Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenTheories 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>
</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/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">
<ul class="field-items">
<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>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
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>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/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>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/46-functional-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46 Functional analysis</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/46-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46-02</a></li>
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<li class="field-item even"><a href="/msc-full/46-cxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46-Cxx</a></li>
<li class="field-item odd"><a href="/msc-full/47axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47Axx</a></li>
</ul>
</span>
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>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
<li class="field-item odd"><a href="/imu/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/30-functions-complex-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30 Functions of a complex variable</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/30-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30-01</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/01a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A99</a></li>
<li class="field-item odd"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
<li class="field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
</ul>
</span>
Mon, 10 Apr 2017 07:36:06 +0000adhemar47613 at http://euro-math-soc.euFinding Fibonacci
http://euro-math-soc.eu/review/finding-fibonacci
<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 name of Fibonacci is connected with the Fibonacci sequence, which hooks up with rabbits and the golden ratio. Most of this knowledge is only partially correct. The man was born around 1175 in Pisa, and died somewhere around the middle of the 13th century. His name was Leonardo of Pisa (Leonardo Pisano), and Fibonacci is a nickname invented by historian Guillaume Libri in 1838 because Fibonacci in his most famous work <em>Liber abbaci</em> (1202) he announced himself as <em>filius Bonaci</em> although his father's name was Guilielmo Bonacci. So instead of "son" he may have meant to say "of the Bonacci family". The rabbit story is only one of the hundreds of examples he uses to illustrate the strength of calculating with the Hindu-Arabic number system as we know it today worldwide. This <em>Liber abbaci</em>, written in Latin, is a true work of mathematics following the Euclidean approach of logic derivation. It also explains many techniques to solve problems like the rule of three, the rule of false position, and so many algebraic recipes we are quite familiar with today. And, most importantly, it contains also many illustrative examples and whole chapters with practical applications from commerce and finance. The rabbit example is only one of them. It was known for centuries by Indians in connection with Sanskrit poetry long before Fibonacci. The name Fibonacci sequence was coined by Édouard Lucas in the 19th century. And Fibonacci never connected the sequence with the golden section φ. Luca Pacioli in 1509 called φ the divine ratio and the news was spread that this number appearing in nature so often should represent perfection and beauty. Devlin debunks also this myth.</p>
<p>
What Devlin admires most in Fibonacci is that he is the initiator and the instigator of spreading the revolutionary system of Hindu-Arabic numerals, after which the world could never be the same. The <em>Liber abbaci</em> is a marvelous piece of didactics, but it is written in Latin, not the language used by bankers and merchants. There is no original copy of the book left, only later transcripts. There are however hundreds of shorter versions written in local Italian dialects, and these are the ones that were used to actually spread the new numeral system and the algebraic methods. Fibonacci referred at several places to a shorter version of his book, his <em>Liber minoris guise</em> or <em>Libro di merchaanti</em> which probably is the primal source of all these vernacular <em>libri abbaco</em>. It had however never been found until in 2003 Rafaella Franci identified a manuscript in a Florence library that directly referred to Fibonacci. This is the missing link between Fibonacci's <em>Liber abbaci</em> and popularization of the method via the <em>libri abbaco</em> and it identifies Fibonacci as the man who was also behind the mechanism for spreading the method. This popularization of mathematics is also dear to the heart of Devlin who wrote several books with that intention. He is also an intensive blogger and columnist, gives expository public lectures, and appears often in the media. This is what Fibonacci probably would have done if he had lived today.</p>
<p>
The main point that Devlin wants to make is that Fibonacci should be glorified not for modeling the reproduction capacity of rabbits but for his insight in the possibilities offered by this new numeral system and the ingenious way in which he helped to spread it in the Western world. In 2011 Keith Devlin published a book on Fibonacci: <em>The Man of Numbers. Fibonacci's Arithmetic Revolution</em> (Walker Publishing Company) which is intended to be a biography of the man, but since so little is actually known, it also is an extensive discussion of his <em>Liber abbaci</em> and the man's legacy and influence. Devlin uses abbaci with double 'b' as Fibonacci did, although a single 'b' is more common as Sigler did in his English translation of the <em>Liber abbaci</em> in 2002.</p>
