Five recent book reviews
This is the second edition of the original text (2006). There are several additions, corrections and some streamlining has been done, but the general idea and structure is maintained. The concept of distance is interpreted in its broadest possible sense. In 29 chapters, the concept of distance is placed in as many different contexts by giving an enumeration of definition and properties.
It was not clear to me what to expect from an encyclopedia of distances counting 650 pages. To me there was the mathematical definition of a distance, and I was wondering what the other 649.5 pages would contain. Well, by interpreting the word "distance" in its broadest possible sense, like two items being different, meaning that they are at a positive distance from each other, then it is difficult to imaging something not related to distance in this sense and you might wonder why this is not the encyclopedia of everything.
The authors however do attach a mathematical meaning to all these possible distances, meaning that there should be some mathematically well defined and computable quantification possible for what is measured. But even with this restriction, a distance can be defined in many contexts like all the subfields of classical mathematics (number theory, geometry, algebra, analysis, probability,...), applied mathematics, computer science, and social sciences. So the authors have grouped the 29 chapters into 7 parts, following more or less these general contexts. The different chapters then single out more specific subfields. For example distances on surfaces and knots, in functional analysis, in graph theory, in systems theory, in biology, ...
Now what does a chapter look like? It is mainly a bullet list of terms that are explained in a few lines. These terms are not only the surprisingly many different kinds of existing metrics, but these include also topics that are for example related to the spaces, graphs, networks etc. on which the distances are defined, or e.g. the surfaces and curves that are defined by distances, or the different meanings of similarity, and the likes of these. These few descriptive lines are mostly very compact, and hence it might require the introduction of many other definitions and concepts as well. For newer topics, or topics further away from mathematics in a stricter sense, there are less formulas and the entries are more verbose. I am certain though that for some items the reader needs to look up more details in the literature. For most (but not all) concepts the authors give (where possible) the original reference. That is the name of the author(s) and a year and the reader is supposed to be able to find the appropriate reference via the Internet. So it might not always be easy to find the intended literature and if it is found, the reference might not be accessible to the reader. In most instances, it will need more than just a few clicks to get it. Of course one does not need the original reference. Sometimes, a topic is better explained in a later reference work. Then of course there is a risk that it might be a generalization or some variation of what is mentioned in this book.
The book is clearly a reference work, i.e., a book in which to look up things. So for the paper version, the extensive index is essential. It is of course much more useful to have an online version where one may search for a word, and where keywords are referring to each other with internal or external links. That might have been possible in the eBook version. However that version is a collection of either a pdf or an html version of the chapters. The html version is however badly formatted, and more importantly, it contains only links to references that are listed at the end of the book, but no cross links or external links. Since readers are now used to wikipedia type of information, I believe this is a missed opportunity. Perhaps an idea for a next edition. The paper version is clearly a welcome asset in a mathematics library. Although many non-mathematical areas are covered as well, I think the approach is too mathematical to recommend it to other libraries. But in any case, libraries are at an increasing speed evolving towards web-based providers of their information, so also for them a true web-version would be most welcome.
The first edition (Springer, 2006) was an outgrow of their Dictionary of distances (Elsevier, 2006). This is the second edition in which corrections are implemented, together with new items and a bit streamlining, but the general structure is largely unaltered. Most changes are situated in sections on graph theory, engineering and the last part on "real world distances".
Corrections to this edition will be maintained at www.liga.ens.fr/~deza/Distances.html
Several possible matlab implementations are given for cubic spline smoothing using GCV (generalized cross validation). The programs are tested and compared for efficiency. The scripts are fully listed in the text. Unfortunately I do not find a place to download the software.
This is an item in the Springer Briefs in computer science. In the Springer Briefs series, the idea is to give short (50-125 pages) reports on a hot topic or a timely snapshot, or an in-depth study of a special case etc. The publication should be self contained and easy to prepare. These issues are primarily intended for rapid electronic publication.
The present volume has only 45 pages and presents several (fully listed) matlab scripts of different methods to compute cubic spline smoothers of a set of measurements of a one-dimensional signal. Smoothing is obtained via estimating a noise level using GCV (generalized cross validation) and choosing an optimal relaxation parameter to get a balance between smoothness and fitting of the data. The numerical implementations are based on a dedicated Cholesky solver, QR factorization, or Fourier transforms. These are used for continuous spline smoothing (the integral of the square of the second derivative is minimized), but Cholesky and Fourier methods are also rewritten for discrete spline smoothing (the integral is replaced by a discrete sum).
For each script, a Monte Carlo simulation tests and compares the method for efficiency, so that the most efficient one can be identified.
The booklet contains a short, practical description of the algorithm and the full listing of the scripts, but one may be more interested in having the software as well, instead of retyping everything. However, neither at the Springer website nor at the author's website could I find a place to download it.
