Book reviews
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Book reviews published on the European Mathematical Society websiteenPower-Up: Unlocking the Hidden Mathematics in Video Games
http://euro-math-soc.eu/review/power-unlocking-hidden-mathematics-video-games
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Matthew Lane is a mathematician who maintains an interesting blog <a href="http://www.mathgoespop.com/" target="_blank">Mathematics Goes Pop!</a> where he links mathematics to popular culture, and this book is perfectly in line with that. Six years before, Keith Devlin in his book <em>Mathematics Education for a New Era: Video Games as a Medium for Learning</em> (A K Peters, 2011), already argued that video games could be a tool for (mathematical) education. Devlin, as well as Lane have some sensible ideas about how to make use of the fact that gaming is so popular among youngsters into a tool or an incentive to learn some mathematics. That this can be obtained by especially designed or pimped versions of existing games is rather obvious, but Lane claims here that any game is suitable. Just analysing the winning strategies or the way in which adventurous problems have to be solved in different games can be concrete examples of a mathematical abstraction. Even, just the creativity that goes into exploring the possibilities within the rules of the game and the endurance with which it is played may promote an attitude of trial and error within the rules of mathematics and a culture of perseverance in solving mathematical problems. Anyway, it would be a waste if the popularity of gaming would not be exploited to serve a higher purpose.</p>
<p>
In different chapters, Lane gives examples of how these ideas can be brought into practice by just relying on some popular video games that were <em>not</em> especially designed with an educational purpose in mind.</p>
<p>
The first chapter introduces several games, in which physical reality is overly simplified. Gravitation and inertia are missing and worlds may be even just two-dimensional. However in the game <em>A Slower Speed of Light</em>, as you may have guessed, the speed of light is lowered so that one moves through the landscape and it will be observed just as relativity theory predicts when you are travelling close to the speed of light. In <em>Miegakure</em> the environment is the familiar three-dimensional setting, but one can move into a fourth dimension to avoid obstacles. Moving to a fourth space dimension is not possible in reality, but there is no problem to experience it in a video game. The gamer can experience a mathematical abstraction or a physical observation that is impossible in real life.</p>
<p>
Chapter two is about guessing games like <em>Family Feud</em> where two teams have to guess the five most popular answers to some question. The popularity of these games dropped drastically after a short time because the number of questions was finite, and hence the questions keep repeating after a while. This can be the hook on which to attach some statistics and combinatorics and to design a procedure to avoid repetition as much as possible by attaching weights to questions that have already been asked. This is like interpolating between drawing balls from an urn with and without replacement, something that simply studying combinatorics mathematically does not offer.</p>
<p>
The pitfalls of voting systems is another popular, yet tricky business to analyse mathematically. This applies not only to politics but also to games in which the user has to grade some components and also to the scores and the ranking of the users themselves. The way in which the player collects his points can be very complicated, and it may not always be clear what will be the score, positive or negative, that can be earned by their actions. Inverse engineering of your final score is not at all a simple problem. But if you succeed, then it should be possible to detect impossible scores, or perhaps screen configurations revealing partial information that is not possible, given the rules of the game. Of course the latter remotely refers to the consistency of a logical system. There are two chapters devoted to this kind of problems.</p>
<p>
Chapter five is all about chasing and shooting. This is the chapter that is the most mathematical or at least the one with most formulas. As far as shooting is concerned, one may consider two kinds of missiles: those that go in a straight line and bounce off walls or the heat seeking missiles that lock in on the target and adapts its trajectory continuously. In the first case, the mathematics involves some simple trigonometry, but still the moving target complicates things, and it becomes really tricky when there are multiple reflections on walls. This is the part that has most of the formulas. The trajectories of heat seeking missiles are not piecewise linear anymore. They can in principle still hit a target that disappears behind a corner. This is a more involved issue and it is worked out to some extent in an addendum. But even a simple interception problem of an enemy missile moving on a straight line towards a target that has to be neutralised by your own missile, also moving in a straight line, is interesting to investigate. A blast with a certain radius can help you still destroying the enemy missile when the interception point is slightly missed. There are some quite interesting mathematics involved here.</p>
<p>
As we progress in the book, the mathematics and the abstraction is cranked up a bit. The next chapter is about computational complexity and the P vs NP problem. These complexity concepts are introduced by explaining Kevin Beacon numbers. This is the distance of an actor to Kevin Beacon measured in coactor-of-coactorship. It is the analogue of the Erdős number which is the co-authorship distance from Paul Erdős, which is quite popular among mathematicians (I wonder why the Erdős number is not even mentioned). Finding these numbers is a shortest path problem in a graph and that is a problem from class P, but finding the longest path or the path of a certain length between two nodes are known to be NP-complete, i.e. easy to check but difficult to solve. So are some problems related to <em>Tetris</em>. Another well known example is the travelling salesman problem. This is a problem a gamer has to solve when he has to pick up some potions, treasures or weapons at fixed places in a maze. Finding a fast algorithm for solving them will earn you instant fame and a 1 million dollar prize from the Clay Mathematical Institute. Games in the class NP are usually the more challenging and perhaps therefore the more attractive ones. There is however little hope that you will crack the P vs. NP problem by playing video games.</p>
<p>
There is a game called <em>Sims</em> which is all about getting (and keeping) friendship relations. Chapter 7 is about modelling such relations between two persons. Several models are proposed in discrete and in continuous time. The latter involves differential equations. It is not explained how to solve systems of differential equations, but solutions are plotted graphically, so that interpretations can be given. This moves seamlessly to the next chapter where nonlinear elements cause chaotic behaviour. For example when a third person competes with the second for the friendship of the first: a three-body problem. Chaotic trajectories may also result when a shell is fired that behaves like a ball on a billiard table. Even when these tables have simple geometries like squares or ovals or when there are a few obstacles inside.</p>
<p>
In a final chapter Lane reflects on how video games can help in solving pedagogical issues. He explicitly refers to Devlin's book mentioned above and to other publications and reports on experiments that have been conducted at several places.</p>
<p>
From this summary, it is clear that this is not about the mathematics of video games which would be much more involved with modelling the physics of the scenes, and the involved mathematics of computer graphics needed for rendering realistic characters. On the contrary, this is all about relatively simple mathematics and logical questions that the gamer could ask spontaneously or with a little help from his teacher. It's the mathematics hidden behind the game, the one not really explicitly visible. The game or its modes of operation can be the hook on which to hang the meaning of some abstraction or it can justify why a certain mathematical concept is useful. The mathematics itself is not really the focus of the book. Differential equations are mentioned but not their solution method for example. Lane just gives some examples of where a game can be an incentive to engage in a mathematical problem, and these problems go well beyond the cuddling mathematics of kindergarten. Lane is certainly convinced of the idea and he has a broad knowledge of the many different games, probably earned with a lot of experience. He does a good job in making his point and the ideas are not naive and they do make sense. If, as a teacher, you are game-phobic and feel like an alien in this virtual world of your students, don't be afraid of this book. Lane does a marvellous job in explaining what all these games do, or at least you are informed about what you need to know, and the book is amply illustrated. We shall not be teaching all our mathematics using games in the near future, but who knows what will happen when the ideas are elaborated further in games especially designed with an educational purpose. It is not unthinkable that they become standard ingredients in our educational toolboxes.