European Mathematical Society - Springer Nature/ Birkhäuser
https://euro-math-soc.eu/publisher/springer-nature-birkh%C3%A4user
enThe 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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/sherif-el-helaly" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sherif El-Helaly</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/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="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/91-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/91b12" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91B12</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/91b14" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91B14</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel49491 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematics-voting-and-apportionment#commentsThe 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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/rafael-g-campos" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Rafael G. Campos</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li><li class="vocabulary-links field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li><li class="vocabulary-links 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><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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/42-fourier-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42 Fourier analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/44a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A05</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel49490 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/xft-quadrature-discrete-fourier-analysis#commentsIndefinite Inner Product Spaces, Schur Analysis, and Differential Equations
https://euro-math-soc.eu/review/indefinite-inner-product-spaces-schur-analysis-and-differential-equations
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is volume 263 of the Birkhäuser series on <em>Operator Theory Advances and Applications</em>. It is devoted to Heinz Langer on the occasion of his eightieth birthday. Two other volumes in this series were celebrating Langer: <em>Contributions to operator theory in spaces with an indefinite metric</em> (OT106, 1995) on the occasion of his sixtieth birthday and <em>Operator theory and indefinite inner product spaces</em> (OT163, 2004) on the occasion of his retirement at the University of Vienna.</p>
<p>
The titles of these three volumes already illustrate that operator theory in indefinite inner product spaces form the focus of Langer's research. Langer was born in Dresden in 1935. After his his PhD and his habilitation at the TH Dresden he became a professor leading the institute of probability and mathematical statistics. His stay in Odessa in 1968-69 where he met M.G. Krein strongly influenced his career and his research interests. He also spent research stays at several Western universities as well, which was not obvious in the time of the DDR. In 1969 he left East Germany permanently to become a professor in Dortmund, later in Regensburg, and finally, in 1991, he accepted a position in Vienna where he stayed until his retirement.</p>
<p>
This is just a very brief summary, but Bernd Kirstein has a much longer, and richly illustrated contribution in this book. It is the ceremonial address on the occasion of the honorary doctorate awarded to Heinz Langer by the TU Dresden in 2016. He received many other prizes among which another Dr. h.c. from Stockholm University in 2015. Kirstein sketches in detail the people that were influential on Langer's career. Many of them became colleagues and friends. Among them are the most important names in the domain: Krein, Nudelman, Iokvidov, Potapov, Sakhnovich, Gohberg (who founded the OT series in 1979), Adamyan, Arov, Potapov, and many more. Kirstein describes this from his own perspective, hence the paper describes also the history of the Schur analysis group in Leipzig that he is leading together with his mathematical twin brother Bernd Fritzsche. Kirstein also illustrates the difficulties in maintaining relationships among mathematicians in an East block country and their colleagues who had left for Israel or another Western country before the fall of the Iron Curtain in 1989.</p>
<p>
A list of the publications of Heinz Langer (op to January 2017) is also included in the biographical part I of this book. A similar list in OT163 in 2006 had 171 entries, while the current one has 203 (the last one from 2017) which illustrates that Heinz Langer at his age is still an active researcher and collaborator. And the latter is what the main content of this book really is: an illustration of the influence that Langer had on other people who worked on topics related to the subjects that are close to the heart of his own research, always prepared to listen and collaborate. These topics include nonlinear eigenvalue problems, indefinite inner product spaces such as Krein and Pontryagin spaces and applications in mathematical physics.</p>
<p>
A collection of sixteen research papers, (some are longer surveys, others are short communications, all together over 420 pages) form the main part II of this volume. The titles of the papers and their authors are available on the publisher's website (see this book's meta-data elsewhere on this page) so that I do not need to repeat them here. The papers are listed in alphabetical order of the first author, but in their introduction, the editors subdivide them into five (overlapping) classes. The largest group falls under the broad title <em>Schur analysis, linear systems and related topics</em>. These papers are about Carthéodory and Weyl functions, Nevanlinna-Pick interpolation, scattering theory, L-systems and an inverse monodromy problem. In the group about <em>Differential operators, inverse problems and related topics</em> which is broad as well, we find papers related to the pantograph delay equation, and spectral and other properties for a selection of other operators. Two papers are explicitly dealing with <em>Pontryagin spaces</em> and one paper is about probability and is classified as <em>Non-commutative analysis</em>. <em>Positivity</em> is a keyword that can be assigned to almost all the papers in the volume, but it groups the remaining three texts where positivity has a key role.</p>
<p>
This volume will of course be of interest to anyone who knows or collaborated with Heinz Langer, but more generally for anyone working in one of the topics that he was, and still is, interested in, and this is a broad field as illustrated by the papers in this volume. So it may be that not all the papers are interesting for a particular reader, but in that case there is of course also the possibility to download an electronic version of a particular paper from the publisher's website, like one would do for a journal paper.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a set of papers collected to celebrate the eightieth birthday of Heinz Langer. The broad research field of Langer can be described by keywords as enumerated in the title of this book. Besides the set of selected papers that fall under these topics, there are also some biographical data like a list of publications of H. Langer and a long and richly illustrated paper by B. Kirstein sketching the career of Langer and his influence on the Schur analysis group in Leipzig.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/daniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">daniel alpay</a></li><li class="vocabulary-links field-item odd"><a href="/author/bernd-kirstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Kirstein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-nature-birkh%C3%A4user" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature/ Birkhäuser</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-68848-0 (hbk), 978-3-319-68849-7 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">116.59 € (hbk); 91.62 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">522</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links 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><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319688480" title="Link to web page">http://www.springer.com/gp/book/9783319688480</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/46-functional-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46 Functional analysis</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/46n99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46N99</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/47a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/93c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">93C05</a></li></ul></span>Tue, 13 Mar 2018 08:03:28 +0000Adhemar Bultheel48324 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/indefinite-inner-product-spaces-schur-analysis-and-differential-equations#comments