These are the proceedings of a conference held in honor of Konstantin Oskolkov at the Georgian Southern University March 11-13, 2011. A first part sketches the work of Oskolkov and has some reminiscences by friends and colleagues. The second part consists of scientific contributions on quite divergent subjects, most of them somehow related to the work of Oskolkov.
This is volume 25 of the Springer Proceedings in Mathematics & Statistics.
For the 65th birthday of Konstantin Oskolkov, a conference was held at the Georgian Southern University during March 11-13, 2011 of which the present book contains the proceedings. Oskolkov was mainly involved in harmonic analysis, convergence of Fourier series, approximation theory, etc.
A first part gives a survey of the work of Oskolkov and his list of publications. Several friends and colleagues add short texts with reminiscences and/or place a spotlight on a particular result. The main part of the book consists of 23 contributions from the conference. The subjects are quite diverse and usually rather specialized, with the exceptions of few (short) surveys. There is not a central theme, unless an overall broad covering of analysis with harmonic analysis as a center of gravity. However besides problems related to Fourier series, orthogonal polynomials and approximation, there are also papers on analysis in finite fields, or analysis on graphs, and all kinds of function spaces are floating around. Because of this diversity, the papers not grouped in themes, but are just listed alphabetically according to the name of the first author (with the exception of the last paper, but without any obvious reason, perhaps a late addition). It would not be appropriate to summarize each individual paper here in this short review. For the table of contents we refer to the publisher's webpage .
Obviously the book will be a natural choice to be bought by a library having a section on analysis. It gives a nice survey of topics that are currently being investigated. The individual researcher might not be as much interested in all of the papers of the book, but he/she will be almost certainly interested in some. Therefore, it might be preferable to have access to an electronic copy so that those particular chapters of interest to the individual can be downloaded separately.
This is a gentle introduction to space filling curves. Emphasis is on the representation, implementation and application in computer science. A situation where they are useful is an (adaptive) subdivision scheme that is represented by a tree, and the space filling curve will then have to visit all the leaves of the tree in some order. Obvious applications are for example computer graphics and the numerical solution of partial differential equations. Additional material and scripts (maple, python) are made available online.
The first chapter immediately makes clear what kind of applications one should have in mind. In computer graphics, as well as in the numerical solution of partial differential equations, some domain in 2D or 3D has to be subdivided into a finer grid, usually in an adaptive way. The parent-child relations are represented by a tree and the actual values one is interested in are located at the leaves of the tree. Hence when one wants to access the overall picture one will have to access all these leaves. Connecting the leaves in the order that they are visited gives a curve that runs through the whole domain. Since it is not clear in advance what the subdivision tree will look like, one needs a curve that is able to visit every coordinate in the domain: a space filling curve (SFC).
With this motivation, the next chapter gives a more formal definition, which is followed by two chapters that learn how to formally identify, and hence compute, the SFC. That can be by a formal grammar, defining strings that say which operations to perform to generate the curve. Once this is formally defined, it is not very difficult to switch to an algorithmic description which actually describes the affine maps that construct the vertices and edges of the SFC. These descriptions define recursive rules, which if they are applied at infinitum will make the curve to fill up the whole domain, which of course should be stopped after a finite depth of the recursion, in which case the curve is a polygon that can also be formally represented as a string of digits.
So far the examples have been the 2D Hilbert and Peano curves in a square. The next chapters extend the set of examples with Sierpinski curves (on triangles), the Lebesgue curve, and βΩ-curves first in 2D and later also in 3D. This brings us about halfway through the book. Assuming that the basics are now assimilated, the author moves on to the actual work. First the link with subdivision is made and it is explained how SFC are used to sequentialize spacetree grids. Then the focus is shifted to parallelization of the algorithm which is really necessary because these applications become large scale problems in practice. This requires an efficient partitioning of the SFC so that each subsection can be processed by a different processor. The load balancing among the processors, based on the SFC, can be quantified by using the Hölder continuity of the SFC which is discussed in the next chapter on locality measures and locality properties of SFC. In many applications, the subdivision is not on a rectangular but on a triangular and tetrahedral grids. So the Sierpinski curve makes a re-entry in a separate chapter. The remaining three chapters discuss further applications. As a case study, algorithms for matrix operations (matrix products) are investigated. The idea is that a SFC-way of traversing the matrix elements can result in a cache-efficient way of computing the result. Of course some elements of the caching mechanism of a computer are explained to justify the approach. Another case study is simulation on spacetree grids using SFC. This closes the loop with the very first chapter. The final chapter is an enumeration with a short guided tour of other possible applications: combinatorial problems, GIS systems, signal processing, and computer graphics.
It is clear that the author has a long teaching experience with this subject. He has found the right balance between motivation, rigor, application, implementation, in just the right pace to take the reader/student along climbing up the hill towards of increasing complexity as academic examples are left and one approaches the real life applications. Each chapter ends with some exercises (a selected set of solutions is added in an appendix) and there is a rather unusual insert-block "What's next", which sketches what could be the next step ahead, now that one has reached the current level of knowledge. This is exactly how one would end a lesson and give a preview of the next one.