</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 point made in this book is that with little effort video games can be used as a hook on which to hang a mathematical problem and hence they can be used for educational purposes. By directing the interest of the pupil for gaming towards questions about the rules of the game, or the winning strategies, or the models that were used in the design of the game, this can serve as an incentive to study its abstracter version or to analyse the sequence of events or to generalize the problem, hence to illustrate the usefulness and the meaning of mathematics</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/matthew-lane" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matthew Lane</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9780691161518 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 29.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">264</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10954.html" title="Link to web page">http://press.princeton.edu/titles/10954.html</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/00a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A35</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/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li>
<li class="field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
<li class="field-item even"><a href="/msc-full/97c70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97C70</a></li>
<li class="field-item odd"><a href="/msc-full/97m70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97M70</a></li>
<li class="field-item even"><a href="/msc-full/97u80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97U80</a></li>
</ul>
</span>
Mon, 19 Jun 2017 06:54:14 +0000adhemar47725 at http://euro-math-soc.euProblem-Solving Strategies in Mathematics From Common Approaches to Exemplary Strategies
http://euro-math-soc.eu/review/problem-solving-strategies-mathematics-common-approaches-exemplary-strategies-0
<div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even"> </div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="http://euro-math-soc.eu/sites/default/files/book-review/2017_ProblemSolvingStrategies_0.pdf" type="application/pdf; length=25740">2017_ProblemSolvingStrategies.pdf</a></span></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/alfred-s-posamentier-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alfred S Posamentier</a></li>
<li class="field-item odd"><a href="/author/stephen-krulik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Krulik</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/world-scientific-publishing-co-pte-ltd" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">World Scientific Publishing Co. Pte. Ltd.</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4651-63-9 </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">188</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>
Wed, 07 Jun 2017 16:48:59 +0000Eugenia47703 at http://euro-math-soc.euLarge Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics
http://euro-math-soc.eu/review/large-truncated-toeplitz-matrices-toeplitz-operators-and-related-topics
<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 volume 259 in the Bikhäuser OT Series <em>Operator Theory Advances and Applications</em>. It contains 30 contributions celebrating Albert Böttcher's 60th birthday.</p>
<p>
Albert Böttcher is a professor of mathematics at the TU Chemnitz in Germany. His main research topic is functional analysis. At his 18th he won the silver medal at the International Math Olympiad in Moscow. He studied mathematics at the TU Karl-Marx-Stadt (now TU Chemnitz) and finished a PhD in 1984 entitled <em>The finite section method for the Wiener-Hopf integral operator</em> under supervision of V.B. Dybin at Rostov-on-Don State University in Russia (this was all while the Berlin wall was still up). Since then he stayed at the TU Chemnitz. At the time of writing he has (co)authored 9 books and over 220 papers. The complete list is in the beginning of the book but one may also consult his <a href="https://www-user.tu-chemnitz.de/~aboettch/" target="_blank">website</a> where he keeps his list of publications up to date.</p>
<p>
The contributions start with reminiscences and best wishes by friends, colleagues and students of Albrecht Böttcher. Besides personal recollections, there is some discussion of his work, some photographs and reproductions of slides he used in presentations to illustrate that he is not only an excellent mathematician but also a passionate teacher and lecturer.</p>
<p>
That leaves about 700 pages of original research papers all of which relate from far or near to subjects that Böttcher has worked on. The Toeplitz operators and Toeplitz matrices of the title are indeed well represented, but there are all the other "Related Topics" which are close to his work too. About fifty renowned authors are involved.</p>
<p>
The Toeplitz operator (and hence also its spectrum) is characterized by a function, which is called its symbol. It features in a multiplication or convolution in the definition of the operator. With respect to a standard monomial basis, Toeplitz operators are represented by (infinite) Toeplitz matrices that have constant entries along diagonals. Of course the spectral and other properties of truncations of the infinite matrices to large finite ones relate to corresponding properties of Toeplitz operators, and similarly it can be related to other operators such as convolution and Wiener-Hopf operators. These matrices and operators have applications in differential and integral equations, systems and control, signal processing, and many more. Depending on the application the symbol may get an interpretation of transfer function of a system, power spectrum or autocorrelation of a signal, the kernel of an integral equation, or just a weight function in a Hilbert space. So, Toeplitz matrices and operators are also related to numerical methods for solving functional equations after discretization. Or to orthogonal polynomials (on the unit circle), which then in turn links to (trigonometric) moment problems, quadrature, and approximation theory (on the unit circle, but in a similar way also to analogs on the real line).</p>
<p>
Obviously this is not the place to discuss every paper in detail. The table of contents is available on the publisher's website and for convenience the research papers are also listed below. From the titles you will recognize the papers on determinants and eigenvalues for Toeplitz matrices, in particular their asymptotic behaviour as their size goes to infinity. Of course circulant and Hankel operators and combinations of these as operators or matrices are not far off the central theme and they are thus also treated in some of the chapters. The majority of the papers present new results. Note that most of them are (functional) analysis. Only a few exceptions are more linear algebra or make a link to physics or explicitly discuss numerical aspects (see [14, 16, 18, 23, 25, 27] below).</p>
<p>
Some of the papers are quite long (more than 30 pages and some even up to 50 pages). They are basically true research papers, sometimes a bit more expository, but they are not of the introductory broad survey type. So this is not the book you should read to be introduced to the subject, but is is more a sketch of the state-of-the-art for who is already famiiar. The style of course depends on the authors, but the book is homogeneous because of the subjects that all somehow relate to Böttcher's work. These topics discussed here are also close to the core idea of this book series <em>Operators Theory Advances and Applications</em>, founded by Israel Gohberg as a complement to the journal <em>Integral Equations and Operator Theory</em>. Only one of Böttcher's books appeared in this series though (<em>Convolution Operators and Factorization of Almost Periodic Matrix Functions </em> (2002) authored with Yu. I. Karlovich, and I. M. Spitkovsky appeared as volume 131) but several of his books are with Springer / Birkhäuser. That these topics are still a main focus of research is illustrated by the successful annual IWOTA conferences (<em>International Workshop on Operator Theory and its Applications</em>), the proceedings of which are also published in this OT series. The IWOTA 2017 is organized by A. Böttcher, D. Potts and P. Stollmann at the TU Chemnitz.</p>
<p>
Thus for anyone interested in the general topics of this book series, this collection will be a worthy addition. For those who are more selective, there is of course still the possibility to get some separate chapters, which is the advantage of having it also available as an ebook.</p>
<p>
Here are the titles and authors of the research papers in this volume:</p>
<p>
<br />
7. <em>Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices, </em> Barrera, M. (et al.), Pages 51-77<br />
8. <em>Echelon Type Canonical Forms in Upper Triangular Matrix Algebras, </em> Bart, H. (et al.), Pages 79-124<br />
9. <em>Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices, </em> Basor, E. (et al.), Pages 125-154<br />
