Book reviews
https://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenLipschitz Algebras. Second Edition
https://euro-math-soc.eu/review/lipschitz-algebras-second-edition
<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 second edition is much more than an update of the previous material that gives a profound insight about the main topics related to Lipschitz spaces and reflects the importance of spaces of Lipschitz functions.</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">Ángeles Prieto</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/WeaverLipschitzAlgebras.pdf" type="application/pdf; length=116934" title="WeaverLipschitzAlgebras.pdf">Weaver. Lipschitz Algebras</a></span></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/nik-weaver" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Nik Weaver</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/world-scientific-publishing-co-pte-ltd" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">World Scientific Publishing Co. Pte. Ltd.</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9789814740630</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">$126</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">462</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>
</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/26a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26A16</a></li>
<li class="field-item odd"><a href="/msc-full/48bxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">48Bxx</a></li>
<li class="field-item even"><a href="/msc-full/81q99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81Q99</a></li>
<li class="field-item odd"><a href="/msc-full/46exx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46Exx</a></li>
<li class="field-item even"><a href="/msc-full/46hxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46Hxx</a></li>
<li class="field-item odd"><a href="/msc-full/46j10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46J10</a></li>
</ul>
</span>
Fri, 19 Jul 2019 14:22:53 +0000Ángeles Prieto49570 at https://euro-math-soc.euUndergraduate Algebra: A unified approach
https://euro-math-soc.eu/review/undergraduate-algebra-unified-approach
<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 book under review consists of two parts and the prerequisites. The prerequisites cover primarily some basic Set Theory and can be skipped by a reader already familiar with the notions; however, this section is important to keep the book self-contained.</p>
<p> The two parts are very different by their content, style, and purpose which make this book quite unique. The first part, The Language of Algebra, consists of chapters Glossary of Basic Algebraic Structures, Examples of Groups and Rings, Homomorphisms, and Quotient Structures. The key novelty and unique feature of the book is that, in this first part, analogous topics on different algebraic structures are considered simultaneously (for example, the section on substructures introduces subgroups, subrings, subfields, etc., subsequently; or another example, the section on normal subgroups and quotient groups is followed by the section on ideals and quotient rings). This organization of the book is coherent and surprisingly efficient; it certainly provides a serious alternative to the traditional one where groups, rings, and fields are treated independently. It is interesting to compare the chapters by their styles; for instance, in Chapter 1, the reader can see only some simple examples that may already be familiar, but Chapter 2 consists completely of examples. Ultimately, the main purpose of Part 1 is to provide some basic language of algebraic structures in order to create necessary machinery for Part 2.</p>
<p> The second part, Algebra in Action, consists of the three chapters: Commutative Rings, Finite Groups, and Field Extensions. This is the more dynamic and mathematically appealing part of the book. The presentation is quite different from the first part, as the structure and exposition is closer to how mathematicians truly work. For example, the proof of the Fundamental Theorem of Finitely Generated Torsion Modules resembles the process in which technically involved results are split into smaller claims (in this particular case, seven claims). The book is written in a lively style and is pleasant to read. New concepts are carefully introduced, starting with an informal discussion and, occasionally, historical comments. Each section ends with exercises which progress from easy to challenging, and give a great deal of insight into the subject.</p>
<p> I highly recommend this book for a standard undergraduate algebra course, as well as to students interested in independent study.</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"> </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/matej-bresar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matej Bresar</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" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</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">2019</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">978-3-030-14053-3</div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li>
</ul>
</span>
Mon, 01 Jul 2019 14:17:10 +0000lcatalano49496 at https://euro-math-soc.euChaotic Dynamics
https://euro-math-soc.eu/review/chaotic-dynamics
<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>Fractal images can be spectacularly complex and raise some aesthetic feelings, which is why it has attracted the attention of non-mathematicians. However the beauty of the images is gained by their complex regularity, recognized as self-similarity. This is only a reflection of their mathematical complexity, that is often the result of the non-linearity, and of the iterative processes that create the curves and images. It is amazing to discover that this mazing complexity is generated by the iteration of sometimes surprisingly simple mappings that describe the dynamical system.</p>
<p>A careful analysis to unravel the complexity requires some mathematical abstraction and topological concepts that are not always very intuitive, or at least deviate drastically from familiar linear approximations. The butterfly effect has popularized the idea of the extreme sensitivity of such systems to their initial conditions in a very clear way. The difficulty to forecasting the weather is the standard example of such a system. These dynamical systems can create remarkable mathematical objects such as Mandelbrot sets and Sierpinski gaskets, or they can generate very realistic images of plants, clouds, or mountains. The theory of these systems has been thought through and has now been reduced to a subject that can be taught at even an undergraduate level to students with some interest in mathematics.</p>
<p>This book is a good example of what is possible as an introduction to this broad material of chaos, dynamical systems, fractals, tilings, substitutions, and many other related aspects. To bring all this in one volume and at a moderate mathematical level is an ambitious plan but these notes are the result of many years of teaching experience and with over 400 pages, something an be achieved.</p>
<p>Part of the trick to make the material accessible is by starting with one-dimensional dynamics, even if these are placed in a context of metric spaces, the many examples are still unsophisticated, simple iterations. So the first chapters are mainly used to introduce terminology (iteration, orbit, cycle, fix point, stability, attractor, chaos) and elementary examples (Newton iteration, logistic maps, tent family,...). Yet in chapter 3, there is an introduction to the Sarkovsky theorem (one of the highlight of the book) with a proof of a special case but the general proof follows only in chapter 12 where it takes about the complete chapter to prove it. Sarkovsky's theorem basically states that a system with a 3-cycle is much more complex and chaotic than a system with only a 2-cycle.</p>
<p>With those introductory teasers in mind, the student is introduced in the next chapters to more abstraction like metric spaces (which is a simple version of a topological space to be introduced later), and other topological concepts like the conjugacy of dynamical systems. These are needed to give a proper definition of what chaos is and to prove that some of the systems that were introduced are indeed chaotic. In some sense two systems are called conjugate if they behave similarly, and thus one can prove that a system is chaotic by showing that it is conjugate with another, perhaps simpler, chaotic system.</p>
