European Mathematical Society - Birkhäuser Verlag
https://euro-math-soc.eu/publisher/birkh%C3%A4user-verlag
enNice Numbers
https://euro-math-soc.eu/review/nice-numbers
<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>
For readers who are used to read books that promote and popularize mathematics, John Barnes may be a familiar name since he published before <em>Gems of Geometry</em>(Springer, 2009, 2012). What he did for geometry there, he is now doing for numbers, another topic commonly used to reach a broad audience. And audience is to be taken in a literal sense because both books are the result of a series of lectures given by the author.</p>
<p>
There are 10 chapters in this book, corresponding to 10 lectures, each one discussing a general aspect related to numbers. Of course many authors have been excavating this topic with the same ambition before and much of what Barnes tells us has been told by others on several occasions. Yet there is always a fresh angle to look at them and there is always some sparkling gem, something unknown before, or an unexpected link or connection to be discovered.</p>
<p>
A quick survey of the chapters.<br />
<strong>1. Measures.</strong> Defining prime numbers and factors leads to an extensive discussion of many non-decimal systems of units for currency, length, volume, time, weight,... In fact 10 has too few factors and therefore other numbers can be more suitable as a base.<br />
<strong>2. Amicable numbers.</strong> Perfect numbers equal the sum of their factors and amicable numbers is a couple where each one equals the sum of the factors of the other. This can be generalized to n-tuples or sociable cycles. Mersenne primes and Fermat numbers are introduced here. <br />
<strong>3. Probability.</strong> Of course this involves coin flipping and dice throwing and normal and other distributions. But as an aside, it is surprising to learn that there are 16 different possibilities to produce a classical die. Furthermore we recognize the Buffon needle problem and the Monty Hall problem. And there is the practical problem of false positives in tests and more playfully several game strategies and gambling theoretical problems.<br />
<strong>4. Fractions.</strong> This is explaining the Egyptian number system, continued fractions, repeating decimal expansions of rational numbers, and a long division-like algorithm to compute a square root (a forgotten skill now that pocket calculators are generally available). <br />
<strong>5. Time.</strong> This discusses the Roman and the Gregorian calendar, but also the astronomical origins of year, season, month, week, and night and day.<br />
<strong>6. Notations.</strong> Thinking of the notation of numbers, this must obviously include our familiar decimal positional system, but also predecessors like the Roman and the Babylonian system. This is also a good place to look at modular arithmetic (also used in other chapters). The repeating expansion of rational numbers is reconsidered in a different base system and connected with Fermat's little theorem.<br />
<strong>7. Bells.</strong> Bell ringing (or tintinnalogia) is a combinatorial problem in which one has to ring a set of bells with certain patterns of repetition. This will involve decomposition of permutations and even some group theory, although the latter is not elaborated here.<br />
<strong>8. Primes.</strong> This involves classical ideas like the Euclidean algorithm and the sieve of Eratosthenes, but also Gaussian (complex) primes, and prime polynomials with coefficients in modular arithmetic.<br />
<strong>9. Music.</strong> This is a relatively long chapter about the different music scales that can be used.<br />
<strong>10. Finale.</strong> This is a roundup of three remaining topics. The most important is an explanation of the working of the public key RSA encryption algorithm. Furthermore possible animal gaits, and finally the towers of Hanoi and the topologically equivalent Chinese ring puzzle.</p>
<p>
In his lectures, Barnes did involve his public actively, which is reflected by the exercises that are given at the end of the chapters. Easy exercises that do not need extra knowledge than what was explained in that chapter. For those who are willing to read more, some references are provided with some advise about their content and their difficulty.</p>
<p>
So, if you are not familiar with popularizing books using `numbers' as a master key to introduce mathematics, this is an excellent start. It is light, entertaining, richly illustrated, and still Barnes has disregarded the advise of many publishes to avoid all formulas since each formula allegedly would cut the sales in half. I tend to disagree with those publishers and I am happy that Barnes did too. I am sure that those who are willing to read this kind of books are not scarred away by a formula. The problem with formulas is that typos easily slip in, like a blatant one on page 70 claiming that $\sqrt{2}$ is a root of the equation $x^2−1=0$. Even if formulas are allowed, the content should still be digestible for anyone, and it should be easy for some of the readers to skip the more technical parts, while these are still available for those who are hungry for more. In this case there are 100 more pages with nine appendices of such marvelous items. Some are indeed more mathematical yet still entertaining (Ackermann function, Pascal's triangle and 3D generalization of triangular numbers, stochastics in game and queuing theory, the Chinese remainder theorem, group theory and an extensive one on the solution of the Rubik cube, etc.). This book is a most enjoyable read. Those who read <em>Gems of Geometry</em> need not be convinced of Barnes' entertaining style and will love to read this book too. Others who read this book without knowing his geometry book will be teased to also check that one out.