10. <em>Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series, </em> Bini, D.A. (et al.), Pages 155-178<br />
11. <em>Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness, </em> Bogoya, J.M. (et al.), Pages 179-212<br />
12. <em>On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II, </em> Bothner, T. (et al.), Pages 213-234<br />
13. <em>Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper's Operators, </em> Bump, D. (et al.), Pages 235-265<br />
14. <em>Fast Inversion of Centrosymmetric Toeplitz-plus-Hankel Bezoutians, </em> Ehrhardt, T. (et al.), Pages 267-300<br />
15. <em>On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses, </em> Fritzsche, B. (et al.), Pages 301-352<br />
16. <em>The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications, </em> Garoni, C. (et al.), Pages 353-394<br />
17. <em>The Bézout Equation on the Right Half-plane in a Wiener Space Setting, </em> Groenewald, G.J. (et al.), Pages 395-411<br />
18. <em>On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem, </em> Junghanns, P. (et al.), Pages 413-462<br />
19. <em>The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts, </em> Karlovich, Yu.I., Pages 463-500<br />
20. <em>On the Norm of Linear Combinations of Projections and Some Characterizations of Hilbert Spaces, </em> Krupnik, N. (et al.), Pages 501-510<br />
21. <em>Pseudodifferential Operators in Weighted Hölder-Zygmund Spaces of Variable Smoothness, </em> Kryakvin, V. (et al.), Pages 511-531<br />
22. <em>Commutator Estimates Comprising the Frobenius Norm - Looking Back and Forth, </em> Lu, Zhiqin (et al.), Pages 533-559<br />
23. <em>Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary, </em> Militzer, E. (et al.), Pages 561-591<br />
24. <em>Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part IV), </em> Pietsch, A., Pages 593-619<br />
25. <em>Error Estimates for the ESPRIT Algorithm, </em> Potts, D. (et al.), Pages 621-648<br />
26. <em>The Universal Algebra Generated by a Power Partial Isometry, </em> Roch, S., Pages 649-662<br />
27. <em>Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals, </em> Seidel, M., Pages 663-680<br />
28. <em>Paired Operators in Asymmetric Space Setting, </em> Speck, F.-O., Pages 681-702<br />
29. <em>Natural Boundary for a Sum Involving Toeplitz Determinants, </em> Tracy, C.A. (et al.), Pages 703-718<br />
30. <em>A Riemann-Hilbert Approach to Filter Design, </em> Wegert, E., Pages 719-740</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 dedicated to Albrecht Böttcher's 60th birthday. The contributions are by friends, colleagues and students. After the impressive list of his publications, many of which dealing with asymptotics of Toeplitz and related operators, the book has some birthday addresses sketching Böttcher as a person and some of his work. The major part however consists of research papers written on invitation by specialists on topics related by far or near to the work of Böttcher. <br />
</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/dario-bini" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dario A. Bini</a></li>
<li class="field-item odd"><a href="/author/torsten-ehrhardt" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Torsten Ehrhardt</a></li>
<li class="field-item even"><a href="/author/alexei-yu-karlovich" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexei Yu. Karlovich</a></li>
<li class="field-item odd"><a href="/author/ilya-matvey-spitkovsky" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilya Matvey Spitkovsky</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/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</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">978-3-319-49180-6 (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">174,89 € (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">766</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/9783319491806" title="Link to web page">http://www.springer.com/gp/book/9783319491806</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/47b35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47B35</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/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li>
<li class="field-item odd"><a href="/msc-full/45e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45E10</a></li>
<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/15b05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15B05</a></li>
<li class="field-item even"><a href="/msc-full/65d15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D15</a></li>
<li class="field-item odd"><a href="/msc-full/65g50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65G50</a></li>
<li class="field-item even"><a href="/msc-full/65j10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65J10</a></li>
</ul>
</span>
Sat, 20 May 2017 12:07:31 +0000adhemar47678 at http://euro-math-soc.euHodge Theory
http://euro-math-soc.eu/review/hodge-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book provides a comprehensive introduction to Hodge theory, written by several authors. It is mainly aimed to graduate students but it can also be very useful to lecturers and researchers in Algebraic Geometry. It is based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy. In its 600 pages, it deals with the following topics:</p>
<p>Chapter 1. Kähler Manifolds, by Eduardo Cattani. It gives a standard introduction to complex manifolds, including the Hodge theory of harmonic forms for Kähler manifolds. It contains an appendix by Philip Griffiths with a proof of the Kähler identities using the symplectic structure of a Kähler manifold. </p>
<p>Chapter 2. Sheaf Cohomology, by Fouad El Zein and Loring W. Tu. It introduces sheaf cohomology making extensive use of Godement resolutions, hypercohomology, spectral sequences, and Cech cohomology. It gives a proof of the algebraic De Rham theorem and Serre's GAGA principle for projective varieties. Then it moves to the algebraic De Rham theorem in general, introducing the tool of compactifying an algebraic manifold with a divisor with normal crossings.</p>
<p>Chapter 3. Mixed Hogde Structures, by Fouad El Zein and Loring W. Tu. It gives a thorough exposition of the notion of Hodge structures and mixed Hodge structures, proving several relevant algebra results, including the use in spectral sequences and hypercohomology. The powerful machinery of mixed Hodge complexes is introduced to prove the theorem of Deligne on the construction of mixed Hodge structures on the cohomology of algebraic varieties. The case of smooth open varieties is done by using the logarithmic complex, and the case of singular complete varieties by using simplicial varieties.</p>
<p>Chapter 4. Period Domains and Period Mappings, by James Carlson. The period domain is the parametrizing space for polarised Hodge structures. For a fibration $X\to C$ with generically smooth fibers, there is a period map from $C$ to the period domain parametrizing Hodge structures of the fibers. Around the singular fibers there is a monodromy map. The cases of Hodge structures of weight $1$ and weight $2$ are addressed specifically.</p>
<p>Chapters 5 and 6. Hodge Theory of Maps, by Luca Migliorini and Mark Andrea de Cataldo. These deal with Hodge theory aspects associated to an algebraic map $f:X\to Y$ between two smooth projective varities. Results given include the triviality of the Leray spectral sequence for smooth projective maps, the theorem of the semisimplicity of the monodromy, the local and global invariant cycle theorems, the definition of the limit mixed Hodge structure for a singular fiber, and an introduction to intersection cohomology. Chapter 6 contains plenty of exercises. </p>
<p>Chapter 7. Variations of Hodge Structure, by Eduardo Cattani. It deals with the algebra associated to families of Hodge structures. For a fibration $X\to B$, the cohomology of the fibers form a local system, which is called a variation of the Hodge structure. The Kodaira-Spencer map and the Gauss-Manin connection are introduced. The situation is also studied in a general context (when the family of Hodge structures does not come from a geometric situation), and it analyses the asymptotic behaviour when we approach a singular fiber.</p>
<p>Chapter 8. Variations of Mixed Hodge Structure, by Patrick Brosnan and Fouad El Zein. This is a long and more technical chapter where the theory of the previous chapter is extended to the case of variations of mixed Hodge structures. Emphasis is put on the notion of infinitesimal mixed Hodge structure.</p>
<p>Chapter 9. Algebraic Cycles and Chow Groups, by Jacob Murre. This chapter, of more elementary nature. introduces the different Chow groups of an algebraic variety associated to algebraic cycles with different equivalences of cycles. The cycle map, intermediate Jacobians and the Deligne cohomology, are reviewed.</p>
<p>Chapter 10. Spreads and Algebraic Cycles, by Mark Green. For an algebraic variety $X$ defined abstractly over some field $k$, one takes an algebraic variety $S$ with $k={\mathbb{Q}}(S)$, so $X$ defines a family ${\mathcal{X}} \to S$ over the rational numbers, called the spread of $X$. The Hodge theoretical properties of this family give geometric information on the Chow groups of $X$, for the different embeddings $k\hookrightarrow {\mathbb{C}}$. This leads to the notion of absolute Hodge classes and the Bloch-Beilinson conjectures.</p>