<p>At this point the reader is ready to be introduced to fractals, the self-similarity of fractals and to fractal dimension, which is illustrated with several classical examples of Newton's method on quadratic and cubic equations and other examples like the Henon map. Most of what has been discussed to far involved real numbers and much was referring to mappings defined on real intervals. The next step is to move to complex numbers. Of course, this is where the Mandelbrot, Julia, and Fatou sets are introduced. Another fractal generating technique is a dynamical system that can be described by substitutions. This is what generates for example a Koch curve, and such curves can be described by strings (which can be interpreted as the expansion of a number in a particular basis). In a more abstract setting (Morse substitution), one chooses an alphabet (which may for example consist of the atoms 0 and 1) and a transform (for example 0 to 10 and 1 to 01) and these transforms are iterated on the growing strings that are the result of these repeated substitutions. This can be given a graphical interpretation of curves or two-dimensional tilings.</p>
<p>These substitutions trigger taking another rung up on the ladder of abstraction with the introduction of compact metric spaces and topological dynamics. The simple concepts of the early chapters can be repeated in this more abstract setting, so that a proper framework is created to deal with spaces of strings (or sequences) which can be interpreted as words in a (formal) language. Sturmian sequences (aperiodic sequences of minimal complexity) is another concept that can be studied in this context. In a concluding chapter, one of the `three pearls of number theory', the van der Waerden theorem, is proved. It says that if the integers are partitioned in a finite number of subsets, then one of these subsets must contain a sequence of arbitrary length where the numbers differ by a constant (i.e. a sequence with arithmetic progression).</p>
<p>From this brief and incomplete summary of the contents, it is clear that a lot of material is covered. The chapters are relatively short and focus each time on a coherent new concept that Goodson wants to introduce. These chapters are sliced into smaller sections that are loaded wit many simple but appealing examples, each time followed by a set of exercises. These numerous little challenges are essential stimulations to keep the student alert and to grind the material down to full understanding. Definitions and theorems are clearly indicated by typography and the proofs are fully included. The extraordinary combination of abstraction linked to simple yet appealing examples is the secret ingredient that is mastered wonderfully in this text. It is a textbook that is accessible for students that have some mathematical background. Some experience with topology and (functional) analysis would certainly help, but it is not necessary since everything is gradually introduced and there are some appendices with extras that are assumed known, but without previous exposure to these subjects it will be harder to assimilate the more abstract material. Clearly, covering everything in one course may be too much, but it is possible to skip some chapters. A clear dependency graph is provided in the introduction, but a teacher can of course make his or her own selection. One could for example skip the substitutions, or skip the chapters at the end that are more abstract, etc. Finally I should mention that mathematica code is provided at the resources for this book at the publisher's website. There you can also find an (already long) list of typos.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book is a textbook for an introductory course to simple dynamical systems, fractals, and chaos. It starts with iteration of (real) functions on an interval, later in the complex plane, or by iterating substitutions, both graphical and in strings. The level of abstraction is gradually increased as one progresses through the text. Theorems and proofs are provided along with many examples and exercises.</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/geoffrey-r-goodson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geoffrey R. Goodson</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">9781107112674 (hbk), 9781316944356 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£55.99 (hbk), 60.00 USD (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">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="https://www.cambridge.org/be/academic/subjects/mathematics/chaotic-dynamics-fractals-tilings-and-substitutions" title="Link to web page">https://www.cambridge.org/be/academic/subjects/mathematics/chaotic-dynamics-fractals-tilings-and-substitutions</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/dynamical-systems-and-ordinary-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dynamical Systems and Ordinary Differential Equations</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/37-dynamical-systems-and-ergodic-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37 Dynamical systems and ergodic theory</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/37-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37-01</a></li>
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<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
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<li class="field-item even"><a href="/msc-full/37fxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37Fxx</a></li>
<li class="field-item odd"><a href="/msc-full/37bxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37Bxx</a></li>
<li class="field-item even"><a href="/msc-full/28a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">28A80</a></li>
<li class="field-item odd"><a href="/msc-full/68w32" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68W32</a></li>
<li class="field-item even"><a href="/msc-full/52c20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52C20</a></li>
</ul>
</span>
Mon, 01 Jul 2019 11:09:27 +0000adhemar49493 at https://euro-math-soc.euFrom Servant to Queen: A Journey through Victorian Mathematics
https://euro-math-soc.eu/review/servant-queen-journey-through-victorian-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>The title of this book is an echo of a very similar title used by E.T. Bell for his book Mathematics: Queen and Servant of Science (MAA, 1997). Bell explains the unreasonable effectiveness of mathematics in the physical sciences. The meaning of this title is that mathematics, has the historic role of being a tool, a servant, for other more applied sciences, but at some point, what is usually called pure mathematics, evolved into a separate discipline, where mathematics is studied for its own right, a science that is the queen of intellect and a source of beauty. And yet, as we have experienced on several occasions already, what was originally assumed to be pure game of abstraction, devoid of any practical use, turned out later to have very practical applications.</p>
<p>In this book, Heard describes how this transition happened in England during the Victorian period (approximately 1830-1900). He sketches how mathematics, traditionally in its servant role for the other sciences, like physics, astronomy, economics, and even the invention of practical mechanical machinery, gradually appeared in its queen role of pure mathematics.</p>
<p>The book starts with sketching the situation in England in the 18th and first half of the 19th century. Since calculus was introduced by Newton and Leibniz, mathematics started to take off in a different direction. A mathematician used to be someone who applied mathematics to practical use, without being a "professional mathematician" in the modern sense of the word. It could be anyone occasionally using some computational technique, and the plural in mathematics may have referred to all these applications. But gradually mathematical knowledge became the ruler of all other sciences, just like queen Victoria was ruling the British Empire.</p>
<p>Britain was still under the spell of Newton, and the controversy with Leibniz had started an aversion for the continental approach to mathematics. There were two main scientific centres in Britain. Oxford which was considered to be the university of choice if you wanted to specialize in the classics. Theology and classics had been the dominant studies for many centuries. And the alternative was Cambridge with its system of of tripos that generated the wrangles, a prestigious honours degree in mathematics, that opened many doors to public positions. So this was the place to be if you were interested in mathematics. Perhaps because of Newton's spirit still dwelling in the premises there, it had more prestige in the eye of some beholders.</p>