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book, like Barnes' previous <em>Gems of Geometry</em> is based on lectures given for a general audience. It discusses in 10 chapters and in nine a bit more advanced appendices some topics from number theory accessible to a general readership. There are mathematical formulas and even an occasional proof, but everything is brought in an entertaining style and there are some pleasantly surprising side stories like patterns for bell ringing and non-decimal subdivided units for all sorts of measurements, or 16 different ways to produce a genuine die. </p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/john-barnes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Barnes</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3319468303 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 39.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">329</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319468303" title="Link to web page">http://www.springer.com/gp/book/9783319468303</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11a51" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A51</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11bxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Bxx</a></li></ul></span>Sun, 19 Feb 2017 09:35:32 +0000Adhemar Bultheel47466 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/nice-numbers#commentsThe Fractal Dimension of Architecture
https://euro-math-soc.eu/review/fractal-dimension-architecture
<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>
With this book, Birkhäuser starts a new book series <em>Mathematics and the Built Environment</em>. The books in this series will focus on "the complex interaction between mathematics and architecture". The present book fits well into this topic as its title already suggests. It is written by experts who have published a lot on this particular subject on the boundary of architecture and mathematics. Michael J. Ostwald is professor and dean of architecture at the University of Newcastle, Australia. Josephine Vaughan is also working at the same university and specializes in the subject of this book.</p>
<p>
Fractal dimensions were introduced by Benoit Mandelbrot. There are however different definitions possible and there are several ways in which to compute it. Here the authors have chosen to use the so called box counting method to compute the fractal dimension of a design of a building. Traditionally this methods looks at some structure at different resolution levels by covering it with smaller and smaller boxes (i.e., squares in a 2D case) from a regular grid. The idea to define the fractal dimension is the following. Count the number N(i) of boxes needed to cover all the details of the picture for grid size s(i). By doing this for different grid sizes, and plotting log(N(i)) versus log(1/s(i)) one gets points of a graph that shows a specific trend. The fractal dimension is then defined as the average slope defined by these points. It is simple and always applicable so that more and more studies (in architecture) become available using this technique.</p>
<p>
Next question is on what structure this method should be applied. A choice is made for two types of graphical representations to characterize the construction: the frontal elevation and the top plan view. Each of these can be given with increasing detail by starting with the outline and successively adding the primary, the secondary, and the tertiary forms of the design, and finally the texture. With these successive levels of detail, one can focus the research and zoom in on the different levels of the design aspects and draw conclusions for each of them.</p>
<p>
However two more components influence the results and hence the conclusions that can be drawn. The first one is the thickness of the lines. The thickness can vary within one graphic or vary over different graphics that one wants to compare. The second one is how the rectangular grid of boxes is placed over the graphic, more precisely where the relevant structure is located in the grid. Is it at the center, or close to the outer boundary of the grid, for example in a corner of the grid? These two components require a pre- and a post-processing step. In the pre-processing one has to decide on the size of the grid and the position of the structure, the line thickness and the resolution of the image (number of pixels). Only then the box counting method can be applied in a uniform way and allow comparison over different graphics. The post-processing then does the statistical processing of the box counting data. All these aspects are tested so that optimal values can be set for all of the relevant parameters.</p>