Additional material including several (maple and python) scripts are available at www.space-filling-curves.org. At the moment of writing this review only maple scripts are available that generate examples that are illustrated in the book.
It is explained with a thorough historical and bibliographic study how Newton arrived at his iconoclastic revision of the chronology of ancient kingdoms of Egypt, Assyria, of the Babylonians and Medes and eventually of the Persian Empire. Newton's ideas were published in his Chronology of Ancient Kingdoms Amended (1728). The authors illustrate how Newton applied text analysis and astronomical observations in his calculations to come to his revolutionary conclusions, as opposed to those who got their ideas solely from the (holy) scriptures.
Newton's notes Chronology of Ancient Kingdoms Amended were published posthumously in 1728. It consists of a chronological list of dates and events ranging from 1125 BCE with pharaoh Mephres reigning over Egypt to 331 BCE with Darius the last king of Persia being slain by Alexander the Great. In six subsequent chapters he discusses the civilizations of the Greek, Egyptians, Assyrians, Babylonians and Medes; he describes the temple of Solomon and ends with the empire of the Persians. Before this 'official' publication of 1728, an abstract made by Newton had been translated into French and was published prematurely in 1725 together with critical comments by Etienne Souciet (which were added anonymously). Because of Newton's iconoclastic views on chronology, these publications unchained a vivid controversy among scientists of the 18th century. Some of them believed in Newton and his scientific methods, others contested his computations with arguments that did or did not rely on science or just on the infallibility of the Bible.
The authors of this book present a thorough analysis of the notes of Newton, to explain how he came to these conclusions about ancient history. Their analysis is both broad and deep, which allows the reader to almost look into the head of Newton as he was constantly revising and improving his ideas along with his analysis of the texts and how he adapted his measurements and computations. Buchwald and Feingold largely rely on unpublished notes by Newton and many other primary sources.
They start with a discussion of Newton's views on the reliability of our senses when doing experiments and how it was possible to obtain accurate results out of multiple experiments. This, and subsequent analysis of Newton's way of thinking is placed in a broad historical perspective. They start from the ancient Greek views and show how this evolved during subsequent centuries to result in the different viewpoints on the matter among Newton and his contemporary colleagues. The authors continue to sketch the ideas about chronology and how it was perceived in the 17th century in which Newton formed his own ideas. In their next chapter, they move to Newton's views on prophecies and idolatry. For Newton, the prophecies were symbols that should be interpreted and used as a starting point for computations that should ultimately result into numbers. The mythology is some kind of sublimated history because the gods refer to kings and rulers of ancient time. Another chapter is devoted to population dynamics. An active dispute was going on about how many people lived on earth before and after the deluge, and how to compute these numbers. Newton then comes to his idea that kingdoms appear only late in history. He gets precise dates by studying ancient written sources about Persia, Egypt and Greek history. From astronomical events that he can find there, he can place a star at a particular position in the zodiac. Taking into account the precession of the earth which shifts the position of the corresponding equinox over the centuries, he was able to pin a date for the event mentioned in the text. Each of these elements that led to Newton's conclusions are elaborated in a separate chapter. How Newton evaluated words and verified them against truth is illustrated with the investigation that he did as the warden of the Royal Mint. The different stories about the leaking of Newton's summary of his notes and the premature publication of the French translation Abrégé de la chronologie de M. le Chevalier Isaac Newton, fait par lui-même (1725) in Paris is told with all details. The authors describe in detail the initial reactions and the lively discussions that broke loose in England and in France, certainly after the publication of the full notes in 1728, shortly after Newton's death the year before.
The more technical details are collected in different appendices: a very useful list of definitions and conventions and the mathematics and computations used by Newton to find the dates. The practical computations by Newton are not mentioned in the published text, but that is amply illustrated by the authors with unpublished notes by Newton. Because of the tsunami of names of persons and sites of all ages that play a role in this book, it is very practical to consult the extensive index, and of course, given the many citations and quotes, there is also an extensive list of references.
Although it is not necessary, it certainly helps to be familiar with the books of the Old Testament, and with ancient kingdoms and the Greek mythology. If not, you will probably need a great deal of wikipedia look-ups. The Chronology and many other of Newton's publications are freely available for example via the project Gutenberg as well as via the project Newton.
With this book Buchwald and Feingold have provided the specialists with an overwhelming source of information. But anyone interested in history, i.e., ancient history but also the history up to the 17th century and before, will love the insightful discussion of ideas and the overwhelming stream of detailed facts and citations. The reader is taken by the hand and is guided through Newton's life and how his ideas have matured. A glimpse in the mind of a genius. The style and vocabulary used is not the simplest, but nevertheless it reads fluently. A highly recommended reading indeed.