<p>Chapter 11. Absolute Hodge Classes, by François Charles and Christian Schnell. Absolute Hodge classes were introduced by Deligne. They stand between the notion of Hodge class and the class of algebraic cycles. This is related to the Standard conjectures of Grothendieck and the Hodge conjecture. Absolute Hodge classes for abelian varieties are reviewed.</p>
<p>Chapter 12. Shimura Varieties, by Matt Kerr. The last chapter gives an introduction to symmetric domains, Shimura varieties, and CM-abelian varieties.</p>
<p>This is an extensive book on the topic of Hodge theory, with content ranging from basic material to very recent progress in the field. Some of the chapters are at the level of graduate students, self-contained and with a long and nice exposition. Other chapters however are addressed to a higher level mathematical audience, with a more sketchy exposition and redirecting to relevant bibliography for details. </p>
<p>Some of the chapters of the book are loosely tied to previous (more basic) content of the book. At some points, the chapters follow too closely the lectures in the way they were delivered. This gives the text some sense of incoherent recollection of topics.</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-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book provides a comprehensive introduction to Hodge theory, written by several authors. It is mainly aimed to graduate students but it can also be very useful to lecturers and researchers in Algebraic Geometry. It is based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy. In its 600 pages, it deals with the following topics: Kähler Manifolds, Sheaf Cohomology, Mixed Hogde Structures, Period Domains and Period Mappings, Hodge Theory of Maps, Variations of Hodge Structure, Variations of Mixed Hodge Structure, Algebraic Cycles and Chow Groups, Spreads and Algebraic Cycles, Absolute Hodge Classes, and Shimura Varieties.</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/edited-eduardo-cattani" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Edited by Eduardo Cattani</a></li>
<li class="field-item odd"><a href="/author/fouad-el-zein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Fouad El Zein</a></li>
<li class="field-item even"><a href="/author/philip-griffiths" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Philip A. Griffiths</a></li>
<li class="field-item odd"><a href="/author/l%C3%AA-d%C3%BCng-tr%C3%A1ng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lê Düng Tráng</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</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">9780691161341</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">$90</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">608</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://press.princeton.edu/titles/10288.html" title="Link to web page">http://press.princeton.edu/titles/10288.html</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/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li>
<li class="field-item odd"><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 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/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/14-algebraic-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14 Algebraic geometry</a></li>
</ul>
</span>
Wed, 10 May 2017 13:48:49 +0000Vicente Munoz47673 at http://euro-math-soc.euChow rings, decomposition of the diagonal, and the topology of families
http://euro-math-soc.eu/review/chow-rings-decomposition-diagonal-and-topology-families
<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 present book originates from lectures that Claire Voisin delivered on topics related to algebraic cycles on complex algebraic varieties. The volume is intended for both students and researchers, and presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures.</p>
<p>The book starts with a nice and comprehensive Introduction to the topics treated on the coming chapters. Chapter 2 gives a review of the theory of Chow groups and Chow motives, and the theory of Hodge structures and mixed Hodge structures, where the Hodge Conjecture and generalized Hodge Conjecture are introduced and put into context. In Chapter 3, one of the central objects of the book is studied, the Chow class of the diagonal of $X\times X$ for a variety $X$. A result on the decomposition of the diagonal, depending on the size of Chow groups, is reviewed. The generalized Bloch conjecture is a converse statement, saying that if the trascendental cohomology of $X$ is supported on a closed algebraic set of codimension $\geq c$, then for any $i\leq c-1$, the map $CH_i(X)_{\mathbb Q}\to H^{2n-2i}(X,{\mathbb Q})$ is injective. This conjecture is a central point for the results treated in the book. Chapter 4 is devoted to the study of Chow groups for complete intersections, and the equivalence of Bloch and Hodge conjectures for general complete intersections. In Chapter 5, the author studies the small diagonal in $X\times X \times X$, which is the appropriate object to understand the ring structure on the Chow and cohomology groups. In this regard, a very particular property satisfied by the Chow ring of K3 surfaces is shown, which leads to conjectures for hyper-Kähler manifolds. The final Chapter is devoted to analysing Chow groups with integer coefficients. It is known that the Hodge Conjecture fails for integer coefficients, a fact that it is reviewed by extracting torsion invariants associated to complex cobordism groups.</p>
<p>This is a dense and very thorough book that reports some of the exciting discoveries that Claire Voisin has made in the study of algebraic cycles. There is a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Vicente Munoz</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 reports some of the exciting discoveries that Claire Voisin has made in the study of algebraic cycles on complex algebraic varieties. The volume presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures.</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/claire-voisin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Claire Voisin</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/annals-mathematics-studies-princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">annals of mathematics studies, princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</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">9781400850532</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">$78,50</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">163</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://press.princeton.edu/titles/10289.html" title="Link to web page">http://press.princeton.edu/titles/10289.html</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>
</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/14-algebraic-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14 Algebraic geometry</a></li>
</ul>
</span>
Thu, 04 May 2017 22:32:03 +0000Vicente Munoz47664 at http://euro-math-soc.euTheories of Everything: Ideas in Profile
http://euro-math-soc.eu/review/theories-everything-ideas-profile
<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>
<em>Ideas in Profile. Small introductions to big topics</em> is a series published by Profile Books that give short introductions to important socio-cultural or scientific topics. The book under review is the only one so far on a mathematical-physical topic: the theories of everything (note the plural!). This has been a popular, yet undeniably difficult, subject in the media since the successes of Einstein's relativity and the mind boggling consequences of quantum physics originating in the previous century. Frank Close is an emeritus physics professor from Oxford University and he has quite some experience in science communication. So he is the right choice to author a book of this kind.</p>
<p>
A theory of everything is a theory that tries to explain everything within the realm of inanimate physics. It should not be speculation but a scientific theory, which means that it should be verifiable in some way by experimental observation. What is illustrated in this book is that the different theories of everything adapt to the scale at which one makes the observation, the scale of the mass, distance, or energy. With more powerful methods usually requiring higher energies, the definition of 'everything' has changed in the course of centuries. We are now even arriving at a point where 'theories' are developed that can probably never be verified by observation since that would require all the energy present in many galaxies. And that is certainly not going to happen in the near future. However, such an 'experiment' has taken place once already, namely at the time of the Big Bang. So our only hope is to rely on cosmic observations. Other theories propose multiverses, and since communication between these is impossible, one could ask whether this can still be called a scientific 'theory' in the usual sense.</p>
<p>
Newton's mechanics could explain what happens to men-size objects on earth. His theory of gravity even explains how planets move around the sun or the moon around the earth, but problems arise when more than three bodies are involved. When many particles are involved, this gives rise to thermodynamics, from which follows the notion of entropy which in turn explains the arrow of time. The electric and magnetic theory were unified in the Maxwell equations. With light as an electromagnetic phenomenon, Einstein introduced his relativity theory which linked space and time in a four-dimensional space-time universe where mass and energy are essentially the same.</p>