<p>Pure mathematics and the mathematical profession was definitely much more accepted on the continent. There was much more exchange of ideas and results were published in professional mathematical journals. England had a tradition of publishing popular science magazines with puzzle sections, but no proper mathematical journals. The more practical notation in the Leibniz approach to calculus, which was more popular on the continent, may have given an advantage. So British mathematics lingered behind, and a lack of communication made that they had difficulties understanding the more advanced continental mathematics. In Britain, only from around 1830, a similar movement of pure mathematicians started to emerge. It became gradually accepted that the square root of a negative number could be studied for its mathematical properties, without the necessity of it representing some physical quantity. Although it was still generally belief that even pure mathematics was developed to the benefit and the advantage of science and technology.</p>
<p>Starting the London Mathematical Society has been a strong driving force in this evolution. Founded in 1964 at the University College London, it officially started a year later with August De Morgan as its first president. At first it was just a local community, but from the beginning it was keeping up a high standard for its members, as well as for the publications in its Proceedings and for the winners of the De Morgan medal that they awarded. The number or members was relatively low, although slowly growing, but its high standard eventually attracted many foreign members. In fact, here the British took a leading role, because the LMS became an example for other societies abroad (SMF, DMV, AMS....).</p>
<p>In the remaining chapters, Heard explains what it actually meant to be a "professional mathematician" for Victorians. In the chapter with the title "The pure mathematician as hero", he introduces the biographies of several British mathematicians who produced some "pure mathematics" and that usually were somehow linked to the LMS. James Whitbread Lee Glaisher who worked on number theory and who was editor of the Messenger of Mathematics (now Quarterly Journal of Mathematics); Henry J.S. Smith (known for example for the Smith normal form of a matrix, and the first to introduce the Cantor set); Percy MacMahon (combinatorics); and others. The British did not have the tradition of seminars as that was usual on the continent. Perhaps for this reason, several of the missionaries of pure mathematics were not the "heroes", the charismatic leaders, that had a school of followers like their continental counterparts had.</p>
<p>Then there is a chapter about the mathematics that was required to solve the problem of light and electromagnetism. Unlike Newton's corpuscular approach to optics, light (and later other electromagnetic quantities) seemed to be propagating like waves. But waves had to propagate in some medium that was termed aether. So there was much ado about the mathematics of the aether. This is a somewhat strange subject to be discussed extensively in the context of this book, but it definitely was a hot topic in those days, and it illustrates that in applied mathematics, Britain was not at all a backwater area. It shows that "pure mathematics" can also be produced by engineers, and non-professional mathematicians. Here we meet big names like Faraday, Stokes, Maxwell, Heaviside, W. Thomson (lord Kelvin), but also mathematicians got involved like Airy, and Clifford. The latter is known from the Clifford algebra, but he is also the one who linked gravity to non-Euclidean geometry which later inspired Einstein for his relativity theory.</p>
<p>The last two chapters are more of a social nature. With G.H. Hardy's A mathematician's apology in mind, (Hardy is the standard example of a thoroughbred pure mathematician abhorring any practical application) there is a discussion about mathematics as a profession, and what it meant to be a professional mathematician. It was quite different from a scientist. A typical British scientist was supposed to be a gentleman, free and independent, whose conclusions were not questionable. If you had a profession, then that was supposed to be at public service (which was not exactly what pure mathematics was pursuing), and you had an organisation that defended your rights (which was not the role of the LMS). So the idea came up that also a pure mathematician can be creative and explore unknown domains, which was also to the eventual benefit of society. This transformed a pure mathematician into an artist striving for beauty. Heard gives an account of the geometer George Salmon who in Nature praises the work of Cayley as an artist which was quite opposite the ideas about aesthetics of Walter Pater (an Oxford essayist).</p>
<p>The book includes many short (in line) and long (displayed) quotes and a few illustrations. There is also a long list of extra literature at the end of the book and many references at the end of each chapter to refer to the source of the quotes in the text. However if you are not an historian you can safely ignore these and keep on reading, because the story is told in a smooth and entertaining way. It is really interesting to read how long the Leibniz-Newton dispute had serious consequences, how ideas changed in about seventy years, and the important role that was played by the LMS in this process. It also illustrates that Britain was (and it still is) an island and that British were in many ways different from the rest of the world. They are probably less so today than they were in the Victorian period. Globalisation has made national identities more fuzzy worldwide, but it will take many more generations before this will be erased completely.</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 this book Heard illustrates how during the Victorian period, pure mathematicians emerged and separated from the practitioners. Scientists who started practising mathematics for the mathematics, and not for its direct application. Especially the role of the London Mathematical Society in this process is highlighted. He also explains what it meant to be a professional mathematician in those days.</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/john-heard" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Heard</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">2019</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">9781107124134 (hbk), 9781108604178 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£34.99 (hbk), £36.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">277</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/academic/subjects/mathematics/history-mathematics/servant-queen-journey-through-victorian-mathematics" title="Link to web page">http://www.cambridge.org/academic/subjects/mathematics/history-mathematics/servant-queen-journey-through-victorian-mathematics</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
</ul>
</span>
Mon, 01 Jul 2019 11:03:28 +0000adhemar49492 at https://euro-math-soc.euThe Mathematics of Voting and Apportionment
https://euro-math-soc.eu/review/mathematics-voting-and-apportionment
<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 mathematics of voting is more than just tallying the votes and the candidate getting most votes is the winner. Not that this book requires higher abstract algebra. In fact, a bit of combinatorics, the notion of a graph, the harmonic mean, and an order relation are the most advanced mathematical concepts that are used in this book and they are properly introduced when needed anyway. However, there are many different voting systems, and many criteria to define the winner. The problem is then, how to design the system in such a way that it is fair, which in turn requires to define what fair is. All this may lead to several definitions of the social model used and what kind of function is to be optimized. Hence theorems can be formulated and proved about which systems will define a winner an/or a loser and what social criterion is optimized. The proofs do not need much mathematics but they are mainly requiring precision and strict logic deduction.</p>
<p>Two chapters are devoted to two different kinds of voting. The first is a social choice model, and the second deals with yes-no voting. What holds for voting systems also holds for apportionment. In this book it is closely related to voting since it refers to the apportionment of the seats in the US House of Representatives that has to be decided every 10 years after the census or it may refer to the representation of different countries in international organizations. This book, is a textbook that is clearly addressing an audience of social science students, in particular in the US, although the general principles hold more generally of course.</p>