<p>
Once all this preparatory analysis is done in Part I of the book, Part II does the field work and analyses 85 designs using the technique described. These are carefully chosen, ranging over the period 1901 to 2007, hence covering different style periods, located in different countries, designed by different architects. All the details of the the buildings and the data of the analysis are reported and discussed. Therefore Part II takes about 2/3 of the book. All these data then lead to answers to three hypotheses that the authors have put forward at the beginning: (1) <em>The complexity of the groupings and functions within the home has reduced over time, and this is reflected in a reduction of the fractal dimension in plans and elevations over time.</em> The data do not convincingly support this hypothesis. (2) <em>The specific character of a movement or genre is reflected in the fractal dimension.</em> This is actually rejected by the data. (3) <em>The fractal dimension characterizes different architects.</em> Again this is not confirmed, but it gives some idea what else could be done in this respect.</p>
<p>
From the content sketched in the above review, it will be clear that this book is in the first place addressing architect students or researchers, but clearly not the mathematicians. Besides some notes on fractals, and a little bit of statistics, there is no definite mathematical content. All the elements of the research methodology, hence also the fractal dimension, are clearly and extensively explained and motivated. Also the analysis of the results and conclusions are carefully described. So it is easy to read and understand for anyone interested in the topic.</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 technique of box counting to compute the fractal dimension is applied to drawings of the elevation and the top plan of buildings, which are represented at successive levels of detail. In part I the methodology is explained and tested. In part II this is applied to 85 designs to test three hypotheses relating the fractal dimension to the complexity of functionality, the stylistic genre, and the individual architects.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/michael-j-ostwald" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael J. Ostwald</a></li><li class="vocabulary-links field-item odd"><a href="/author/josephine-vaughan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Josephine Vaughan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-32424-1 (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">422</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links 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><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/9783319324241" title="Link to web page">http://www.springer.com/gp/book/9783319324241</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/28a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">28A80</a></li></ul></span>Fri, 02 Dec 2016 11:36:37 +0000Adhemar Bultheel47306 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fractal-dimension-architecture#commentsIntegral Equations with Difference Kernels on Finite Intervals (2nd ed.)
https://euro-math-soc.eu/review/integral-equations-difference-kernels-finite-intervals-2nd-ed
<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 a revised and extended revision of the previous edition from 1996 that appeared as volume 84 of the series <em>Operator Theory Advances and Applications</em> (OTAA). The chapter on operator Bezoutiants and roots of entire functions has been removed since it was treated in another of Sakhnoviches books: <em>Lévy Processes, Integral Equations, Statistical Physics: Connections and Interactions</em> (2012, OTAA vol. 225). On the other hand, the part on Lévy processes is largely extended in the form of new chapters. Also a number of open problems has been added.</p>
<p>
As the title says, the book considers the solution of the equation $Sf=\varphi$ with $Sf=\sum_{j=M}^N \mu_j f(x−x_j)+\int_0^\omega k(x−t)f(t)dt$ with as a special case $\mu_j=0$ for $j\ne0$ and $x_0=0$. By modifying the kernel, this can be transformed into</p>
<p>
\[ Sf=\frac{d}{dx}\int_0^\omega s(x−t)f(t)dt \quad\mbox{where}\quad \left\{\begin{array}{ll}s(x)=\int_0^x k(u)du+\mu_+,& x>0\\ s(x)=\int_0^x k(u)du+\mu_−,& x<0\end{array}\right. \quad\mbox{and}\quad \mu_++\mu_−=\mu. \]</p>
<p>
</p>
<p>
This relies on a relation like $S\frac{d}{dx}=\frac{d}{dx}S$, which is a special case of more general operator identities of the form $AS−SB=Q$. It are the identities of this kind that are used in this book to solve the equation, i.e., to construct an inverse $T=S^{-1}$ which obviously has to satisfy $TA-BT=TQT$. For example when $Af=i\int_0^x f(t)dt$ and $A^∗f=−i\int_x^\omega f(t)dt$ then $(AS-SA^∗)f=i\int_0^\omega(M(t)+N(t))f(t)dt$ with $M(x)=s(x)$ and $N(x)=-s(-x)$. It then follows that the knowledge of $N_1$ and $N_2$ satisfying $SN_1=M$ and $SN_2=\mathbb{1}$ allows to find the structure of $T$ or even construct $T$ explicitly in some cases.</p>
<p>