<p>
By joining Maxwell's theory to Dirac's quantum theory the quanta radiating at an atomic scale (1eV) can be described in accordance with general relativity. It took however quantum electrodynamics (QED) to match experiments properly. This insight was only possible after it was understood that terms in a seemingly divergent series cancelled so that it did converge indeed. It is all depending on mathematics after all. While QED describes the exchange of energy, quantum flavourdynamics (QFD) includes the exchange of electrical charge. However, when looking at a subnuclear particle scale (108−109</p>
<p>
eV = 100MeV-1GeV), we are dealing with a strong nuclear force, and then the appropriate quantum field theory is quantum chromodynamics (QCD). Like we live in an electromagnetic field, it was conjectured in the 1960's that we are also surrounded by an electroweak plasma. This is only recently proved by the detection of the Higgs boson, which is its quantum excitation. Its energy is about 125 GeV, which is just within reach for the Large Hadron Collider (LHC) in CERN.</p>
<p>
This brings us to the so called standard model, and this is where the present theories of everything are conceived. Now we are dealing with the next step up the scale, which is the Planck scale (energy: $1.25\times 10^{19}$GeV, length: $1.6\times 10^{−35}$m, time: $0.5\times 10^{−43}$sec). Both relativity theory and quantum theory have reached their limits here. In the core theory gravity does not matter because it is 40 orders of magnitude less than its electromagnetic counterpart and hence not observable. Observations with this amount of energy are not conceivable and it would imply weird situations, since black holes would be created making observations impossible and quantum theory predicts an unmeasurable space-time foam of black holes. The big challenge to combine quantum field theory and general relativity is to understand dark matter, and to know what prevents the fluctuation of the Higgs field. Possible ways out are string theory (but there turn out to be many), superstring theory (based on symmetry considerations), and multiverses (not verifiable but it would postulate the precise values of the fundamental constants just right for us to exist).</p>
<p>
In a final chapter Close hints that some answers could be found in cosmological observations and that the quantum theory, built on the Heisenberg uncertainty principle, is only an approximation. If one could apply energies well above the Planck scale, observations could be made at smaller intervals of space and time and these would decrease indefinitely as the energy keeps increasing. But this is of course speculation as most theories at this scale are for the moment.</p>
<p>
Close has done a good job, faithful to the objective of the series. No formulas and no technical details. No mathematics either, although it is clear that it is the driving force in the background of all these theories. I do not think that this is the place where you should learn what relativity theory or what quantum theory really is. When it comes to particle physics, it would be difficult to keep track of all the terminology of the different actors if you never heard of them before. Thus I think, you should not start reading this booklet unprepared. The point that Close makes quite clear is that the quest for the theory of everything is chasing a moving target. As long as one stays within a certain interval of the scale, some phenomena are perfectly negligible, and a theory of everything within that interval can be designed that matches the observations. However close to the boundary of that interval, deviations can be seen and things get mixed up like for example space and time are connected or mass and energy when the speed of light is approached. Then a new, more general theory, has to be designed that explains the phenomena on a much larger interval of the scale. Close guides the reader at a high level to the cliff where we are now standing. The cliff where gravity at a Planck scale has to be incorporated is the competing theories of relativity and quantum dynamics. And he sheds some light on what might be possible roads to a solution.</p>
<p>
For those interested in this topic, note that other physicists have published books that were written with the intention. They all explain in their own way to the interested non-specialists the evolution that has brought us from the discovery of relativity theory and quantum physics in the previous century to the current state of the art in mathematical physics. Often these emphasize the personal view of the author. Here are just a few (in alphabetical order).</p>
<ul><li>
Michio Kaku, <a href="/review/hyperspace" target="_blank"> Hyperspace</a>. A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension (1994)</li>
<li>
Roger Penrose, <a href="/review/emperors-new-mind" target="_blank"> The Emperor's New Mind</a>. Concerning Computers, Minds, and the Laws of Physics (1989)</li>
<li>
Roger Penrose. <a href="/review/fashion-faith-and-fantasy-new-physics-universe" target="_blank"> Fashion, Faith, and Fantasy in the New Physics of the Universe</a> (2016)</li>
<li>
Ian Stewart, <a href="/review/calculating-cosmos-how-mathematics-unveils-universe" target="_blank"> Calculating the Cosmos</a>. How Mathematics Unveils the Universe (2016)</li>
<li>
Max Tegmark, <a href="/review/our-mathematical-universe-my-quest-ultimate-nature-reality" target="_blank"> Our mathematical universe.</a> My quest for the ultimate nature of reality (2014)</li>
<li>
Frank Wilczek, <a href="/review/beautiful-question" target="_blank"> A Beautiful Question</a>. Finding Nature's Deep Design (2015)</li>
<li>
Anthony Zee, <a href="/review/fearful-symmetry-search-beauty-modern-physics" target="_blank"> Fearful Symmetry</a>. The Search for Beauty in Modern Physics (1986)</li>
</ul></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>
Close explains how a theory of everything has evolved since Newton detected gravity. He illustrates how such a theory is valid within a certain interval of the scale at which physics is considered. As soon as one shifts to a different scale, a new, broader theory has to be developed. He brings the reader up to the point where modern physics is facing the problem of incorporating phenomena at a Planck scale which is out of reach for observations in any foreseeable future. Physicists have therefore no indication in what direction the solution can be found and how gravitation can be incorporated in quantum physics, while not contradicting general relativity. Close gives a glimpse of possible directions in which to look for a solution.</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/frank-close" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Frank Close</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/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</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">978-1781257517 (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">£8.99 (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">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://profilebooks.com/theories-of-everything-ideas-in-profile.html" title="Link to web page">https://profilebooks.com/theories-of-everything-ideas-in-profile.html</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/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</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>
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<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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li>
</ul>
</span>
<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/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li>
<li class="field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li>
<li class="field-item even"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li>
</ul>
</span>
Thu, 20 Apr 2017 13:59:19 +0000adhemar47636 at http://euro-math-soc.euThe Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love
http://euro-math-soc.eu/review/calculus-happiness-how-mathematical-approach-life-adds-health-wealth-and-love
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The subtitle explains more clearly what the book is about: <em>How a mathematical approach to life adds up to health, wealth, and love</em>. It is thus one of these books showing to the layperson how mathematics <em>can</em> be used in everyday life (not necessarily how it <em>is</em> used in practice). Therefore the mathematics are really elementary. Unlike similar books, written with the same purpose, here the health, wealth and love take up some serious part of the pages, and give only little mathematics in return.</p>
<p>
Let's start with the health. That subject has two chapters: one on the calories you take in and burn and one about the composition of your diet. What you get to digest for mathematics is a weighted linear sum of components such as your age, weight, and height that are influencing your metabolic rate, your calorie burning, or your cholesterol ratio. A simple quadratic defines your maximal heart rate as a function of age, and the expected years of life loss as a function of waist to height ratio.</p>