<p>In the first chapter, a society has to choose between two or more alternatives. A social choice procedure (or function) has to be designed that will define who is the winner or who are the tied winners and who are the losers. The problem is to define the choice procedure in such a way that a majority of the voters see their vote reflected in the result of the choice procedure. That is rather vague and thus leaves much freedom, and therefore many different possibilities, to organise the voting system and to define who are the winners and who are the losers. The chapter starts by explaining the difference between the plurality procedure (the group voting for the winner is the largest) or the majority (the group voting for the winner is larger than all the other groups together or larger that half the total number of votes).</p>
<p>Then, with this distinction in mind, the procedures can be complicated by organising several rounds, eliminating some candidates in every round, which may lead to a last round with only two candidates. Voters can perhaps give a ranking (like ranking a top three on the ballot with or without ordering them). Important is that the social choice procedure is monotone, which means that earning more votes should not turn a winner into a looser or conversely. This already gives many different systems, but when surveying the global "feelings" of the voting community towards the results, one may define a social welfare function which will define a ranking among (groups of) candidates. Important is that the relative ranking of two candidates by such a welfare function should be independent of the rest of the ranking. It should be independent of irrelevant alternatives (IIA). With all these restrictions, this approach to a voting system comes close to an axiomatic definition, which can have properties like neutrality, anonymity, or it can be dictatorial (i.e., where one voter or a group of voters can get a powerful dictatorial role). One has to change the axioms to turn the system from a dictatorial regime into an oligarchy and in such a way that it can not be manipulated. As proved by the many theorems in the text, it is difficult to find an ideal voting system.</p>
<p>In yes-no voting, the voter has only these two alternatives to vote: a yes or a no. This system is common practice when a candidate has to be selected for an important position or to accept or reject a motion or referendum. Here are fewer different procedures and hence the chapter is shorter than the previous one. Voters are grouped in coalitions. It now becomes important to define a power score of each individual voter. Since that depends on his/her position in the whole system of coalitions. Finding the power of a voter requires some combinatorial calculus. It even involves some probability (which is just a matter of counting) and magic squares (here only 3 x 3) to arrange the possibilities. It becomes more complicated when trades among coalitions are involved and when one needs to define the robustness of such a trade.</p>
<p>In the chapter on apportionment, the representation of a state can be proportional to it population, but it can only be represented by an integer number of persons. Thus some rounding (to the nearest integer) is required. Problems arise when the nearest integer is zero, or when the number of seats is larger than the number of groups to be represented (there are surplus seats to be distributed). Paradoxes can occur when a state looses a seat while its population has increased. Here again some monotonicity of the quota procedure should be imposed. Several criteria can be proposed, like for example looking at the per capita representation, that is the number of people in a state that are represented by each seat. One might for example minimize the difference so that each seat is representing (approximately) the same number of people. Other divisor procedures look at harmonic means, and there are many other possibilities.</p>
<p>The proofs of the theorems in the book are usually relatively simple, just relying on logical deduction rules applied to the definitions. It they are a bit more complicated, they are subdivided in a sequence of partial results. It should be noted that the author has definitely taken into account that his readership consists of non-mathematicians. So for a mathematical audience, the book could have used much more of the usual mathematical language and notation. Most of the text consists of examples that illustrate the possibilities of criteria that can be used and what result they give, which can be sometimes paradoxical. Each chapter has a list of exercises (answers to most of them are listed in an appendix). Thus the author has brought some mathematical rigour into a mainly non-mathematical subject, yet avoiding mathematical notation and formulation as much as possible, to transfer all this to non-mathematical students who would certainly not appreciate a more typical mathematical approach. El-Helaly has based this text on two decades of teaching experience. It is not only a textbook for his students, but it brings together a lot of material that is not easily found in this compact form and as such it will be of interest to any politician or anyone who is generally interested in the subject.</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 textbook for social science students that brings some mathematical rigour into the different voting systems and apportionment systems. Mathematical notation and concepts are avoided as much as possible, and yet there are definitions and theorems to illustrate how the different systems work with all their advantages and disadvantages.</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/sherif-el-helaly" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sherif El-Helaly</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-nature-birkh%C3%A4user" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature/ Birkhäuser</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">978-3-030-14767-9 (pbk); 978-3-030-14768-6 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€ 36.91 (pbk); € 26.99 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">279</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.springer.com/fr/book/9783030147679" title="Link to web page">https://www.springer.com/fr/book/9783030147679</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/91-game-theory-economics-social-and-behavioral-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91 Game theory, economics, social and behavioral sciences</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/91-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91-01</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/91b12" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91B12</a></li>
<li class="field-item odd"><a href="/msc-full/91b14" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91B14</a></li>
<li class="field-item even"><a href="/msc-full/91b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91B15</a></li>
</ul>
</span>
Mon, 01 Jul 2019 10:55:21 +0000adhemar49491 at https://euro-math-soc.euThe XFT Quadrature in Discrete Fourier Analysis
https://euro-math-soc.eu/review/xft-quadrature-discrete-fourier-analysis
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The Fourier transform is without any doubt an essential tool in applications such as signal and image processing. It is an integral transform that can be discretized in an algorithm called the Discrete Fourier Transform (DFT). Basically, that means replacing the integral by a numerical quadrature formula. A smart implementation turns this into a Fast Fourier Transform (FFT) procedure that has become a widely used numerical computational technique that is stable (because it is an orthogonal transform) and fast (the typical order is N log N for signals of size N). So it may be surprising that after decades of intensive use and analysis, something new is added.</p>
<p>The one-dimensional Fourier transform can be considered as a rotation over ninety degrees in an orthogonal time-frequency representation of the data. The original signal given as a function of time on a time axis is transformed into its spectral contents that is as a function of frequency and the frequency axis is orthogonal to the time axis. In optical systems, certain lens combinations can transform (rotate) the data over any angle, which is called a fractional Fourier transform (FrFT). The classical Fourier transforms is a special case of the FrFT, which in turn is a special case of an even more general linear fractional transforms (LFT). The latter kind of transforms has applications in quantum theory. The FrFT is like computing a fractional power of the ordinary Fourier transform. The extended Fourier transform (XFT) discussed in this book emerged by defining a discrete version of the Fourier transform in a slightly different way than what is done in the classical DFT. In the XFT, it is implemented as a special case of a discrete version of the FrFT. The result is a procedure that is as fast as the FFT, but slightly more accurate.</p>