<p>
The present book <em>Finding Fibonacci</em> describes Devlin's quest to collect the sources and information needed to write Fibonacci's biography. The title that may be inspired by Aczel's <em>Finding zero</em>, an account about his quest for the first appearance of 0 to represent zero in a number. In the present book Devlin summarizes what is already in <em>The Man of Numbers</em>. It is a "the-making-of" version with a lot of background information and told as a first-person narrative. If he is more objective in <em>The Man of Numbers</em>, he lets his admiration for the man who caused this revolution in the Western world run more freely in this book.</p>
<p>
In fact gathering all the information went with a lot of lucky coincidences and unfortunate setbacks. Sometimes the situations are really funny when English-Italian communication was not optimal or when he had to deal with the Mediterranean laid-back attitude. But we also learn of his emotions when he is finally paging through these very old manuscripts. The buildings in which the manuscripts are kept, the people that he interviewed, his search for the statue of Fibonacci, his pictures of street signs referring to Fibonacci, and much more are described. You might as well be interested in seeing <a href="http://www.maa.org/external_archive/devlin/Fibonacci.pdf" target="_blank"> some pictures</a> available on the website of the MAA related to his visits of the cities and the libraries and copies of some pages in the old manuscripts.</p>
<p>
It is also an amazing story how Sigler's English translation of the <em>Liber abaci</em> finally appeared in 2002 just 800 years after Fibonacci finished the original. In fact the translation was finished in 1997 with only some editorial details left when Laurence Sigler died of cancer. His wife Judith decided to handle the last details, but then the project was abandoned from the publisher's side. The computer of Sigler had to be hacked to recover most of the text, but the typesetting was lost. Springer then got interested in publishing the book but it required to do the typesetting all over in LaTeX. It took Judith about five years to finalize the work.</p>
<p>
There are also a few chapters referring to what happened after the publication of <em>The Man of Numbers</em>. For example his consultation of the manuscript in Florence that was discovered by Rafaella Franci and identified as "the missing link". He also includes a short chapter in which he draws a parallel between the arithmetic revolution caused by Fibonacci and the computer revolution initiated by Steve Jobs. He has some vimeo links about that: <a href="https://vimeo.com/93390473" target="_blank">Leo & Steve (part 1)</a> and <a href="https://vimeo.com/93532834" target="_blank">Leo & Steve (part 2)</a>. And finally, he learned from William Goetzmann that much of the mathematical analysis that governs the international financial markets has its origin in the <em>Liber abbaci</em>. In particular the computation of the present-value, which means that with this method one may compare the relative economic value of differing payment streams, taking into account the changing value of money over time. The present value of a euro is less than its future value because of its investment and interest potential.</p>
<p>
All in all a book to be recommended. If you already read <em>The Man of Numbers</em> it is most informative to read this "behind the scenes" version and know how it came about (and what happened after its publication). If you didn't know <em>The Man of Numbers</em>, you at least get a summary of what is in there too. Only it is told in a much more personal and lively version. It is working to some kind of climax with the consultation of the "missing link" book in Florence. Nevertheless, it made me go back and read <em>The Man of Numbers</em> too, which has much more information about the contents of the <em>Liber abbaci</em>, and that made me look up Sigler's translation in the library. It illustrates well how Devlin can motivate his readers.</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>
In 2011 Keith Devlin published <em>The man of Numbers</em>, a biography of Fibonacci, a discussion of his most important work the <em>Liber abbaci</em> and most of all to express his admiration for the man who has introduced the Hindu-Arabic numeral system to the Western world. In this book he tells his personal quest while preparing that book. The lucky coincidences, the hilarious failures, and the deep emotions while paging through the old manuscripts. Some chapters tell what happened after the previous book was published. In one of them a parallel is shown between Leonardo Pisano (Fibonacci) and Steve Jobs, who both changed society without recognition, and in another one Fibonacci is pictured as the father of modern finance.</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/keith-devlin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Keith Devlin</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">9780691174860 (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">24.95 £ (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">256</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10950.html" title="Link to web page">http://press.princeton.edu/titles/10950.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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</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/01a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A35</a></li>