<p>
The second part has the promising title <em>A mathematician's guide to manage your money</em>. This also has two chapters. One is about managing your budget and the second about financial transactions like saving and investing. The mathematics we learn here is that taxes are computed in a linear way but only within certain intervals, so that it is actually a piecewise linear function. Also we learn what a compound interest rate is (or inflation rate in this case) and this leads to Euler's constant e and consequently also to the logarithm. A glimpse at the financial markets is the occasion to introduce some statistical concepts like average and standard deviation.</p>
<p>
The 'love' part introduces a formula to compute the number of possible dating candidates, and the well known 37 percent rule which states that if you need to select the best one (for example partner among the candidates) in a sequence, then you should first register the best candidate among the first 37% of the sequence and then take the first one that is better than that one. It also describes the Gale-Shapley algorithm to solve the stable matching problem. The last chapter is mathematically the most involved one of the book and analyses the relation between two persons as a dynamical system described by two simple differential equations. Also the Nash bargaining problem is discussed in which the optimalization of the quadratic Nash product has to be found when the couple has to come to a joint decision.</p>
<p>
Most of the mathematical derivations and computations are removed from the text and are summarized in appendices and if you want to apply it to your own life, you don't even need a pocket calculator because the publisher's web page has a link to online apps that will evaluate the formulas for you when you introduce your data. Each chapter also ends with a summary of the mathematical and nonmathematical takeaways. If you are interested in one of the topics, further reading is provided. Indeed, all the equations and methods described here are abstractions and usually drastic simplifications of reality. Therefore I would also like to refer to a don't-try-this-at-home type of warning that Fernandez provides in the introduction: if you want to implement major changes in your life based on the methods presented in this book, be sure there is an expert (like for example your medical doctor) to assist you and give good advise.</p>
<p>
I doubt that the noble hope of the author, which is that by reading this book the reader will adopt a mathematical approach to life, shall be fulfilled. The mathematics are really precalculus, while the problems like composing a diet, financial investment, and finding a partner for life, do not seem like the problems one is facing at the age one is brought in contact with the required precalculus. Somehow I think that the level of the applications and the level of the mathematics do not match well. There are however still wise lessons to learn from the book which anybody (certainly journalists and politicians) should know. For example one should have the numeracy to know that doubling the price of a sandwich over 10 years, does not mean that the inflation is 10% per year. Also the mathematical techniques shown here do not only apply to the three main topics enumerated above, but they are also applicable in many other situations, like an optimal selection of a secretary or the best way to subdivide a pizza among a number of hungry children.</p>
<p>
I believe it would take a student already interested in mathematics to be sincerely attracted to reading the book. On the other hand, teachers may find inspiration in some of the examples to use these as illustrations in their teaching. Or perhaps the mathematics that are used in the book may be an inspiration for them to apply it in perhaps similar applications that are more adapted to their particular set of students.</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 another book showing the use of mathematics in everyday life. The mathematics are rather elementary and include simple functions like linear, quadratic, or cubic at most (in relation with calorie consumption, or composing a diet), the computation of interest or inflation and the logarithm as well as mean and standard deviation (in connection with managing a budget or investment) and the 37% rule for making an optimal selection in a sequence, an algorithm for the stable matching problem and the Nash bargaining problem (to solve partnership and relational problems).</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/oscar-e-fernandez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Oscar E. Fernandez</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9780691168630 (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">19.95 £ (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">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="http://press.princeton.edu/titles/10952.html" title="Link to web page">http://press.princeton.edu/titles/10952.html</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>
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<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
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<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
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</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
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Thu, 20 Apr 2017 13:35:20 +0000adhemar47635 at http://euro-math-soc.euBeyond Infinity: An Expedition to the Outer Limits of Mathematics
http://euro-math-soc.eu/review/beyond-infinity-expedition-outer-limits-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Eugenia Cheng is a professor of mathematics whose research field is higher dimensional category theory. She has made it one of her missions to counter mathphobia. Her credo is that mathematics is not the difficult part to deal with in life, but that on the contrary it is life that is difficult and mathematics helps us to make it simpler and manageable. She has tried to illustrate this by combining her love for cooking and her passion for mathematics in her previous book <em>Cakes, Custard + Category Theory</em> (reviewed <a href="/review/cakes-custard-category-theory" target="_blank">here</a>). In that book she gave attention to mathematics alright, but there were also proper recipes for cooking. The latter are interesting if you love cooking yourself and they are a springboard to make a link towards mathematics, but they do not really help to understand category theory.</p>
<p>
In the present book however she is explaining a really important mathematical concept: infinity, and it is far from being the simplest one to explain for non-mathematicians. The approach here is that she does just that. Not like in her previous book where she placed cookery next to the mathematics. Here of course Chen is still Chen and she still can't hide her love for cooking and category theory. However cooking is now only used as an anecdote or as and introduction to a chapter, just like perhaps a hiking experience, of a concert she attended, can be.</p>
<p>
So what is this book about? The first part is intended to explain what infinity really is, and it soon becomes clear that it is not as simple as saying it is larger than any number one can imagine. It cannot be a number since the usual arithmetic rules do not work as with finite numbers. And then there are the paradoxes like the well known Hilbert hotel with infinitely many rooms that can always accommodate infinitely many more guests, even when it is fully booked. So Chen uses a more systematic approach introducing the simplest number systems: natural numbers, integers, and rationals. She goes even further and defines the natural numbers in the set theoretic style of Frege, only she does not use the abstract concept of a set, but she uses 'bags' instead. So 0 corresponds to the empty bag, 1 to the bag containing only the empty bag, 2 to the bag containing the two previous bags, etc. Also concepts like injection, surjection, and countable are introduced here.</p>
<p>
Then a stumble stone is blocking the development. It turns out that there are more than countably many real numbers. The reals are not properly defined yet, but using Cantor's diagonal argument, and using a binary representation, Chen shows that there are more irrational numbers than natural numbers. Thus there are gradations of infinity, at which point $\aleph_0$ is introduced. The 'smallest' infinity of a countable set, but there exist higher forms like $\aleph_1=2^{\aleph_0}$ the number of reals, and this can be iterated $\aleph_k=2^{\aleph_{k-1}}$. The continuum hypothesis is briefly touched upon, and it is noted that it can't be proved (Cohen) or disproved (Gödel). The distinction between ordinal and cardinal numbers clarifies the difficulty that infinity gives with the usual arithmetic operations.</p>
<p>