<p>In the first introductory chapter, the ordinary DFT is recalled and it is illustrated that for non-periodic functions and when the N is not very large, some errors will occur as a consequence of the discretization. The DFT occurs as a unitary matrix that multiplies the data vector of the sampled signal to generate the frequency vector representation of the same signal. In this introduction also the two-dimensional transform is considered. It is illustrated that the effect of a translation in the transform, is a cyclic shift of the image, but it also has some other side effects.</p>
<p>The second chapter introduces the XFT, which requires a discussion of the Hermite functions. That are the eigenfunctions of the Fourier transform. It is shown how they feature in discretized versions of the transform, i.e., in the alternative quadrature approximation of the Fourier transform. This quadrature was introduced by the author and his co-workers in their paper A new formulation of the fast fractional Fourier transform SIAM J. Sci. Comp. 34(2) A1110–A1125 (2012). Instead of equidistant nodes, it uses the zeros of the Hermite function of order N. The (matrix representing the) XFT has its own eigenvectors, which can be seen as discretized versions of the Hermite functions. Since the Hermite functions are orthogonal, one wants to make sure that the eigenvectors are orthogonal as well. Also the even and odd properties of the Hermite functions are preserved in the discretized eigenvectors. This discrete XFT version has a differentiation matrix with interesting properties, which in fact allows to obtain also fractional powers of the differentiation operator. The core of the eventual XFT implementation is an ordinary FFT that is applied to a scaled and re-sampled data vector. The FFT result is then scaled back to the original setting. Classical issues like the discrete cosine transform, problems of sampling and aliasing, and two-dimensional versions are briefly discussed.</p>
<p>The next two chapters give a survey of many applications where the XFT can be used, mainly (partial) differential equations, and that includes usual derivatives as well as fractional derivatives (and integrals), and other fractional transforms (Laplace, Hilbert, derivative,...) and the generalization to fractional Fourier transforms and linear canonical transforms. The two appendices describe the mathematica and the matlab codes for the implementation of the XFT.</p>
<p>The book gives a concise survey of problems with classical DFT and introduces the XFT as an alternative. The performance is clearly illustrated with many applications and above all, the code to compute the XFT (and related algorithms) are provided both in mathematica and matlab, so that it is possible to immediately start experimenting with the methods. Recommended for everyone who uses Fourier transforms in a computational context and wants to learn about its extended XFT alternative and the theory behind it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">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 extended Fourier transform (XFT) is, like the discrete Fourier transform (DFT), an approximation of the continuous Fourier transform by a quadrature that approximates the integral of the transform. The XFT first published in 2012 by the author and his co-workers differs slightly from the DFT by an appropriate choice of the nodes of the quadrature. The result is a discrete transform that is as fast as the DFT, but that performs slightly better. The method is explained, several applications illustrate the method and the codes in mathematica and matlab are provided.</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/rafael-g-campos" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Rafael G. Campos</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/springer-nature-birkh%C3%A4user" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature/ Birkhäuser</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">978-3-030-13422-8 (hbk); 978-3-030-13423-5 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€ 88.39 (hbk); € 51.16 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">248</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.springer.com/gp/book/9783030134228" title="Link to web page">https://www.springer.com/gp/book/9783030134228</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/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li>
<li class="field-item even"><a href="/imu/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial Differential Equations</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier 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/42-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42-02</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li>
<li class="field-item odd"><a href="/msc-full/44a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A05</a></li>
<li class="field-item even"><a href="/msc-full/65txx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65Txx</a></li>
</ul>
</span>
Mon, 01 Jul 2019 10:50:07 +0000adhemar49490 at https://euro-math-soc.euCalculus Simplified
https://euro-math-soc.eu/review/calculus-simplified
<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>Oscar Fernandez is the author of <a target="_blank" href="/review/everyday-calculus-discovering-hidden-math-all-around-us">Everyday Calculus: Discovering the Hidden Math All around Us</a> (2014) and <a target="_blank" href="/review/calculus-happiness-how-mathematical-approach-life-adds-health-wealth-and-love">The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health</a> (2017) which are extracurricular texts to illustrate the applications and usefulness of calculus and mathematics in general. With the current book he provides the actual lecture notes, or, as he defines it: a calculus supplement. He starts the book with an extensive commercial, as if he has to justify that he is adding "yet another calculus supplement" to what is already available. His main argument is that he uses the "goldilocks approach", i.e., he provides everything in "just the right amount": just the right level of abstraction, details, insight, intuition, applications, number of pages,... Fernandez characterizes his book as a goldilocks average of a proper calculus text, a calculus supplement, and a calculus teacher. Thus the book provides all the goodies of the usual calculus texts in combination with the help that a teacher would add to it: highlighted text, frames, summaries, take home messages, tips and tricks, many exercises and solutions, and a PUP <a target="_blank" href="https://press.princeton.edu/titles/13351.html">website</a> for interactive content.</p>
<p>All this information is what you usually expect at a publisher's website or on a flyer advertising the book, so it is a bit strange to read it in a book that you already have purchased, but it has the advantage that you are well informed about what and what not to expect, even before you start reading. In fact, "reading" is not the right verb as Fernandez correctly advises the reader to "work through" the book rather than just read it. The level is elementary, somewhere between pre-calculus and first year calculus. The difference between both approaches is static versus dynamic as Fernandez explains: for example pre-calculus just gives the formula for the volume of a sphere (static), while calculus explains the formula as the limiting value as a sum of discs that become infinitesimally thin (dynamic). The infinitesimal concept (something becoming infinitely small without ever being zero) is essential for practical concepts such as instantaneous speed, slope of a curve, and area of a region, which relate to the calculus concepts of limit (the foundation of it all), continuity, derivative, and integral. These are the traditional mathematical topics to be expected but Fernandez manages to cover this in only 109 pages (excluding exercises). The exponential, logarithmic and trigonometric functions are optional, so that everything can be treated using only algebraic functions. At many places a section entitled Transcendental Tales is inserted where the general theory is applied to these transcendental functions. If you skip these sections, then only 89 pages suffice. In fact the main text ends after 158 pages (that includes the exercises). This means that about one third of the main text consists of exercises. The rest of the book is mainly a set of appendices surveying all the material that is supposed to be known in advance (algebra, geometry, functions), answers to exercises, additional applications, bibliography, and index, all together these add an extra 90 pages.</p>
