. He or she will undoubtedly make errors but learn from them by experience. Similarly the supervisor will comment of first drafts of their paper and refer to the corresponding chapter or chapters. I do not think that giving a formal course on the topic is very helpful, if not experienced by the student.</p>
<p>Of course this is not the only book written on this topic. The list of the first 391 references in the book contains many examples, among which the text by Halmos and the book by Higham that I also mentioned at the end of my review of <a target="_blank" href="/review/how-write-and-publish-scientific-paper-8th-ed">How to Write and Publish a Scientific Paper</a> (B. Gastel and R.A. Day, 2017); the texts by Krantz and Tao given there are also not include in this book. Nevertheless, this book is a great asset for any PhD student or a fresh researcher in one of the STEM disciplines, and there should be a book like this on the (virtual) shelf of the library.</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 is a collection of guidelines for communication in STEM disciplines. A poster presentation, a lecture, a report, a journal paper, or a dissertation. Besides general principles and specific guidelines for each of these, the reader is also enlightened about the publication process and general ethical issues. Everything is amply illustrated by good as well as by bad examples, all citations from actually published papers.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-h-foster" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David H. Foster</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">978-0-1987-0424-9 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 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">408</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/a-concise-guide-to-communication-in-science-and-engineering-9780198704249" title="Link to web page">https://global.oup.com/academic/product/a-concise-guide-to-communication-in-science-and-engineering-9780198704249</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/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</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/97a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A10</a></li>
<li class="field-item odd"><a href="/msc-full/97u99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97U99</a></li>
</ul>
</span>
Tue, 20 Feb 2018 17:54:13 +0000adhemar48283 at http://euro-math-soc.euMathematics Rebooted
http://euro-math-soc.eu/review/mathematics-rebooted
<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>Lara Alcock has a degree in mathematics, but her research is about mathematical education and mathematical thinking. She has written two books on these subjects already and won several awards for teaching mathematics. The present book is a bit difficult to classify. It is not a book about her research, neither is it a textbook in mathematics. It is also not a popular math book, or is it? That depends on how you define a popular book about mathematics. It is not a collection of puns, puzzles, paradoxes, and all these other topics that are usually found in such books. On the other hand, you could call it a popularizing book about mathematics because it does not require much prior mathematical education to read. In her introduction she describes the readership as those people who have some affection for mathematics, but who lost track at some point in the past. It could for example be a help for teachers or parents who have to help or teach children and who themselves are missing some mathematical way of thinking. Some people, usually at a later stage of life, start to learn a foreign language, or read history books, or biographies, etc. Why should they not learn some mathematics? If they want to pump up their literacy, they could as well improve on numeracy or polish their knowledge of mathematics.</p>
<p>So what does the book contain? It is some kind of a textbook, but it is a freewheeling kind, not restricted by any kind of prescribed rules of what should or should not be included. There are no delimiting containers of algebra, calculus, or geometry. A simple idea brings along another idea, which leads to a different topic etc. Neither is it just entertainment. There are claims, even some sporadic theorems, and there are proofs (sometimes graphical or geometrical to make them more intuitive).</p>
<p>The book has five chapters, which are five threads of ideas. The first chapter is called Multiplying. It starts with the simple idea that by visualizing n rows with m elements makes multiplication easy and immediately proves the commutativity and distributivity. Also identities like $(a+b)^2=a^2+2ab+b^2$ are easily verified geometrically. The area of a triangle and of course also the Pythagoras theorem have geometric proofs as well. But then there is a trail from the Pythagoras theorem to Pythagorean triples and this in turn leads to the last theorem of Fermat. Along the way, suggestions are made for exercises to be elaborated further (Alcock refrains from formulating them as formal exercises, she just suggests to think a bit longer about some problem).</p>
<p>The other chapters are somewhat similar in nature. In the second chapter, entitled Shapes, the starting point are tessellations of the plane. Again by a geometric proof, it follows that the sum of the angles in a triangle is 180°, and by subdividing a polygon in triangles, the formula can be generalized to polygons. But considerations of regular and semi-regular tessellations also lead to symmetries and Penrose tiling. An interesting remark here is that Alcock also stresses the fact that, even with all the formulas included, the mathematics read as sentences. Thus that one is reading mathematics, just like one would be reading some other formula-free text. Another lesson learned is that mathematicians are often interested in generalizing some result, more than just applying it.</p>
<p>Chapter three is called Adding up. The goal is to arrive at infinite sums, but the starting point is adding fractions. But once more there are easy visual ways to show how to add a finite number of integers. This gives classical formulas for the sum of the first n integers, or the odd or the even ones. This is also the place to introduce proofs by induction. Furthermore she tackles convergence and divergence of series, with the geometric and the harmonic series as a particularly interesting cases.</p>
<p>The chapter on Graphs is about plotting functions in a coordinate system. The motivation here is a word formulation of an optimization problem with several (linear) inequality constraints. Plotting the constraints shows easily where the target function will be optimal. Further explorations lead to circles and polar coordinates, but curiously enough the sine and cosine do not appear because only Cartesian equations are used. A glimpse is given at these topics in three dimensions.</p>