</ul>
</span>
Mon, 10 Apr 2017 07:25:57 +0000adhemar47612 at http://euro-math-soc.euThe Best Writing on Mathematics 2016
http://euro-math-soc.eu/review/best-writing-mathematics-2016
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the seventh volume in the series for which Mircea Pitici selects every year a set of papers that discuss popular mathematics-related topics. This genre, if it can be called so, has gained an increasing interest, and the number of pages in each volume seems to be a nondecreasing sequence. Pitici's task may become more difficult every year in making some selection within the scope of the book taking into account the limitations of space and copyright restrictions. On the other hand, with a larger supply to choose from, the quality of the eventual selection is easily kept as high as in previous years.</p>
<p>
In the first four volumes some <em>big shot</em> wrote an introduction, but since the 2014 volume, the series has established a reputation of its own and Pitici writes his own introduction with a long list of books that fall into the same class as the papers in the collection. As usual, there is also a long enumeration at the end of the book with other publications that were not included but that are still highly interesting. That list also has references to notable book reviews, interviews, and special issues of journals. Pitici stresses in his introduction that each volume is an integral part of the whole series. He announces the compilation of an index over all volumes to be produced in the near future. That would indeed be a nice tool to browse through the whole series more easily.</p>
<p>
There are 30 papers selected for this book, all published in 2015. The numbering is indeed a bit misleading, yet reasonable and correct if you think of it practically. This collection is published in 2017, with the title <em>Best writing on Mathematics 2016</em>, since 2016 is the year in which the papers of 2015 are harvested and the book is being prepared. It takes indeed some time to deal with copyright and to typeset the papers in a uniform format.</p>
<p>
What exactly should be understood by fitting under the umbrella of <em>popular mathematics</em>? The last contribution by Ian Stewart, who is a prolific writer of this kind of books, gives a possible answer. Besides children's books, topics treated can be philosophy, history and biography, fun and games, big problems, pure versus applied, or links with arts and culture. He also gives a lot of good advice for anyone who feels the urge to write a book in this class. We find papers in this collection that represent most of the topics that Stewart enumerates, to which we should probably add education and teaching. Not exactly a "popular" topic, but it certainly addresses a public much broader than just mathematicians. Therefore also this topic has in the collections of this series found a settled place.</p>
<p>
Remarkable in the current collection are the three contributions that debunk or at least place in proper perspective what has been a firm folk belief: Wigner's "the unreasonable effectiveness of mathematics in physics" and Hardy's "defense of pure mathematics" and Leibniz's formulation of "the fundamental theorem of calculus". The authors if these papers show that these statements seem not to be as accurate as generally accepted. Another historical contribution is about the most illustrious constant in mathematics. The constant <em>π</em></p>
<p>
is often defined as the ratio of the circumference of a circle over its diameter, but if we consider this as a theorem, then its origin is surprisingly very fuzzy. One more generally accepted expression is to say that a theorem is "deep". It is used lightly by many, but difficult to define. What exactly does it mean when a theorem is called deep? You can find some answers to all these puzzling issues in this book.</p>
<p>
Of course the other topics mentioned are also represented: games and recreation (design of a card deck for Spot It!, stacking wine bottles, billiards), art (mathematics in the collection of the Metropolitan Museum of Arts in NY), and the big problems (The monster group, and Mochizuli's "proof" of the abc conjecture). And there is much more of course. Each paper is selected among the best papers that were published. Not only are the subjects teasing, but the stories are told with vivacity that just gets you hooked after reading the first paragraph. If you do not know where to start, you can find a 3 page survey of all contributions in Pitici's introduction. It is clear that each paper can be read independently, but there is some loose logical ordering. For example the five papers with a statistical flavor are placed together.</p>
<p>
Once again a highly recommended collection that saves you the time to search for the papers yourself and finding out whether it is top quality or not. You do not need to be a mathematician. Mathematical technicalities are totally avoided. These are papers <em>on</em> mathematics, not mathematical papers. It also aims at politicians, managers, philosophers, and whoever has a broader interest in science or society. It presents mathematics in its broadest cultural and social context.</p>
<p>