All this work in the first part of the book, leading to an understanding of what infinity actually is, is like a journey uphill. In a second part Chen points to the sights that are possible from the top of the hill. With the recursive definition of the natural numbers, a proof by induction is within reach and one can solve all sorts of counting problems and even evaluate infinite sums. Although the latter needs more careful consideration. She also introduces higher dimensions, i.e., larger than 2 or 3. It may even grow to infinity for a continuum. When a relation or a property is associated with a dimension, this brings her to her beloved research subject: higher dimensional category theory. Perhaps, this doesn't fit so well in the otherwise rather elementary exposition, but it is a nice, be it a somewhat unusual example, of a higher dimensional mathematical object.</p>
<p>
The move is then from the infinitely large to the infinitely small, leading back to infinite sums of diminishing terms and Zeno paradoxes. What is needed here is the concept of a limit. She however explains it essentially avoiding to use that name. Instead she illustrates the idea with hitting a target that becomes smaller and smaller. This way it can be explained what infinitesimals are and how they are applied. It can now be proved that the harmonic series diverges, and eventually also that irrational numbers do exist, which is done by approaching Dedekind's definition of the reals.</p>
<p>
I find this a very pleasing way of introducing some elementary, but also some less elementary, mathematical concepts to the layperson. Taking infinity as the carrot to lead the reader uphill is an interesting choice. This is the most essential concept needed when moving from algebra to analysis. Chen is an excellent guide to show the reader the way uphill. With many analogies and illustrations and reformulations it seems like the reader is carried to the top, no toiling required. The story is told fluently. Side remarks, historical notes, or a slightly more advanced remark are inserted as a framed boxes in the text. I guess it will be too elementary for mathematicians of mathematics students, but it is warmly recommended for secondary school pupils. In fact anyone who has the slightest interest in what infinity actually means should read it. The word is used lightly in common language, but you will learn what it means in a more exact sense and thus what it means to a mathematician. It turns out that it triggered the development of calculus and it has shaken the foundations of mathematics as recently as in the early 20th century.</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>
Eugenia Cheng continues her crusade against mathphobia. In this book she explores the meaning of infinity. To properly define infinity she has to define the cardinality of the natural numbers, and thus also the definition of the latter. That includes solving the inconsistencies with arithmetic operations and the paradoxes that result. However it turns out that the real numbers are not countable, so that there are gradations of infinity and hence also the definition of the reals is needed. That requires to consider the infinitely small, which leads to infinitesimals that form the onset of calculus. All is brought to the reader avoiding the usually boring technical approach of mathematics, but using many analogies and elementary everyday language.</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/eugenia-cheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eugenia Cheng</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/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</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">978-1781252857 (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 12.99 (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">316</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://profilebooks.com/beyond-infinity.html" title="Link to web page">https://profilebooks.com/beyond-infinity.html</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>
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<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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Thu, 20 Apr 2017 13:12:50 +0000adhemar47634 at http://euro-math-soc.euElements of Hilbert Spaces and Operator Theory
http://euro-math-soc.eu/review/elements-hilbert-spaces-and-operator-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Books on functional analysis and operator theory appear regularly, but they are often dealing with a special topic such as differential operators, integral equations, spectral theory, or particular classes of operators. Not so remarkable since the topic is relatively new and mainly developed during the previous century by some of the best: Hilbert, Banach, Riesz, von Neumann, etc. The subject has grown very rapidly. Many of the general introductions therefore date back to the twentieth century as well. Only few are published after 2000. Given that the subject is still growing and has become a standard tool in theory and applications from analytic number theory to dynamical systems, a new modern introduction such as this one is very welcome.</p>
<p>
In its most elementary form engineers only need finite dimensional vector spaces, and if we restrict to linear operators, there is linear algebra to solve all the problems. However these problems are in many cases approximations for a phenomenon taking place in an infinite dimensional space, and when infinity is involved the mathematics become tricky and one needs good foundations to develop the proper theorems. Of course the subject is immense as is illustrated by the classic 3 volume set <em>Linear Operators</em> (1958-1971) by N. Dunford and J. Schwartz that contains over 2500 pages. So, it is a difficult balance to be kept between an encyclopedic work and an introduction that is both complete in the theoretical foundations and yet has attention for the applicability. This requires a lot of craftsmanship that is achieved remarkably well in this book.</p>
<p>
The author has made a selection in the massive amount of material available to provide a very readable introduction to the subject. To study the operators, one first needs to understand the spaces on which they operate. These are in practice mostly Hilbert spaces and Banach spaces. Then the linear operators can be defined and their spectral analysis can be studied. This defines the structure of the book shaped by the 5 chapters: a short general introduction recalling the necessary preliminaries, then inner product spaces, the linear operators, their spectral theory, and finally the Banach spaces. There is an extra chapter with hints and solutions to the many exercises that are amply sprinkled throughout the text.</p>
<p>
The emphasis is definitely on linear operators on Hilbert spaces and the spectral analysis of special classes of operators. So a first target is to introduce inner product spaces, that is the spaces of $L^2$ type, so that one can talk about orthogonality of a basis, define projections and discuss approximations. The most important operators are the normal, unitary, and isometric operators. The special classes for which the spectral analysis is studied in more detail include compact operators, trace class, self adjoint, and Hilbert-Schmidt operators. In the Banach space chapter, topics include the Hahn-Banach theorem, the Baire category theorem, and the open mapping and closed graph theorems. There is also a section on unbounded operators and at some point also invariant subspaces are discussed.</p>
<p>
The style is the typical mathematical approach of definition-theorem-proof kind of sequencing. However this is made lightly digestible by including many examples, remarks, and illustrations of what these formal definitions or theorems mean in practice and what the applications can be. Thus, although this is a quite mathematical subject, I think also engineers of a more theoretical kind will certainly appreciate this book very much. Among the applications we can mention Fourier analysis, orthogonal polynomials, approximation and convergence, Müntz theorem, Browder fixed point theorem, the mean ergodic theorem, numerical range, and much more. All the proofs are fully worked out. Only few theorems are mentioned without a proof if it is really too long and complicated and thus beyond the scope of these notes. Of course such a book cannot be read without an appropriate preparation which should include analysis and linear algebra. Most sections are followed by a set of exercises. These include often applications and examples, or ask to prove some extra properties. The level of difficulty nicely matches the level of the text. To assimilate the material, one should solve at least some of these exercises. As mentioned above, some 100 pages with hints and solutions are summarized in the last chapter.</p>
<p>