A special role is also played by a right-hand side equal to $e^{i\lambda x}$ with solution $B(x,\lambda)$. This allows to make a link with scattering theory since the function $\rho(\lambda,\mu)=\int_0^\omega B(x,\lambda)e^{i\mu x}dx$ has the interpretation of a reflection coefficient. The link with roots of entire functions can be made because the operator $T$ is the analogue of a Bezoutiant, which, in the classical case is used to find common zeros of two polynomials.</p>
<p>
Several applications sometimes come to the forefront like hydrodynamics, radiation, communication and antennas. Other applications like elasticity and diffraction lead to even more general situations where the one interval $[0,\omega]$ is replaced by a system of intervals. The physical interpretation of the results is sometimes given but not always.</p>
<p>
Different cases are discussed in successive chapters like difference kernels belonging to different $L^p$-spaces or with a power of logarithmic behavior with and without particular right-hand sides. As we mentioned already, Lévy processes take a particular place in this second edition (chapters 7-10). This is basically a stochastic process with zero initial condition and with independent and stationary increments. The transition operators form a continuous semigroup and it is shown that its infinitesimal generator is of convolution type. By introducing a quasi potential an analysis is made of the process staying within a certain domain when the measure is not integrable. Several examples are given for triangular factorization when $S$ is a positive definite operator. Finally, if the measure of the process is integrable, the quasi potential takes a particular form and again it is analyzed when the process will stay within a domain either after a finite number of steps or asymptotically.</p>
<p>
The coherence of the different chapters is not very strong. They give results for variants and particular cases of the equations that all fall under the title of the book, but that are somewhat independent from each other. Notwithstanding the practical applicability and the importance of these convolution type equations for engineers and applied sciences, the applications are not the prime interest of this book. Numerical and computational aspects are not considered at all. This is a theoretical study of these equations anchored on a strong Russian tradition referring to problems investigated by Krein, Kac and others. Also the list of references consists mainly of items from the Russian literature. For the mathematician, the open problems at the end of the book may be some good challenges in operator theory or stochastic processes to consider.</p>
<p>
Let me mention that at about the same time of this second edition, another volume was published in the OTAA series as volume 244 <a href="reviews/recent-advances-inverse-scattering-schur-analysis-and-stochastic-processes" target="_blank"><em>Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes</em></a> with 10 research papers dedicated to Lev Sakhnovich on the occasion of his 80th birthday. It also contains some biographical notes and a list of his publications.</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 revised and extended revision of the previous edition from 1996. The extension concerns the chapters on Lévy processes. The chapter on operator Bezoutiants and roots of entire functions has been removed.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/lev-sakhnovich" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Lev Sakhnovich</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-16488-5 (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">90.09 €</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">226</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li><li class="vocabulary-links field-item odd"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319164885" title="Link to web page">http://www.springer.com/gp/book/9783319164885</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/45-integral-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45 Integral equations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/45e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45E10</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/45h05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45H05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/45c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45C05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/60g51" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60G51</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/44a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A35</a></li></ul></span>Mon, 20 Jul 2015 15:53:07 +0000Adhemar Bultheel46312 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/integral-equations-difference-kernels-finite-intervals-2nd-ed#commentsRecent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes
https://euro-math-soc.eu/review/recent-advances-inverse-scattering-schur-analysis-and-stochastic-processes
<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>
Lev Aronovich Saknovich was born on February 24, 1932 in Lugansk, Ukraine. Almost agains all odds as a Jew in the communist Russia of those days, be became a mathematician and had teachers such as D.P. Milman, V.P. Potapov, M.S. Brodskij, and M.S. Livšic and his PhD was supported a.o. by I.M. Gelfand, M.G. Krein, and M.A. Naimark.</p>
