The European Mathematical Society

With the format of this book (8,5 x 9,5 inch) with wide margins often containing images, and with numerous larger illustrations (including inserted color plates) one immediately realizes that this is not just an ordinary book about mathematics or a dull enumeration of data about historical buildings and architects. This may well be considered an art book. Two narratives are marvelously intertwined over a time span from the cave dwellings to the modern architecture of the 20th century. Architecture is of course one of the narratives and the other one is about the evolution of mathematics, both placed in the appropriate social and cultural environment. The two stories are connected in a very natural way. It would be difficult to read only the part about architecture and skipping the mathematics or vice versa. Of course geometry and symmetry play an important role, but also the forces in the structure that need to be diverted to keep everything stably in place is easily explained using modern vector calculus. The mathematics is often the mathematics needed for describing the statics of the structure. It is amazing to realize that many of the historic buildings were constructed before the mathematics was available to do all the appropriate computations and yet survived for many centuries.

The mathematics are however kept very elementary. Some basic calculus is amply sufficient. In fact the last chapter, which is the most advanced on a mathematical level is an introduction to the necessary calculus to compute for example the length of an arc, the center of mass or the moments. On the other hand, also architecture has its own vocabulary that not every reader may be familiar with. The author has solved this by including a glossary of architectural terms. Every chapter is followed by a number of diverse problems and discussions. These can be mathematical (a proof or a computation) but can as well be a question about the forces in an arch by looking at a picture of a cathedral, or open questions such as commenting on a shape appearing in a figure

In the first 6 chapters successive historical periods and some of their characteristic architectural monuments are discussed. Sometimes the mathematics are a pretext to discuss properties of the buildings, sometimes it is the other way around. Never is it a dull enumeration of historical or architectonic facts, and neither is it mathematics, just for the sake of mathematics. This is not an encyclopedia and the author has definitely made a choice in which buildings to include and which not. Most of the obvious ones, i.e., the ones known by the majority of the readers find some place. The choice is however mostly confined to the ancient cultures of the Mediterranean area and later Europe with few exceptions as may be clear from the enumeration below.

To give an idea about the contents, I include a quick summary of the first six chapters (without being exhaustive).

1. Human awakening (Stonehenge, papyrus, Egypt, theorem of Pythagoras)
2. Greek geometry and Roman engineering (Parthenon, round arches and forces that keep them in place, Colosseum, Pantheon)
3. Architecture inspired by faith (Hagia Sophia, Dome of the Rock in Jerusalem, Mosque of Cordoba, Gothic churches, Milan, Venice, Pisa, and the ways of deviating the forces from the dome or the roof to the walls, and of course the symmetry of Islamic ornaments)
4. Transition in mathematics and architecture (conic sections, coordinate systems and algebraic geometry, the dome of Florence)
5. The renaissance (music, harmonics, Venice, Brunelleschi and the Basilica di Santa Maria del Fiore in Florence, Da Vinci, Michelangelo and Bernini with St. Peter's in Rome)
6. A new architecture (domes of St. Paul's in London and the Capitol in Washington, analysis of structures, the Sydney opera house, Guggenheim museum in Bilbao, Sagrada Familia in Barcelona)

It is not only a picture book but also a book that is a pleasure to read from cover to cover and I can imagine that after reading it, after a while one will pick it up again and again to just enjoy the illustrations or reread sections and chapters.

Julian Havil does it again. Like his previous book Gamma: Exploring Euler's Constant (2009) in which a key role was played by Euler's constant 0.57721..., this epic story follows the trace of the irrationals throughout the centuries. His pleasant narrative style, larded with, citations, anecdotes and historical facts keeps your attention going, following the attacks of mathematicians fighting like knights trying to unveil the secrets from an ever metamorphosing irrational dragon.

However, it is not just a superficial story leading you through the history of mathematics from the Greek till the present, it is also a serious book about mathematics with proofs and derivations. Some of them are easy at an undergraduate level, but others are at a much more advanced level.

Starting with the Greek where Pythagoras' theorem bears the seed of the first irrational √2, which was called incommensurable in those days. Square roots and surds kept playing an important role while algebra and calculus were being developed. It was however by the introduction of continued fractions that a new tool came available to stir the world of irrationals. Lambert proves the irrationality of $\pi$ (in 1761) (although a modern proof was only published in 1947 by Ivan Niven) and also Euler's number e enters the scene (proved to be irrational by Fourier in 1815). More intrinsic studies of the irrationals were undertaken. What combinations of irrationals are irrational? Niels Abel in 1929 and Evariste Galois in 1930 prove that there are algebraic numbers not expressible in radicals. One of the remarkable achievements of the 20th century is Apéry's proof of the irrationality of ζ(3) and how well it can be approximated. Enter transcendental numbers (Liouville's number $\sum_{r=1}^\infty 10^{−r!}$ seems to be the first and Hermite proved in 1863 that $e^r$ is transcendental). In fact a whole hierarchy of irrationality emerges (starting a search for the "most irrational number"). Then Cantor, Dedekind and others take the scene and come to an axiomatic definition of the reals.