To conclude, I think this is a marvelous introduction to the topic. Certainly applied mathematicians and engineers who need a stronger mathematical background, or for mathematicians with an interest in applications, will appreciate this most. Obviously it can be used, or at least parts of it, as a perfect set of lecture notes for a course on the subject. Note however, that it is a general introduction. Let me give two examples of what it is <em>not</em>. It does <em>not</em> discuss in any detail the solution of differential or integral equations (Sobolev spaces are too specialized and out of the picture). Neither is there a direct link to the extensive literature on systems theory. The books of the Birkhäuser series on <em>Operator Theory Advances and Applications</em> for example are much more specialized. However this book introduces the preliminaries to engage in all these topics, just because it is paying special attention to the operators that are most common in applications such as these.</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 an excellent introduction to linear operators on Hilbert and Banach spaces. Definitions and properties are introduced. In particular the spectral theory for particular classes of operators is discussed. Special attention is given to the applicability with many examples and illustrating the theory with several applications. Many exercises are provided with an extensive chapter giving hints and solutions at the end of the book. The book is perfectly fit to be used as a basis for a course on the topic.</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/harkrishan-lal-vasudeva" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Harkrishan Lal Vasudeva</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">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">978-981-10-3019-2 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">116,59 € (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">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/9789811030192" title="Link to web page">http://www.springer.com/gp/book/9789811030192</a></div></div></div>
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<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>
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<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>
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Thu, 20 Apr 2017 12:58:13 +0000adhemar47633 at http://euro-math-soc.euDr. Euler's Fabulous Formula
http://euro-math-soc.eu/review/dr-eulers-fabulous-formula
<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>
Nahin published this book originally in 2006. This copy is a reprint in the <em>Princeton Science Library</em> of the revised paperback edition of 2010. It is a sequel to the author's <em>An Imaginary Tale: The Story of √-1</em> (see <a href="/review/imaginary-tale-story-√-1" target="_blank">here</a> for a review). The series brings reprints in cheap paperback and eBook format of classics and bestsellers and makes them available for a new generation of a potential readership. It covers a broad spectrum of science books and among them many are about mathematics.</p>
<p>
Euler's fabulous formula of the title is of course the extraordinary, and (think of it) amazing formula $e^{i\pi}+1=0$. Thus the square root of $-1$ is not far off. Complex numbers and complex functions being already introduces in his previous book, Nahin can concentrate on specific applications of this Euler formula. Although in principle all the material covered can be read and understood by mathematics or engineering students at an advanced undergraduate level and the material is covered in a leisurely almost pleasant discourse, there is a lot of serious mathematics that is covered. A good mathematical background is necessary. Big chunks of a (complex) analysis course are covered. The difference with a classical course is that in lecture notes one has a strict target list of concepts, properties and theorems that have to be covered. The reader is then guided through all these topics in most efficient and matter-of-fact kind of way. Here the same landscape is explored but there is no urgent target. The guide is the Euler formula and the reader is leisurely exploring some topics that are from far or near related to it without a strict travel plan or compelling arrival time.</p>
<p>
Because the Euler formula is known as the most beautiful formula in mathematics, there is an introductory support act contemplating what it means when a mathematical formula or a proof is generally accepted to be "beautiful". During the main mathematical dish, of the subsequent chapters, often some historical background is given, if not in the text, then it is in the extensive list of notes at the end of the book. As a dessert we can read a biography of Euler in the last (unnumbered) chapter.</p>
<p>
So what is then the main dish? There are five chapters. The first is about complex numbers, but it goes beyond the first elementary steps that were already in <em>An Imaginary Tale</em>. By interpreting multiplication with a unimodular complex number as a rotation that can be represented by a matrix multiplication, a Cayley-Hamilton theorem can be found. Furthermore formulas of De Moivre, Cauchy-Schwartz, infinite series, the construction of n-gons, and its relation with Fermat's last theorem, and Dirichlet's integral of the sinc function. The same mixture of exploring and digressing is maintained in the other chapters. The next chapter is called vector trips. The interpretation of complex numbers in the plane allows a geometric interpretation of summing power series, leading eventually to the solution of some differential equations. Another chapter proves the irrationality of $\pi^2$, and a thicker one introduces Fourier series. The idea already lingered in Euler's time where Euler, D'Alembert, and Daniel Bernoulli were contemplating the solution of the wave equation. The wiggles appearing in Fourier approximations near discontinuities is well known and nowadays identified as the Gibbs phenomenon. It is noteworthy that Nahin, as for most topics discussed in this book, gives the historical background of this phenomenon and discloses that it was actually discussed in a 1848 paper published fifty years before Gibbs by a forgotten Englishman Henry Wilbraham. An equally thick chapter is devoted to the Fourier integral and the continuous Fourier transform, including the Dirac delta function, the Poisson summation formula, the uncertainty principle, autocorrelation and convolution. The closing section here is about a difficult integral discussed by Arthur Schuster (1851-1934) in connection with optics. Hardy got interested and evaluated the integral, which is another instance where Hardy helped solving an applied problem, something he rejected in his <em>A Mathematician's Apology</em>. The final chapter is about applications in electronics: signal processing, linear time invariant systems, filters, and more.</p>
<p>
It is clear from the interpretation, the wording, and the examples that Nahin's background is in electrical engineering. Not that this is diminishing the value of his treatment of all the mathematics in this book. There is however a bias. It is also a typical engineering hands-on attitude to check the validity of some formulas with a numerical simulation, even if they were mathematically proved already. The title of Euler's biography in the trailing chapter is <em>Euler: The Man and the Physicist</em>. Despite Hardy's attitude towards applied mathematics, one has to admit that historically mathematics has developed also, and probably mainly so, because of the applications. In this sense, the book stays close to the spirit of Euler's approach to mathematics who made no proper distinction between pure and applied mathematics, and therefore the whole book is also a tribute to Euler.</p>
<p>
Let me give a quote from chapter 3 to illustrate the way Nahin tells his story. "Thus we have at last [some integral expression for $R(\pi i)$]. The reason I say <em>at last</em> is that we are not going to evaluate the integral. You probably have two reactions to this —first, relief (it is a pretty scary-looking thing) and, second shock (why did we go through all the work needed to derive it?). In fact, all we need for our proof that $\pi^2$ is irrational are the following two observations about $R(\pi i)$." But don't be mistaken, there is a lot of serious mathematics and formulas. If this book falls under "popular mathematics", it can only be popular for the readers literate in at least some more than elementary calculus. Many classical mathematical issues are discussed, but often using an original approach. This makes it also a recommendable read for professional mathematicians</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a paperback reprint in the new <em>Princeton Science Library</em> of the bestselling original from 2006. It is a sequel to the author's <em>An Imaginary Tale: The Story of √-1</em>. More complex analysis is presented in a pleasantly entertaining way. That includes the geometric interpretation of complex numbers, differential equations, the irrationality of <em>$\pi^2$</em>, Fourier analysis, and signal processing, which can be considered a tribute to Euler and his approach to mathematics. Besides all the mathematics, readable with an (advanced) undergraduate level of mathematics, there is also a discussion about beauty in mathematics and the book concludes with a biography of Euler.</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/paul-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paul Nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9780691175911 (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">18.95 £ (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">416</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://press.princeton.edu/titles/9438.html" title="Link to web page">http://press.princeton.edu/titles/9438.html</a></div></div></div>
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Mon, 10 Apr 2017 07:36:06 +0000adhemar47613 at http://euro-math-soc.eu