<p>
This book is compiled on the occasion of his 80th birthday. It starts with a short biography and a list of his publications. Lev Saknovich himself gives an account of his studies and teachers. The main part of the book consists of 10 scientific papers that are of course related to the work of Sakhnovich. They fit very well in the Birkhäuser series on <em>Operator Theory Advances and Applications</em> founded by I. Gohberg in 1979 and more particulary in the subseries <em>Linear Operators and Linear Systems</em>. Contributors to this book are well known is this domain. Several of them published books in this series before like for example D. Alpay, V. Dubovoy, A. Kheifets, A.E. Frazho, M.A. Kaashoek, B. Fritzsche, B. Kirstein, J. Rovnyak, and by Lev Sakhnovich himself. Volume 84 in the OTAA series <a href="/review/integral-equations-difference-kernels-finite-intervals-2nd-ed" target="_blank"><em>Integral equations with difference kernels on finite intervals</em></a> by L.A. Sakhnovich got a second revised edition in 2015 along with the present book.</p>
<p>
The papers are listed alphabetically, but they can be organized in 4 groups. Three papers deal with interpolation and moment problems. Infinite product representations of reproducing kernels are used to study iterated function systems, harmonic analysis, and stochastic processes (Alpay et al). Commutant lifting and solution of Riccati equations are used to generate all rational solutions to a Leech problem in state space form (Frazho et al). It is the continuation of a previously published paper generating minimum entropy solutions. The solutions for the truncated matrix Hamburger moment problem are traditionally treated separately for the even and the odd case. Here Schur analysis is used to treat both simultaneously via a Schur-type algorithm (Fritzsche et al).</p>
<p>
Two papers fall under the flag of indefinite inner product spaces. The paper on quaternionic Krein spaces is a continuation of a published paper that treated quaternionic Pontryagin spaces (Alpay et al). In a joint paper Rovnyak and Sakhnovich continue exploring the relation between interpolation problems, operator identities and Krein-Langer representation of Carathéodory functions. In their previous paper they had treated the case of Nevanlinna functions.</p>
<p>
Different aspects of operator-valued functions are treated in four papers. One deals with operator-valued Q-functions with positive definite boundary conditions (Arlinskii et al). Another paper gives proofs for the Radon-Nikodym theorem for measures that are vector- or operator-valued (Boiko et al). Semi-separable iintegral kernels in infinite dimensional spaces, and in particular their Fredholm determinants are analyzed using a Jost-Pais type reduction (Gesztesy et al). The relation between certain analytic function classes, their closedness under addition and multiplication, and other properties is the subject of another paper (Makarov et al).</p>
<p>
Finally a paper by Lev Sakhnovich and his son discusses stability of nonlinear Fokker-Planck equations in inhomogeneous space.</p>
<p>
From this short sketch of the contents, it is clear that, besides the brief biographic part, the essence is a collection of research papers that will be of interest to the mathematcians and engineers that feel at home in this Birkhäuser series. Most of the papers could as well have been published in journals, and are not of introductory of survey type</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of research papers dedicated to Lev A. Sakhnoviches 80th birthday. It also has a short biography and a list of publications and some reminiscences of Lev Sakhnovich on his teachers and studies. The research papers cover (matrix valued) moment problems and interpolation, indefinite inner product spaces, operatr valued functions, abd nonlinear Fokker-Planck equations.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/daniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">daniel alpay</a></li><li class="vocabulary-links field-item odd"><a href="/author/bernd-kirstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Kirstein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-10334-1 (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">100,69 €</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">394</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319103341" title="Link to web page">http://www.springer.com/gp/book/9783319103341</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00b30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00B30</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/46c20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46C20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/46e40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46E40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/35q84" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35Q84</a></li></ul></span>Mon, 20 Jul 2015 15:10:38 +0000Adhemar Bultheel46311 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/recent-advances-inverse-scattering-schur-analysis-and-stochastic-processes#commentsAnd yet it is heard. Musical, Multilingual and Polycultural History of Mathematics (2 vols.)