<p>The title of the last chapter is Dividing. It starts with explaining our positional number system. This can lead to rules about divisibility like a number is divisible by 3 if and only if the sum of its digits is divisible by 3. When rational numbers are represented as ratios of integers, the prime factorization is needed to simplify them. A bit more advanced is a discussion of rational and irrational numbers and issues of countability.</p>
<p>In some concluding remarks Alcock reflects on what has been learned by reading this book (she has already summarized the main points after each chapter). She also gives suggestions for further reading, many of them are the more mathematical popularizing books, but also some books inspired by research on and experience in mathematical education.</p>
<p>Anybody interested in mathematics (those who want to learn and those who want to teach) will benefit from reading this book, but in my opinion it is particularly of interest for beginning teachers in secondary schools, especially when they were not explicitly trained in teaching mathematics. Parents too who want to follow up their children can be classified in this category. Furthermore there are those who are sincerely interested in jacking up their long forgotten mathematical knowledge.</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>With her experience in teaching mathematics and her research in mathematical education, Lara Alcock has composed this book discussing some simple mathematics for the layperson or a fresh university student. As she is not constrained by any curricular prescription, she is freewheeling by association through the mathematical topics. The content pays special attention to those issues that she has experienced as stumbling stones for students.</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/lara-alcock" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lara Alcock</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">978-0-198-80379-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 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">256</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/mathematics-rebooted-9780198803799" title="Link to web page">https://global.oup.com/academic/product/mathematics-rebooted-9780198803799</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/97a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A99</a></li>
<li class="field-item odd"><a href="/msc-full/97d70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D70</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>
Tue, 20 Feb 2018 17:54:03 +0000adhemar48282 at http://euro-math-soc.euA Friendly Approach To Functional Analysis
http://euro-math-soc.eu/review/friendly-approach-functional-analysis
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Many books were written on this topic already. The current text can be called friendly because when the author gives a definition, he usually connects it to what the reader is supposed to know. These prerequisites are basically not much more than finite dimensional vector spaces. Since this is what functional analysis is about: calculus lifted to the level of infinite dimensional vector spaces. A lot of effort goes into pointing out the similarities and the differences. Since also Lebesgue integrals are used, an appendix introduces this concept in the one dimensional case.</p>
<p>But friendly does not mean that the reader is spared from effort. The text is peppered with many (197) exercises. The formulation of the exercises and the solutions provided at the end form an essential part of the book and they actually fill the larger part of its pages. Thus the reader who is new to the topic is supposed to work hard to properly grasp all the ideas. By including the solutions, the book is suitable for self study, although the text grew out of lecture notes used by the author while teaching at the London School of Economics.</p>
<p>On the other hand, the subject is vast, and since this is only an introduction not all the most difficult proofs are provided in full extent and not all the most complicated issues are discussed in all detail. It is important to mentioned that, even though the subject is abstract, attention is also paid to the application aspect. In fact the introduction starts with an optimal control problem as a motivation to embark on functional analysis. In a later chapter when differentiation and its application to optimality conditions is discussed, the Euler-Lagrange equation is derived and it is applied to classical Hamiltonian and Poissonian mechanics. This forms also the basis for a discussion of quantum mechanics in the chapter on Hilbert spaces. Compact operators are a reason to go into the subject of finite dimensional Galerkin approximations of the operators, which is important for numerical computations. Weak solutions of differential equations are obtained using distributions, discussed in the last chapter. This means that also physics and engineering students will appreciate this approach.</p>
<p>The text is rather dense and to the point and covers an enormous variety of topics, even though the main text has only six chapters packed on 258 pages. Here is a superficial sketch of the contents. The first chapter makes the step from vector spaces to normed spaces and to Banach spaces. The next obvious things to tackle in any calculus course are continuity and differentiation, which are also here the subjects of chapters 2 and 3. These include some operator theory with the open mapping theorem, some spectral theory and even a proof of the Hahn-Banach theorem. The application of the Fréchet derivative in optimization and the application in mechanics was mentioned above. The chapter on inner product and Hilbert spaces may be the easiest ones to deal with since these spaces behave mostly like finite dimensional vector spaces when separable, and they are frequently used in all kinds of applications: approximation, Gram-Schmidt orthogonalization, generalized Fourier series, problems involving self-adjoint operators, etc. The next chapter shows that to study operators, things become easier when we restrict ourselves to compact operators. The final chapter on distributions (to stress that it is only an introduction to the subject, its title is A glimpse of distribution theory) has as one of its applications the weak solutions of differential equations but also allows to extend Fourier analysis. A few of the topics that are considered to be more advanced (like the open mapping theorem, the dual space and the Hahn-Banach theorem, and the spectral theory of compact self-adjoint operators) are marked by a star, so that they can be skipped it needed.</p>