For reviews of previous volumes see <a href="http://www.euro-math-soc.eu/review/best-writing-mathematics-2012" target="_blank">2012</a>, <a href="http://www.euro-math-soc.eu/review/best-writing-mathematics-2013" target="_blank">2013</a>, <a href="http://www.euro-math-soc.eu/review/best-writing-mathematics-2014" target="_blank">2014</a>, and <a href="http://www.euro-math-soc.eu/review/best-writing-mathematics-2015" target="_blank">2015</a>.</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 the seventh volume in the series for which Mircea Pitici selects every year a set of papers that discuss popular mathematics-related topics. As previous volumes we get a high quality selection of papers originally published in 2015 discussing mathematics and its relation to education, history, philosophy, games and recreation, and art. Once more a marvelous selection.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/mircea-pitici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Pitici</a></li>
<li class="field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</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">9780691175294 (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">27.95 £ (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">408</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10953.html" title="Link to web page">http://press.princeton.edu/titles/10953.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/00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00b15</a></li>
</ul>
</span>
Mon, 10 Apr 2017 07:21:29 +0000adhemar47611 at http://euro-math-soc.euAll sides to an oval
http://euro-math-soc.eu/review/all-sides-oval
<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>
It is not difficult to define mathematically what an ellipse is. Its Cartesian equation is well known. It is however less clear what an oval is. Most people will come up with the condition that it looks like an ellipse. It is a smooth convex closed curve of the plane with two orthogonal symmetry axes. But how to be more precise? Since antiquity, ovals have been used in architecture. So what was the construction used by the architects?</p>
<p>
There are the Cartesian and the Cassini ovals, that have a simple Cartesian equation, but they do not always have the symmetry (in th first case) or are not convex (second case). Historically however the ones that have been used most in arts, especially in architecture, are the polycentric ovals that consist of circular arcs that are stitched together in a smooth way. This is the only kind of ovals that is considered in this book.</p>
<p>
The easiest and most popular one that has been studied thoroughly is the so-called four center oval. It consists of four circular arcs that fit together smoothly. Two arcs with smaller radius are at the tops of the long symmetry axis and the ones at the end points of the shorter symmetry axis have lesser curvature because they belong to a circle with a longer radius. The crux is of course to choose these four centers of the circles in such a way that the arcs fit smoothly together at the four connection points. How does one have to select these centers and how large should the sector angle be that supports the arcs so that one does indeed get this smooth transition? Because of the symmetry, two centers are located symmetrically on the long axis and two on the shorter one. So it suffices to consider only a quarter of the oval and find two of the centers to define the arcs and the connection point where the arcs meet in that quadrant. Once the length of the axes are given, an ellipse is completely defined. For an oval, one needs at least one more parameter, like the distance from one of the centers of the circles to the center of the oval or the distance of the connection point to one of the symmetry axes.</p>
<p>
Once these arguments have been formulated, it needs some analysis of the geometry of the problem. And that is where this book gets started. The author has, besides other interests, a knack for polycentric curves like eggs or ovals. This book is restricted to ovals, and the first chapter analyses the properties that will enable us to relate the different parameters. Once this is cleared out, the construction with ruler and compass of an oval (actually a quarter of an oval, because the rest follows by symmetry) is given step by step depending on which parameters are prescribed. So one might choose three of the six possible parameters in many different combinations and that gives rise to twenty different ways to define and construct an oval satisfying the data. Some are more complex and some have more restrictive conditions than others. The solution may not always be unique. Everything is clearly explained and the many illustrations produced with geogebra are crystal clear. It might however be interesting to have a look at the associated website <a href="http://www.mazzottiangelo.eu/en/pcc.asp" target="_blank">www.mazzottiangelo.eu/en/pcc.asp</a> where you find links to YouTube videos showing animated geogebra constructions. The link goes both ways: you may consider this book as a manual for the online site, or the online site as an illustration for the book.</p>
<p>
Besides the parameters described above, one might also choose for one of the radii of the arcs or the ratio of the axes or the angle formed by a symmetry axis and the line joining the circular centers of the arcs. With all ten parameters, there are a total of 116 possibilities to construct the ovals, many of which, but not all, reduce to the twenty constructions mentioned before. Some of the constructions are historical and often pretty old, but others are surprisingly recent. For particular choices of the parameters, the construction may simplify considerably or the oval may have especially pleasing esthetic properties, which are discussed in a separate chapter.</p>
<p>