<p>Elementary as the subjects may be, what has been treated has all the rigour that one would expect. There are definitions and there are theorems, but proofs are skipped or hidden in an appendix or in an exercise or it is replaced by several illustrating examples. Examples and applications are main ingredients of the text. Especially optimization as an application of differentiation gets a separate chapter and is rather well elaborated.</p>
<p>I do not think this is suitable for mathematics or engineering students. These definitely need more depth. Unless they are at a pre-calculus level and are so eager that they want to learn more calculus in advance on their own. However, for students that will need some mathematics, and are required to take a calculus course, even if they do not like it, then this book is a nice approach, indeed for the reasons given by the author in his introduction. Many glossy calculus books of up to a thousand pages are a major overkill. The many examples here should stimulate intuition before rigour. Not including the proofs is a practice that has gained popularity, probably not to the liking of mathematicians, but it might help students that are abhorred by the required formal and technical details of a proof. The PUP website is not spectacular but it works nicely and smoothly. There the "dynamic" approach of calculus is lively illustrated by the animations. The text can be "personalized" by skipping for example the sections Transcendental Tales and/or some applications. It will require the guidance of a teacher to help make the proper decisions. Making a selection is however something that one can do with every text. The most interesting property of this book is in my opinion the conciseness (which need to be taken with a grain of salt, since it depends on how much one is prepared to skip, and how many of the exercises are considered to be essential). The abundance of examples instead of proofs is another distinct property, but I believe that also exists in some other texts (or one could just skip the proofs if they are present). Thus I believe there is indeed a wide potential readership for this text since the chosen ones among the students that love mathematics with all its rigour, proofs and technicalities is still a minority with respect to all the students that are submitted to a calculus course because they just need a minimal amount of 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>This is a calculus supplement using a goldilocks approach to help the student in his or her transition from pre-calculus to a first year calculus course (limit, continuity, derivative, integral). Most obvious characteristics of the text: concise, many examples and applications, many exercises, avoiding proofs, possibility to avoid transcendental functions (i.e., exp, log, and trigonometric functions).</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-fernandez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Oscar 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">2019</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">9780691175393 (pbk), 9780691189413 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 19.00 (pbk), £ 14.99 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">272</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://press.princeton.edu/titles/13351.html" title="Link to web page">https://press.princeton.edu/titles/13351.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/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A35</a></li>
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<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/97i10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I10</a></li>
<li class="field-item odd"><a href="/msc-full/97i40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I40</a></li>
<li class="field-item even"><a href="/msc-full/97i50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I50</a></li>
<li class="field-item odd"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li>
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</span>
Mon, 01 Jul 2019 10:43:37 +0000adhemar49489 at https://euro-math-soc.euBernard Bolzano: His Life and Work
https://euro-math-soc.eu/review/bernard-bolzano-his-life-and-work
<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>Bernard Bolzano (1781-1848) is well known among mathematicians. He was however a philosopher and logician in the first place. As a product of the Enlightenment, he had very modern, almost revolutionary, ideas about science, church, and society. This was however not appreciated by the ruling political upper class, since these ideas endangered their absolute power over people.</p>
<p>The first author of this book is specialised in the philosophy of mathematics and the second is a philosopher, both ardently interested in Bolzano and his work. They started working together at the end of the 1990's, which resulted in this massive book project. They could complete it because the majority of the work of Bolzano became available as it is compiled in <a target="_blank" href="https://www.frommann-holzboog.de/editionen/20"><em>Bernard Bolzano: Gesamtausgabe</em></a> (Fromann-Holzboog, 1969ff), which is still an ongoing project since 102 volumes are currently (2019) published, and 29 are still in preparation.</p>
<p>Bernard Bolzano was born in Prague in Bohemia, which the authors want to explicitly distinguish from (modern) Czechia, and not only for geographical reasons. More important in this context is its political history. The fact that this region was part of the Holy Roman Empire, and later of the Austrian Empire with oppressive rulers resulted in a smoldering anti-catholic and anti-German sentiment. By the end of the 18th century, the Czech language and culture experienced a revival as a consequence of widespread romantic nationalism, and Bolzano was an exponent of this movement.</p>
<p>Bolzano was raised in a pious catholic family (12 children but only two reached adulthood). His health has been weak throughout his life. Although his father wanted him to become a merchant like he was himself, Bolzano started studying mathematics and theology and became a catholic priest in 1804. He started teaching the philosophy of religion at the Charles University in Prague where he became a popular teacher, loved by his students. However, Bolzano's rather liberal opinions about the church and about politics, made him not popular among his superiors. The Austrian government however was suspicious of his ideas spreading among his students. They put some pressure on the local authorities, which led to Bolzano's suspension as a professor in 1819 and he was put under house arrest. Later he was also tried by the church and, since he refused to recant his "heresy", he retired from his chair and spent the summer with his friends the Hoffmanns outside Prague, and only returned during the winter period. This gave him ample time to work on his mathematical and philosophical texts. However many of his writings had only limited distribution because of his conviction official publication was almost impossible. Other texts remained unpublished until 1962 or even much later. This explains why he had relatively little direct impact and some of his original ideas were later rediscovered by others.</p>
<p>Among his major publications were his Rein analytischer Beweis. Here he tried to remove infinity from calculus, and hence had to define limit, continuity, derivative, and convergence without it. In this context we find his treatment of the intermediate value theorem and a definition of a Cauchy sequence. So he developed these ideas some years before Cauchy. He used the Bolzano-Weierstrass theorem many years before Weierstrass did. His Grossenlehre is his attempt to start setting up a logical foundation of mathematics, which he generalized in his Wissenschaftslehre to a complete theory of knowledge. In his Paradoxien des Unendlichen (Paradoxes of the infinite) the word "set" is used and he also has the bijection between the elements of an infinite set and an infinite subset. Many other results stayed unpublished for a long while until in the 20th century, and hence are attributed to other mathematicians. For example in his discussion of continuous functions, he had some fractal-like sawtooth monster-function, that became known as a Weierstrass function, and there are several other examples discussed in the book.</p>