<p>This is a nice and modern introduction to the topic. The impatient or more advanced reader can just read the text, skipping the exercises (perhaps skip only their solutions, not all their statements since they often are formulations of additional properties to be used in further proofs), and so get a quick idea. Where possible, examples clarify some basics and some subtleties of the definition or the property. If, on the other hand, the reader is a student who wants to become proficient in the subject, then solving the exercises, or an many as possible, is an excellent way to acquire the necessary skills. The flexible possibilities provided —the text can be used as lecture notes for a course, or as a tool for self study, and even as a handbook to look up some definitions or theorems— is another of its great advantages.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a modern introduction to functional analysis for mathematics or engineering students who are familiar with finite dimensional vector spaces. There are many exercises that form an integral part of the text. Topics discussed are Banach spaces, continuity and differentiation, Hilbert spaces, compact operators, and distributions. Applications include classical and quantum mechanics and optimization problems. The book can be used for self-study, as a guide for a course, and even as a reference work.</p>
</div></div></div></div>
<span class="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/amol-sasane" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amol Sasane</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">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-78634-333-8 (hbk), 978-1-78634-334-5 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 98.00 (hbk), GBP 56.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">396</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/Q0096" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/Q0096</a></div></div></div>
<span class="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/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/46-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46-01</a></li>
</ul>
</span>
Tue, 20 Feb 2018 14:08:08 +0000adhemar48277 at http://euro-math-soc.euA singular mathematical promenade
http://euro-math-soc.eu/review/singular-mathematical-promenade
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is a small jewel for a mathematical library. As the author defines it, it is a promenade into the mathematical world. With the excuse of a simple mathematical question, the author enters into many realms of mathematics, some of them almost recreative, other bits are truly deep mathematics concepts, all of that flavoured with historical review of many developments in topology, geometry, algebra and analysis. </p>
<p>First the question proposed by the author: if one has a collection of real polynomials $P_1(x),\ldots, P_k(x)$, that are ordered in some way for small $x>0$, and in other way for small $x<0$, how this ordering may change? This is rewritten as how the branches of real algebraic curves are cyclically ordered at a point. Surprisingly, there is a neat characterization (not any order is allowed), and one may count how many posibilities are, with nice combinatorics popping up. The book explores very different questions related to this problem, and follows on different ramifications. Questions in combinatorics, algebra, number theory, geometry and topology, algebraic topology, algebraic geometry, complex variables, analysis, come to the surface at every moment. The book is at no point shallow in the content: there are very deep mathematics told at a fairly easy level. A professional mathematician can (and will) learn a lot from this text. On the other hand, the book does not contain hard edge-cutting research with new theorems, although the last chapters have a more modern flavour directing to the proof of the main result on the order of the branches of a real analytic singularity. The divulgative nature of the book does not mean that one will encounter the typical stories for spreading maths to the general public. Certainly, the book is addressed to a person with knowledge at the level of undergraduate student in mathematics, increasing slowly and steadily where the last part contains results at the graduate level and beyond.</p>
<p>One very strong aspect of the book is the review of historical matters. The author has done good work in accounting on the history of some very classical notions (just to name a few, the fundamental theorem of algebra, or the theory of Puisseux series, the linking number of knots, but there are very many), and he has made a very loable effort (and very sucessful indeed!) to transmitting this in a friendly, amusing and rigorous manner. Notions of combinatorics, discrete mathematics (like the analysis of trees), algebra (operads), algebraic geometry (resolution of curve singularities, complex singularities), are explained in an enlightening way, changing mathematical rigour for clarifying drawings in a wise choice.</p>
<p>The layout of the book is also impressive: it contains many nice photographs, pictures and drawings, also scans of manuscripts, historial bibliographic references, references of papers and books on mathematical concepts, clarifying remarks, all distributed in a column at the right of the page (so that one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our "classics" (Newton, Gauss, Hipparchus, are among hundreds of other names mentioned along the text). </p>
<p>One last word. The book is available for free at the arxiv: <a href="https://arxiv.org/ftp/arxiv/papers/1612/1612.06373.pdf">https://arxiv.org/ftp/arxiv/papers/1612/1612.06373.pdf</a><br />
It is published with a price of 27 euros. Just let me say that, after reading the copy of my library for writing this review, I will buy a personal copy for myself. I really enjoyed it!</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Vicente Munoz</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is a small jewel for a mathematical library. As the author defines it, it is a promenade into the mathematical world. With the excuse of a simple mathematical question, the author enters into many realms of mathematics, some of them almost recreative, other bits are truly deep mathematics concepts, all of that flavoured with historical review of many developments in topology, geometry, algebra and analysis.</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/%C3%A9tienne-ghys" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Étienne Ghys</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/ens-editions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ENS Editions</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">978-2-84788-939-0</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">27 euros</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">302</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.ens-lyon.fr/editions/catalogue" title="Link to web page">http://www.ens-lyon.fr/editions/catalogue</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/algebraic-and-complex-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebraic and Complex Geometry</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li>
</ul>
</span>
Wed, 14 Feb 2018 12:44:28 +0000Vicente Munoz48259 at http://euro-math-soc.eu