Towards more practical applications of stadium design, one may consider ovals circumscribing or inscribed in a rectangle. If the symmetry axes are the middle-lines of a rectangle and the diagonals of a rhombus, then all previous constructions circumscribe the rhombus and are inscribed in the rectangle. For a stadium one should find an oval circumscribing the inner rectangular field (for example a soccer field) and surround it by ovals like running tracks, all inside an outer rectangle defining the limitations of the stadium. Modern constructs however have straight parts for the running tracks along the long sides.</p>
<p>
While the constructions are mostly obvious, it takes more algebra and more formulas to express some parameters as a function of others. This is a short chapter, but essential to find ovals that are optimal in some sense. For example finding the "roundest" oval with given axes. They are also needed in geogebra animations when slider rules are provided allowing to see the effect of changing a parameter.</p>
<p>
The last two chapters discuss ovals in two famous architectures in Rome: the dome of the church <em>San Carlo alle Quattro Fontane</em> by the architect Borromini and the ground plan of the <em>Colosseum</em>. A careful study is made of the ovals of the base of the dome in the church, the rings of coffers, and of the lantern. It turns out that there are small defects making them deviate from perfect mathematical ovals. This has long been a mystery. It is suggested that the starting point was a mathematically perfect oval, but that practical restrictions entailed heuristic corrections. The solution that Mazzotti proposes here corresponds remarkably well with Borromini's original drawings.<br />
For the Colosseum, we have to leave the simple ovals with four centers and go to quarter ovals consisting of more than two arcs. Because of symmetry there have to be always $4n$ centers. Again constructions of such ovals are considered. In the case of the Colosseum, $n=2$, i.e., ovals consisting of eight circular arcs seem to match the ground plan perfectly well.</p>
<p>
This is a very nice geometric application that requires only simple algebra and that can be easily experimented with. You do not need to be a mathematician to enjoy it. It that sense, it might be interesting to have the geogebra source available somewhere, which is unfortunately not the case. Also historians might be interested in the last two chapters about historical buildings. For the mathematician, it is invaluable because it brings together so much information that was either not known or never writen down or if it was, then at least it was scattered in diverse publications. The graphics are very readable since they use colors (except for the pictures in the last two chapters, only red, green, and blue suffice for the mathematical constructions). As a LaTeX purist, I cannot resist mentioning my irritatin when seeing variables mentioned in roman font when in a sentence, while they are in a different font when used in a formula. Also, I do not understand why the ratio of the half symmetry axes is denoted at least twice as $\frac{p=\overline{OB}}{\overline{OA}}$ (p.20 and 148) and when at the end of a line $p=\frac{\sqrt{2}}{2}$ is split into $p$, which is left dangling at the end, and $=\frac{\sqrt{2}}{2}$ at the beginning of the next line (p.102). These are however minor flaws in an otherwise nice text, and as I am sure, these will disappear in a next edition. Do not let this prevent you from reading this most enjoyable book and you should certainly try out some of the constructions for yourself, either with ruler and compass or with geogebra.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book restricts to polycentric ovals, which means that they concist of circular arcs that fit smoothly together. Some properties are derived to allow for many different ruler and compass constructions. The major part of the book is about the case of simple ovals, i.e., ovals concisting of four arcs. They can be constructed when 3 parameters are given (like location of the four centers, the length of the symmetry axes or the location of the points where the arcs meet). The book ends with the discussion of ovals in two historic buildings in Rome: the dome of the <em>San Carlo alle Quattro Fontane</em> church by Borromini and the ground plan of the Colosseum.</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/angelo-alessandro-mazzotti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Angelo Alessandro Mazzotti</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-international-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-3-319-39374-2 (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">31,79 € </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">170</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/9783319393742" title="Link to web page">http://www.springer.com/gp/book/9783319393742</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</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/51m04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51M04</a></li>
<li class="field-item odd"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li>
<li class="field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li>
<li class="field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li>
</ul>
</span>
Wed, 22 Mar 2017 08:49:13 +0000adhemar47565 at http://euro-math-soc.euThe Mathematics of Secrets
http://euro-math-soc.eu/review/mathematics-secrets