<p>Those strict mathematical subjects in this book are discussed in a relatively short chapter, but the other aspects of Bolzano's work, mostly philosophical, are even more thoroughly discussed in the other chapters. That includes his opinions about ethics, political philosophy, philosophy of religion and the catholic church, about aesthetics and a science of beauty, as well as about ontology and metaphysics. A large chapter is devoted to logic and another one to his Wissenschaftslehre (Theory of science). Concerning logic, Bolzano was of the same idea as Leibniz who was convinced that logic had an important part in the philosophy of science, and he is at the origin of the interaction between logic and mathematics. This was opposed to Kant, who thought there was no role for logic in philosophy, and that what was used in mathematics was a completely different thing. Logic became a core aspect of Bolzano's philosophical work and it is a hidden precursor of what later became known as analytical philosophy, for which Gottlog Frege is usually considered to be the founding father.</p>
<p>His Theory of Knowledge, just like his logic, starts from the concept of truth by which he means the "truth in itself". Several such propositions do exist outside our mind. We do not have to reason about these. This sounds Cartesian, but he considered it not to be fundamental but as a way to refute scepticism. Then these propositions are brought in relations, inductions, etc., which is the logic in a narrow sense. Only after eleven hundred pages he comes to the theory of knowledge: ideas that can be conceptual or empirical and they are subject to judgement. New propositions can be obtained by logic, probability, or what is called an entailment relation (a set of propositions can entail a new proposition).</p>
<p>This book contains a thorough analysis of the philosophical work of Bernard Bolzano. Much of his work, has for a long time been unpublished, so that this book comes at a good time with the complete and very extensive scientific and philosophical production of Bolzano becoming more generally available in the Gesamtausgabe. The discussion of his mathematics and of his ideas on all aspects of science and society, are strictly documented with many citations from the original sources (in English translation) and from other authors that have studied his work. It is also placed in relation with other philosophers and mathematicians. This book is in the first place about the philosophy (of mathematics) as it can be found in Bolzano's writings. The strict mathematics in a narrow sense are omnipresent but at a second level.</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>After a sketch of the time and the life of Bernard Bolzano, the book is mainly an analysis of his work, which is mostly of a philosophical nature, even the chapter on mathematics. The chapters on logic and on his theory of knowledge are the most extensive ones, but also his philosophical ideas about religion, ethics and aesthetics, ontology and metaphysics, and politics, are amply discussed.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/paul-rusnock" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paul Rusnock</a></li>
<li class="field-item odd"><a href="/author/jan-sebestik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jan Sebestik</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</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">978-0198823681 (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">£ 80 (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">704</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/bernard-bolzano-9780198823681" title="Link to web page">https://global.oup.com/academic/product/bernard-bolzano-9780198823681</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/03-mathematical-logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03 Mathematical logic and foundations</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/03a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A30</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/03a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A05</a></li>
<li class="field-item odd"><a href="/msc-full/03a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A10</a></li>
<li class="field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
<li class="field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li>
</ul>
</span>
Mon, 03 Jun 2019 09:01:48 +0000adhemar49422 at https://euro-math-soc.euThe Universe Speaks in Numbers
https://euro-math-soc.eu/review/universe-speaks-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>The title of the book may suggest that this is about numbers, but there are nu numbers in the book, at all. So the subtitle: "How modern mathematics reveals nature's secrets", is a better description of the content. Because the book is about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" as Eugene Wigner formulated it back in 1960. Especially with physics, there has been a close and successful interaction with mathematics. But Farmelo explains that this is not so unreasonable and in fact it goes also the other way around since there is also "A Reasonable Influence from Theoretical Physics on Mathematics". Perhaps, there can even be a superstructure that has mathematics and nature as two of its realizations that we humans, with our limited intellectual capabilities, are able to experience, without yet understanding the superstructure, an idea that has been proposed before by Max Tegmark in <a target="_blank" href="/review/our-mathematical-universe-my-quest-ultimate-nature-reality">Our Mathematical Universe</a>.</p>
<p>Clearly both communities, physicists and mathematicians, have their own culture. Mathematics expands in the minds of mathematicians that are driven by abstraction and a mathematical result is true and remains true beyond discussion after it has been proved once and for all. The (traditional) physicists are driven in their urge to explain nature. They care somewhat less about rigour and their models are inspired by observation, and perhaps less by strict abstract deduction. Their models are accepted if they are confirmed by nature itself after appropriate experiments, but acceptance of the model is only guaranteed until it is contradicted by new or better experiments. The latter situation seems to have changed for current theoretical physics that has moved to the mathematical approach.</p>
<p>Since antiquity mathematics and physics have developed in parallel. Even mathematics developed and progressed often driven by practical physical problems. Yet, Greek philosophers already discussed whether mathematics was created by humans or provided by nature and left for humans to be discovered. Farmelo sketched in the first half of the book this on-off relationship between mathematics and physics up till about the 1970's. He tells the history by staging the people who have contributed to the major steps in the evolution of both mathematics as well as physics.</p>
<p>Of course Newton and the scientists of the Enlightenment who envisioned a mechanical clockwork world. It was promoted by Laplace that if all the details of the current state age given, then this would allow to predict the future perfectly. Much of this resided on an atomic idea about the world consisting of particles subject to forces.</p>
<p>The electromagnetic theory of Maxwell introduced the concept of a field. His theory is condensed in the Maxwell equations named after him, but written down by Heaviside. The "beauty" and symmetry in the equations were a source of inspiration for later developments.</p>
<p>Gravity, the source of inspiration for Newton, coupled to a geometric vision, and thought experiments, were the instruments used by Einstein to develop his special relativity theory, only to be confirmed afterwards by practical experiments. At first he was not convinced that physics required advanced mathematics but he had to abandon this idea when he got in trouble for the development of his general relativity theory. Early twentieth century were turbulent times, shaking the foundations of both physics and mathematics. This was promptly followed by a steep increase in our understanding of the world.</p>
<p>Heisenberg and Schrödinger developed quantum mechanics, but it was Dirac who could link this new model, which inherently involved uncertainty, with classical world view of Laplace. He endorsed Einstein's revised view that mathematics is unavoidable for the development of physics. But then, during the war and in the post war period came, what Farmelo calls, "The Long Divorce" where mathematics and theoretical physics each went their own way for about two decades. However, physicists in search for their unifying theory got stuck and were confronted with a zoo of subatomic particles. Feynman made some progress, and Yang and Mills tried to generalize the symmetry of the Maxwell equations, but Freeman Dyson in his 1972 talk to the AMS pointed to the missed opportunities because the physicists were not aware of the most recent developments in mathematics and mathematicians were not interested in physics. Until in November 1974 (hence called the November Revolution) the psion (J/ψ meson) was discovered, a particle that lived much longer than other elementary particles, and gauge theory became the common interest of physicists and mathematicians, also because of the Atiyah-Singer theorem which had shown the power of differential geometry in explaining subatomic quantum mechanics. The Standard Model was realized in the 1970's.</p>