https://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In these two volumes of a thousand pages, Tonietti gives a very personal selection of the history of mathematics, and in particular the pieces where music and mathematics meet. He has some strong views on certain aspects that are not always mainstream and these are strongly put forward. One of his pet peeves is that many of his colleagues approached the subject with an eurocentric bias. Another one is that mathematics and science in general is too often considered to be an abstract universal entity besides or above the socio-cultural soil in which they are rooted. That means for example that also the language spoken and or even more so, the <em>lingua franca</em> used by the scientists and philosophers has had its influence which often goes hand in hand with cultural and religious foundations. This is his contribution to prove the Sapir–Whorf hypothesis in the case of mathematical sciences. Every culture generates its own science, and qualifications like superior and inferior or questions of precedence are often absurd. And of course, probably the main reason for writing this book, is his conviction that the mutual influence of music on the development of mathematics and vice versa is grossly underestimated. In this context he brings several contributions to the forefront that were wrongfully neglected (Aristoxenus, Vincenzo Galilei, Simon Stevin, Kepler,...). Thus there are many provocative viewpoints that some historians will disagree with, yet his arguments are extensively documented. Note however that this is not really a book on the history of mathematics, and neither is it a history of music. The reader is supposed to be familiar with music theory and should have some background in mathematics too. The book is a long plea and an extensive argumentation to underpin the viewpoints of the author, like those just mentioned. There are practically no formulas in the text, and relatively few illustrations, but the number of citations is overwhelming. These are almost always given in the original language with translation in brackets. The Chinese citations are written in pinyin, but full Chines characters are added in an appendix. This illustrates the importance that Tonietti is attaching to the language, since indeed, the translation is always an approximation and often an interpretation of what the original text is meant to say.</p>
<p>
Let's go quickly though some of the contents to illustrate what has been said above. Volume 1 contains Part I: The ancient world, and Volume 2 consists mainly of Part II: The scientific revolution, and a shorter Part III: It is not even heard.<br />
Part I treads the ancient cultures united around their language used: The Greek, Chinese, Sanskrit, Arabic, and Latin. For the Greek, music was part of the <em>quadrivium</em> and hence coexisted at the same level as mathematics and astronomy in the schools of Pythagoras, Euclid, Plato and Ptolemy. The search for harmony in music was reflected in the music of the spheres, all based on an orthodoxy of commensurability, hence integers and rationals. The music theory was developed on the basis of length of strings. The often neglected unorthodox outsiders here are Aristoxenus (who does not restrict to rationals) and Lucretius (although the latter wrote in Latin, he is Greek in spirit) who get special attention.<br />
The Chines on the other hand developed a theory of music studying the length of pipes (the <em>lülü</em>), bells, and chime stones. The cultural essense of <em>qi</em> is an energetic flow, a continuum which is apposite to the discrete orthodoxy of the Greek. Another difference is the lack of an equivalence for the verb "to be". This implies a different way of doing mathematics like for example the way in which they proved the Pythagoras theorem.<br />
Indian rules and regulations stem from religion. Precise prescriptions of how to build an altar show mathematical knowledge. Of course there was music, mostly by singing mathras, but most curiously, there is no trace of a music theory left. Musicians had `to trust their ears'.<br />
The Arabs are the saviors of the Greek culture. Most of what we know about the Greeks comes to us through them. The <em>Syntaxis mathematica</em> of Ptolemy came to us in Arabic as the <em>Almagest</em>: `the greatest' Greek collection of astronomical data. So they inherited the orthodoxy in music and mathematics from the Greek. They brought us our number system, but also terms like algorithm and algebra.<br />
Meanwhile in Europe, Latin had conquered the scientific scenary. This brought about a clash between the people, like Fibonacci promoting the introduction of the new Indo-Arabic number system against the Roman numerals. The Greek orthodoxy prevails, with Euclid being the reference for mathematics. Music theory florishes (Beothius, Guido D'Arezzo, Maurolico, Cardano,...). Tonietti gives special attention to Vicenzo Galilei, the father of Galileo, who picked up some ideas of Aristoxenus again.<br />