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The subtitle of this book <em>Cryptography from Caesar ciphers to digital encryption</em> illustrates that cryptography has played a role in society of all times. Since antiquity, passing secret messages has been mainly of military interest. The simple cipher that is attributed to Caesar is probably much older. This military privilege lasted till around the middle of the 20th century. Then digital computers became part of society, first for big companies like banks, who needed failsafe communication between their computers, and later, as the digital computer conquered the daily life of individuals it became a standard procedure for practically every digital communication. Authentication and privacy has become everybody's concern. Often, mathematicians were involved in the design of cryptographic systems, even though for most of them, the mathematics are not complicated. Some modular arithmetic often suffices and for attacks, some statistics can help. In more recent systems prime numbers and elliptic curves enter the scene and there of course the mathematical challenges are a bit more substantial, but still manageable. History has shown that it has been a leapfrog pattern of copper and thief, of designer and attacker. If a new cryptosystem is designed it is used until it has been cracked by attackers, and then a new smarter system has to be designed. The newest threat for current systems is quantum computing, and designers are preparing to jump to a new system that can cope with that kind of attacks.</p>
<p>
Analyzing cryptosystems involves three characters: Alice, Bob and Eve. Alice wants to send a secret message to Bob and Eve is eavesdropping, trying to read the message. So there are three stories to tell: How will Alice encrypt the message, how will Bob decrypt it, and what can Eve do to attack the system and intercept the message. Hiding the message involves codes and ciphers. In principle these are different. A code is a collection of many (perhaps thousands) of words that are used to replace words in the plaintext while a cipher describes how to replace letters (or short groups of letters) in the plaintext. A code is a kind of dictionary, while a cipher is more an algorithm. Steganography is another way of hiding a message by making it invisible (for example using invisible ink), but that is not covered in this book.</p>
<p>
The contents of the book is pretty complete in surveying cryptography from antiquity till the current state of the art. In his introduction, the author explains that his first intention is not to be complete in all the details and in all the systems that ever existed, but he primarily wants to bring the mathematics involved to the foreground. And since these mathematics are relatively simple, the book should be accessible for anyone with a basic secondary school algebra course. The development of the contents is historical. So the first three chapters discuss historical ciphers that are not in use in their simplest form anymore, but they introduce the basic concepts that will be needed in the rest of the book when more complicated techniques are introduced. That begins around halfway the twentieth century when computers (mechanical or electronic) became involved. Also in each chapter the development is historical. Respecting the chronology throughout the book and in the chapters is a logical choice since it brings the increasingly complicated ideas as they were introduced historically. This makes it easy to understand why some modifications are needed and where the complication comes from. Each chapter also ends with a section <em>Looking forward</em>, linking it to future chapters and giving an outlook on what the potential use and challenges are for the techniques just introduced.</p>
<p>
Describing in detail all the systems that are discussed in the book is not possible in this review. There are too many. Here is a crude outline. As said above, the first three chapters are mainly historical and introduce the reader to substitution ciphers. The Hill cipher (1929) is still in use as part of more complicated modern systems. Since frequency analysis can be used in attacks, a polyalphabetic version is a better option. Combination of such systems were used in the German Enigma machines during World War II where rotors applied successive ciphers to the plaintext. The next tool is a transposition cipher. The ciphertext is written in one or more tables row by row and read off column by column scattering patterns throughout the ciphertext. Claude Shannon (1916-2001), father of modern information theory, introduced a mixing function to create diffusion and confusion in cryptography. This should make it difficult to use a frequency analysis on the ciphertext or to detect a key if statistics are available. The systematic professional cryptosystems based on these principles are Feistel and SP-networks (a sequence of substitutions and permutations are applied during encryption), Examples are the DES standard (approved by NSA and published by NIST in 1976), and its improved version AES (2011).</p>
<p>
The next chapter introduces stream ciphers. Here encryption of a block may depend on what has happened to previous blocks. Typical examples are the linear feedback shift register systems, which apply a linear filter to the bitstream. The problem here is the linearity. So, exponential ciphers try to introduce nonlinearity. Here the mathematics come to the foreground and we meet for example Fermat's little theorem, and the discrete logarithm problem. The drawback of these exponential ciphers is however that they are computationally demanding.</p>
<p>