<p>This is where the first part of the book ends, surveying about 3 centuries from Newton till the Standard Model. The second half deals with the 4 decades that follow. Veneziano had written down the formula forming the model for string theory on a napkin already in 1968. It was however abandoned because quantum chromodynamics and quantum gravity (the quantum mechanical approach to study gravity near black holes) had stolen the hearts and minds of physicists. But strings were later reinstalled as the road to take for a Theory of Everything. String theory introduced extra dimensions because supersymmetry is the only way to extend the symmetry between space and time as in Einstein's special relativity theory. Farmelo continues to illustrate the intense interaction between theoretical physicists and mathematics. They mutually helped each other to make progress, with Witten, Deligne, Seilberg, Penrose, Arkani-Hamed and many others as main contributors. However the Large Hydron Collider (LHC) of CERN didn't provide the many particles that were predicted by the theory, the detection of the Higgs boson in 2012 being the last success.</p>
<p>However the state of affairs have brought mathematicians and theoretical physicists closer together than ever before. They are collaborating in the new emerging field of mathematical physics. In the last chapter Farmelo concludes his arguments defending what he has illustrated in this book: it is predestined and the fate of mathematics and physics to work together. He even makes some predictions about ideas that will stand the test of time like space and time are not fundamental but are aspects of a more fundamental concept. He also believes that supersymmetry will be verified experimentally thereby affirming the beauty of mathematics to be basic. And he has a few more like those. So with this book he contradicts Sabine Hossenfelder who in her book <a target="_blank" href="/review/lost-math-how-beauty-leads-physics-astray">Lost in Math</a> complains about the state of affairs that physics is at a dead end caught up in theoretical imaginations remote from reality and just because this vision happens to be mainstream, it absorbs all the research money. Who is right? Time should bring the answer, but in my opinion it doesn't look like it will come in the very near future, despite what mathematicians and physicists may think or hope for.</p>
<p>It is remarkable that this book, that is from the first till the last page about mathematics and mathematical physics, has no formulas (well almost none, I counted five very simple ones and that includes E = mc²). Farmelo does not go into technical details in the sense that he avoids confusing the reader with technicalities. If that reader does not know the exact meaning of the terms (gauge theory, quark, gluon,...) then it does not really harm the basic story that he wants to tell and it does not hinder reading on. He keeps the reader hooked. This alone is a tour de force. Farmelo's style is very entertaining, describing the moments and the circumstances when it was realized that some breakthrough had been found. This is only possible because he interviewed the people involved or he himself was a witness of the events in the second half of his book. He also frames the time and the setting by referring for example to the fact that some physics event took place "in the year that the Beatles produced their first LP" or "when nearby a large group of music lovers flocked together" (referring to Woodstock), or "a few weeks after Obama was inaugurated". He discusses throughout the book the, sometimes subtle, interplay between the mathematics and the physics, and he is as generous about mathematics as he is about physics. I also appreciated how he analyses the important lectures of Dyson, Witten, and others where they made some important statements about the state of affairs. He claims that it is the book he has been writing since his childhood, and I can believe that. A recommended read, and Hossenfelder can be a comparable complementary read to keep the balance.</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 describes without being technical, the close collaboration between mathematics and physics in the course of history. In particular the mutual influence of mathematics and theoretical physics since the 1970's till now.</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/graham-farmelo" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Graham Farmelo</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/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic 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">2019</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">978-0465056651 (hbk), 9781541673922 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.00 (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">336</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.basicbooks.com/titles/graham-farmelo/the-universe-speaks-in-numbers/9781541673922/" title="Link to web page">https://www.basicbooks.com/titles/graham-farmelo/the-universe-speaks-in-numbers/9781541673922/</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>
<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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum theory</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
<li class="field-item odd"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li>
<li class="field-item even"><a href="/msc-full/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</a></li>
<li class="field-item odd"><a href="/msc-full/83-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-03</a></li>
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</span>
Mon, 03 Jun 2019 08:43:09 +0000adhemar49420 at https://euro-math-soc.euIntroduction to Malliavin Calculus
https://euro-math-soc.eu/review/introduction-malliavin-calculus
<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 Malliavin calculus is an infinite-dimensional differential calculus on Wiener space, that was first introduced by Paul Malliavin in the 70’s, with the aim of giving a probabilistic proof of Hörmander’s theorem. This theory was then further developed, and since then, many new applications of this calculus have appeared.</p>
<p>This textbook provides an introductory course on Malliavin calculus intended to prepare the interested reader for further study of existing monographs on the subject. Moreover, it contains recent applications of Malliavin calculus: density formulas, central limit theorems for functionals of Gaussian processes, theorems on the convergence of densities, noncentral limit theorems, and Malliavin calculus for jump processes.</p>
<p>Recommended prior knowledge would be and advanced probability course.</p>
<p>The book is organized as follows. Chapters 1 and 2, give an introduction to stochastic calculus with respect to Brownian motion. Chapters 3, 4, and 5 present the main operators of the Malliavin calculus. Chapters 6, 7 and 8 are devoted to different applications of the Malliavin calculus for Brownian motion. Chapter 9, 10 and 11 develop Malliavin calculus for Poisston random measures.</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"> </div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-nualart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Nualart</a></li>
<li class="field-item odd"><a href="/author/eulalia-nualart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eulalia Nualart</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">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107039124 1107039126 9781107611986 1107611989 </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">236</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/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</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/49-calculus-variations-and-optimal-control" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49 Calculus of variations and optimal control</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/stochastic-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stochastic Analysis</a></li>
</ul>
</span>
Mon, 20 May 2019 08:40:10 +0000MarFenoy49364 at https://euro-math-soc.eu