Most results that can be proved in a reasonable number of lines are included. Sometimes it is a reconstruction of the historical proof, sometimes it is a more modern one. As the reader advances in the book, the proofs become on the average more advanced as well. Some items are included as an appendix (e.g. the spiral of Theodorus, equivalence relations, continued fractions, mean value theorem, etc.)

As a conclusion it is a book that can be warmly recommended to any mathematician or any reader who is generally interested in mathematics. One should be prepared to read some of the proofs. Skipping all the proofs would do injustice to the concept, leaving just a skinny skeleton, but skipping some of the most advanced ones is acceptable. The style, the well documented historical context and quotations mixed with references to modern situations make it a wonderful read.

In an historical perspective, the author recalls how paradoxes were the impetus for leaps of improvement in the evolution of mathematics. The $\sqrt{2}$ introduced the irrationals beyond the rationals, 0 was the missing link between positive and negative numbers, $\sqrt{-1}$ introduced the complex numbers beyond the reals, $\infty$ allowed to study divergent series, and the quaternions $\mathbb{H}$ (for Hamilton) related to the space-time computations of relativity theory. The latter can be extended to octonions $\mathbb{G}$ (for Graves) and this is the start of a sequence of Dickson algebras: $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{G}\subset\cdots$, or more generally a sequence $A_k=A_{k-1}\times A_{k-1}$ of algebras equipped with a recursively multiplication. This definition doubles the dimension in each step so that $A_k$ has dimension $2^k$ (Dickson 1919). As k increases, more and more classical properties of the multiplication are lost: the square may be negative in $A_1=\mathbb{C}$, commutativity is lost in $A_2=\mathbb{H}$, associativity is lost in $A_3=\mathbb{G}$, zero divisors occur, etc. The sequence of Dickson algebras is what Chatelin calls Numberland where hypercomputation takes place. Working in $\mathbb{R}$ is what Chatelin calls thought or one-dimensional thinking, but moving to $\mathbb{C}$, this becomes intuition or two-dimensional thinking. Together with $\infty$ they form Reason $ = \{\mathbb{R},\mathbb{C},\infty\}$.

The first chapters explore the calculus, i.e., all the computational rules in $A_k$. Leaving the strict computational conventions of familiar grounds gives the freedom to choose on how to define or compute things. Thus loosing properties for increasing k means opening up for many more possibilities. For example classical causality is based on the ordering in $\mathbb{R}$. A new linear concept of causality or derivability is given via a particular linear map (a derivation) in a Dickson algebra and the nonlinear core of the Dickson algebras is the part that is out of reach of all possible derivations.

The next chapter explores the norm and the singular values decomposition of the multiplication maps $L_x$ and $R_x$, i.e. left or right multiply with an element $x\in A_k$. The different possible definitions of a norm in the Dickson algebras give rise to different geometries. Complexification of a Dickson algebra is the generalisation of $\mathbb{C}=\mathbb{R}+i\mathbb{R}$, i.e. $A_k=A_{k-1}\times 1 \oplus A_{k-1}\times\tilde{1}_k$ where $\tilde{1}_k$ is the hypercomplex unit of $A_k$. It is illustrated and related to the dynamics of Verhulst's logistic equation.

The Dickson algebras have a dimension that is a power of 2. For the algebra for which the dimension is not a power of 2, one needs to resort to addition instead of multiplication. As an application the spectrum of the perturbed matrix $A(t)=A+tE$ is investigated for varying $t\in\mathbb{C}$.

When Dickson algebras are defined over the integers or in $\mathbb{Z}_r$ (in particular r = 2), several possible applications open up like number theoretic problems, floating point representation (the probability of the first digit in the representation, known as the Borel-Newcomb paradox), Sharkovski's theorem and the ordering of the natural numbers, etc.

More number theoretic applications are possible in the first four of the Dickson division algebras mentioned above ($A_0=\mathbb{R},\ldots,A_3 \mathbb{G}$), because they have no other zerodivisor than zero. When these algebras are considered as rings (addition and multiplication), then as an application, number theoretic theorems of (2, 4, and 8) squares can be analysed, i.e., which natural numbers can be written as a sum of 2,4, or 8 squares. This results in a quest for possibilities of 8-dimensional arithmetic.

Besides the discrete/continuous dichotomy, there is also the real/complex dichotomy. How these different dichotomies interact in computation is illustrated in the next chapter which analyses two applications. The first is about the relativity of the concept of inclusion. Think of fuzzy sets, but also about the dynamics of chaotic systems. The second one is about Fourier analysis and complex signals.

The computation of e.g. an SVD, which we know as a concept in linear algebra, leads to paradoxes when it is applied in a nonlinear environment of nonassociative Dickson algebras ($k \ge 3$). Classical logic is deductive and tries to avoid any paradox (Russel, Turing). Chatelin however sees these paradoxes as an opportunity to leave the classical deductive logic and escape to a more organic logic. That is a logic that allows to reason about hypercomputing. An alternative (organic) representation of complex numbers and higher dimensional complex vectors is given and it is illustrated how these are used in computation.