The crypto universe changed with the introduction of public-key systems. Bob has a private and a public key. Alice uses Bob's public key to encrypt the message. Bob's private key is needed to decrypt it. Eve does not have it and hence cannot read the message, and neither can Alice after she encrypted it. The idea is pioneered in the 1970's by R. Merkle, W. Diffie, and M. Hellman, but the breakthrough came from R. Rivet, A. Shamir, and L. Adelman with their RSA system (1978). These systems are based on a one-way function. This means that it represents a problem extremely difficult to solve, but once a solution is proposed, it is easily checked that it is indeed a solution to that problem. The problem acts like a safe. The public key can lock it but only the private key can unlock it. Hellman's problem was based on the discrete logarithm and in the RSA case it is prime number factorization for very large primes. It was only in 1996 that it was disclosed that J. Ellis had discovered public key cryptography already in 1969, but since he worked for the British equivalent of the NSA, he was bound by secrecy and could not communicate about it. There are other public key systems like the three pass protocol, or systems based on elliptic curves, and the same ideas can also be used for digital signatures (authentication problem).</p>
<p>
In the last chapter the reader is briefly introduced to quantum computing. In this case the one-way functions, i.e., the "difficult problems" that hide the secret message are not based on prime factorization or discrete logarithms, but on much more difficult problems such as solving multivariate polynomial systems, or finding the closest point on a skew grid to some given point, not on that grid. If the dimensions are high enough, these problems are still hard, even with a quantum computer. The idea of the latter problem on grids, also called the closest vector problem, is used on a two-dimension grid as an example. Another technique is the BB85 (1985) protocol of C. Bennett and G. Brassard, elaborating an idea of S. Wiesner. This protocol uses the polarization of photons to encrypt the message.</p>
<p>
There is an extensive appendix with references and notes. It contains suggestions for further reading, often taking a different approach to cryptography; it includes an extensive bibliography; and many pages with notes. The notes are references, quotes, or explanations, and they have a link to the page on which they comment, but there are no references on these pages to the notes. Thus while reading, you are not tempted to interrupt the flow of the text to look up the notes.</p>
<p>
This is a marvelous way of illustrating the use of simple mathematics in an important application that has triggered the wit of the designers and the ingenuity of the attackers since antiquity. As the application became more and more important after computers entered the scene, mathematics became more and more involved. Someone with an elementary background, even if (s)he does not reach the end of the book, can come a long way on the path to where modern cryptography involves more advanced mathematics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">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>
Using only some elementary arithmetic, Holden introduces the reader to the secrets of cryptography. He gives a nice survey of how the field has involved since antiquity to public key cryptography and even the future challenges of quantum computing. Although the mathematics are usually simple, the idea is to bring especially these mathematical aspects of the design and the attacks to the foreground. The most difficult mathematics are some elements from discrete logarithms and from elliptic curves, and there is also some quantum computing in the last chapter. But that should not scare away the lesser mathematical reader since understanding some elementary principles suffice to grasp the main cryptographic ideas.</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/joshua-holden" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Joshua Holden</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">9780691141756 (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">£22.95 (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">392</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/10826.html" title="Link to web page">http://press.princeton.edu/titles/10826.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>
<li class="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</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/94-information-and-communication-circuits" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94 Information and communication, circuits</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/94a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94A60</a></li>
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<li class="field-item even"><a href="/msc-full/81p94" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P94</a></li>
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Thu, 16 Mar 2017 18:03:14 +0000adhemar47549 at http://euro-math-soc.euWavelets. A Student Guide
http://euro-math-soc.eu/review/wavelets-student-guide
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
There are many ways in which wavelets can be introduced, depending on the mathematical knowledge of the student's or readers' background: one may take a linear algebra approach, a signal processing approach, an approach starting from approximation theory, in particular splines, or Fourier analysis, or a general analysis approach, or even use a background from theoretical physics. And there is the digital algorithmic approach versus the continuous analytic vision. There are almost as many ways to introduce wavelets as there are books written with that purpose. The present one, is one I like very much if it is to be used to bring mathematics students at the level of their first or second year at the university into contact with wavelets. If I had to teach such a course, this would definitely be my first choice.</p>
<p>