Besides the appendix with Chinese characters mentioned above, three other appendices are texts related to music translated from Chinese, Arabic, and Latin.</p>
<p>
In Part II chapters are named again afer the main languages (mostly European) used to disseminate scientific results. The interplay between geometry, astronomy, and music becomes explicit in work by Stevin and Galileo, but most of all in Kepler's <em>Harmonices Mundi Libri Quinque [Five books on the harmony of the world]</em> in which he completed Ptolemy's <em>APMONIKA</em> and interwaves geometry, astronomy, music and geometry, reflecting the music of the spheres. Tonietti does not shy away from critique on colleagues who had different interpretations of Kepler's work. People started using national languages besides Latin in their writings and (perhaps because of that) mathematical symbolism increases like writing music on staves was adopted before. Transcendent symbolism was mixed with music, God, and natural phenomena in work of Mersenne, Descartes, Wallis, and Huygens (Constantijn and Christiaan). The latter was not only a musician and composer, he used the newly invented logarithms and Leibnitz's differential calculus in his music theory. All this, according to Tonietti, shows that the status of music should be reinstalled as an essential element that contributed to the development of mathematics. Also Leibnitz and Newton worked on music for some time, but of course their main contribution here is the mathematical symbolism that allowed to deal with the infinite and the infinitesimal. With the use of the twelfth root of two in the equable temperament, the Pythagorean-Plato orthodoxy was definitely finished. The music of the spheres had degenerated and became intense discussions about God and creation. Again Tonietti analyzes interpretations of other historians, philosophers, or theologians on these topics sometimes rebutting them with his own vision. While in the 18th century French became the prominent language in Europe, music theory received its last flares. The French composer Rameau wrote a treatise on harmony using a theoreical basis, but he was opposed by Euler who declared tones more pleasing when they could be represented more simply. He based his analysis on prime numbers, with reminiscences of Pythagoras. Another opponent was d'Alembert discussing the harmonics of the vibrating string and also the other illuminists compiling the <em>Encyclopédie</em> had their explanation for musical terms. Still musicians wrote of science and scientists wrote of music. Entering the 19th century, Lagrange and Fourier, and later Riemann entered the discussion about vibrations, while von Helmholtz also did the physical experiments to analyze sound. Max Planck not only wrote about music but even composed some and also Einstein loved playing the violin and had correspondence with the composer Schönberg.</p>
<p>
Part III is very short. It gives a short round-up of things not discussed like Africa, Cental and South America, and more extensively the music and navigation skills of Polynesians. A last chapter briefly touches upon the science of acoustics, in which music is largely neglected. A quote from one of the final paragraphs that renders explicitly what Tonietti has allowed to emerge in his book:</p>
<p>
<em>The decision to move away from musicians and their music impoverished both natural philosophers, first of all, and then mathematicians and physicists. This influenced and facilitated the development of their research in those main directions which are known to everybody, but which continue to deserve to be criticised for their limitations and their (negative?) effects on our life.</em></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 history of music theory and its relation with mathematics, placed in a general cultural framework, and by doing so, giving a less usual, less eurocentric approach. Besides the well known historical framework, Tonietti selects and extensively discusses some less known sources and gives arguments for his critique on the views or interpretations of some of his colleagues. He comes to the conclusion that although mathematics and natural sciences has taken big steps forward in recent history, music theory was detached and has lost the interest of mathematicians, and this is a regrettable impoverishment for natural philosophers, mathematicians and physicists.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/tito-m-tonietti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tito M. Tonietti</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</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">978-3-0348-0667-0 (hbk - vols.)</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">253,34 €</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">1020</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/birkhauser/mathematics/book/978-3-0348-0677-0" title="Link to web page">http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0677-0</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li></ul></span>Wed, 13 Aug 2014 07:09:30 +0000Adhemar Bultheel45674 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols#comments