The concluding chapter is about Euler's $\eta$ function. This is explored as a tool to give weight, or meaning, or depth to hypercomplex numbers, or as Chatelin calls it, organic intelligence.

This is a book unlike anything I have read before. The potential reader who is looking for philosophical aspects should be warned that there are hard mathematics involved, but the mathematician should be warned as well, that he/she should be willing to abandon familiar grounds and follow the ideas and philosophy behind the mathematical exposition. This book is almost a paradox in itself. The reader is guided around some of the phenomena at the boundaries of Numberland which is much like an experience Alice must have had when she explored Wonderland. I do not think the book will become the computational bible of the future, but as an exercise in out-of-the-box-thinking it has overwhelmingly succeeded. It is far from giving a solution to all problems posed by nonlinear computational problems. It is not even giving a definitive solution to the most elementary partial problems. As Chatelin writes herself, the right choice to make among the many possible choices that can be made in higher dimensional Dickson algebras, can only be validated by experience.

This is a selection of papers presented at the the conference Blaschke products and their applications held at the Fields Institute in Toronto July 25-29, 2011. It is volume 65 of the Series Fields Institute Communications.

Blaschke products were introduced by Blaschke in 1915 to study analytic functions in the complex unit disk. These (infinite) products are examples of inner functions, i.e., functions analytic in the unit disk with boundary value 1 a.e. A similar definition exists for functions analytic in the upper half plane. Blaschke products may collect the zeros of of functions in Hardy spaces and are therefore an essential element of inner-outer factorization of these functions.

Blaschke functions and their derivatives appeared in several applications like approximation theory, solution of extremal problems, operator theory, differential equations, geometry, and even computability theory. Some of these applications are treated in the contributions of these proceedings. Among the fifteen papers are a few survey papers which are usually longer, but most of them are shorter research-type papers. The titles with a one-sentence description are as follows. Even if the title does not mention it explicitly, all these papers involve Blaschke products one way or another.

1. Applications of Blaschke Products to the Spectral Theory of Toeplitz Operators (30 pp.) (Sergei Grudsky and Eugene Shargorodsky)
   A survey-type paper with some open problems.
2. Approximating the Riemann Zeta-Function by Strongly Recurrent Functions (12 pp.) (P.M. Gauthier)
   Research paper. The approximant shares properties of the zeta-function.
3. A Survey on Blaschke-Oscillatory Differential Equations, with Updates (56 pp.) (Janne Heittokangas)
   A survey of the study of the differential equation f"+A(z)f = 0, with A analytic and its generalizations.
4. Bi-orthogonal Expansions in the Space L²(0,∞) (14 pp.) (Andre Boivin and Changzhong Zhu)
   Research paper about expanding a function in an exponential basis and its biorthogonal counterpart.
5. Blaschke Products as Solutions of a Functional Equation (6 pp.) (Javad Mashreghi)
   Study of the solution of ψ(φ(z))s = ψ(z) with φ given, but ψ unknown. The problem is not completely solved.
6. Cauchy Transforms and Univalent Functions (13 pp.) (Joseph A. Cima and John A. Pfaltzgraff)
   Study of Cauchy transforms of a measure on the unit circle which give univalent functions.
7. Critical Points, the Gauss Curvature Equation and Blaschke Products (25 pp.) (Daniela Kraus and Oliver Roth)
   A survey of results that are known about points where the derivative of a Blaschke product vanishes and the functions having these as zeros.
8. Growth, Zero Distribution and Factorization of Analytic Functions of Moderate Growth in the Unit Disc (15 pp.) (Igor Chyzhykov and Severyn Skaskiv)
   This is a short survey about the topic described by the title.
9. Hardy Means of a Finite Blaschke Product and Its Derivative (12 pp.) (Alan Gluchoff and Frederick Hartmann)
   A research paper with a conjecture.
10. Hyperbolic Derivatives Determine a Function Uniquely (6 pp.) (Line Baribeau)
    It proves that the sequence of hyperbolic derivatives defines a function mapping the unit disc to itself.
11. Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane (16 pp.) (Hans G. Feichtinger and Margit Pap)
    A rational orthogonal wavelet is constructed generating a multiresolution for H² of the upper half plane.
12. Norms of Composition Operators Induced by Finite Blaschke Products on Möbius Invariant Spaces (14 pp.) (María J. Martín and Dragan Vukokotić)
    The norms of these operators on Bloch and Dirichlet spaces are computed.
13. On the Computable Theory of Bounded Analytic Functions (26 pp.) (Timothy H. McNicholl)
    The theory of bounded analytic functions is reexamined from the viewpoint of computability theory.
14. Polynomials Versus Finite Blaschke Products (26 pp.) (Tuen Wai Ng and Chiu Yin Tsang)
    This research paper shows analogy of properties between (Chebyshev) polynomials and (Chebyshev) finite Blaschke products.
15. Recent Progress on Truncated Toeplitz Operators (45 pp.) (Stephan Ramon Garcia and William T. Ross)
    Another survey paper giving the advances in the theory of Toeplitz operators since 2007.

The book is of interest both to students and researchers who need the theory of complex functions and especially those who are working in Hardy and related spaces.