Not only does it bring the subject in a most suitable and systematic way that, I am sure, mathematics students are used to and probably appreciate most. It is also following some good rules of didactics taking the students by the hand and bringing them to a higher level of understanding, ensuring that at least the bulk of the students does not declutch. A lot of effort is put into taking the rungs of the ladder at just the right pace, not boringly slow or not frighteningly fast, and always placing a chapter in the proper context: what has been achieved, and where do we want to go?</p>
<p>
Faithful to these principles, the first chapter is just a survey, summarizing the content of the whole book, even introducing the Haar wavelet, which being the simplest possible wavelet, does not require much analysis, but it does illustrate the idea. The rest of the first half of the text does not really deal with wavelets at all but introduces vector spaces, inner products, projections, etc., first in $\mathbb{R}^N$, but once this has been explained, all the concepts are shown to be not much different when it is generalized to sequences $\ell^2$ or square integrable functions that form $L^2$. Of course the latter are Hilbert spaces which requires some more advanced elements such as convergence, measure theory, integration, etc. It is not an in-depth analysis of all these topics, but just what is needed to move on is introduced. For example there are some considerations about a basis, density, and orthogonalization in an infinite dimensional space, but the concept of a frame is not introduced. Most of the proofs are included, some parts and some proofs are given as exercises, but again the most difficult ones are left out.</p>
<p>
The second half of the book then treats the wavelets. First the Haar wavelet is revised for which all the wavelet concepts are introduced such as a multiresolution analysis and all its properties, the scaling and the wavelet function, the scaling (or dilation) equation. The next step is to lift this to the more general situation of a general wavelet (assuming it has a finite support), the vanishing moments and the smoothness of the wavelet and the orthogonality properties and how all these properties can be formulated as conditions on the coefficients of the scaling equation. In the next chapter all this is made more concrete by deriving, drawing, and analysing the Daubechies wavelets for $N=2$ (for $N>2$ and other families the analysis is much shorter). In a last chapter, the Fourier-domain treatment of all this is discussed. Fourier analysis is again only introduced at a level just sufficient to do the computations, which avoids the deeper analysis requiring the massive body of Lebesgue integration and the subtleties of Fourier analysis.</p>
<p>
This survey illustrates the level of the approach and also the content is purely mathematical, avoiding algorithms, applications, linear algebra, etc. Each chapter is concluded with a long list of exercises (there are about 230 exercises in the whole text). They respect the level of the text and are not trivial nor exceptionally demanding. A remarkable feature of the book is the use of something like ▶ earmarking many sections typeset in a slightly smaller font. They give some extra information or warning, not really essential to follow the flow of the exposition. It is as if the authors whisper some extra information into the ear of the reader while he/she is studying the text. The authors give also several suggestions in the introduction on how a selection can be made from the text to cover a shorter course, and in an appendix they discuss pointers to the literature and they do this chapter by chapter and in particular also for the exercises. I think this is a book perfect for what it is intended to be and it is obviously prepared with great care for precision, level of complication, and it has very good didactical qualities. </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 text to be used when the topic of wavelets is to be introduced to undergraduate mathematisc students. Half of the text forms an introduction to inner product vector spaces and Hilbert spaces. The second half is introducng multiresolution first for the Haar wavelet, then it is generalized and worked out for second order Daubechies wavelets. An elementary Fourier analysis approach is the subject of the last chapter. The booklet contains many exercises, all of a similar theoretical level. Algorithms, and applications are not considerd.</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/peter-nickolas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Peter Nickolas</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/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge 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">9781107612518 (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">£39.99 (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">274</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.cambridge.org/be/academic/subjects/mathematics/abstract-analysis/wavelets-student-guide" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/abstract-analysis/wavelets-student-guide</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/41-approximations-and-expansions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">41 Approximations and expansions</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/42-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42-01</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/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li>
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Sat, 04 Mar 2017 12:48:37 +0000adhemar47500 at http://euro-math-soc.eu