Encadrement

Is there a particular word in English for a string of two inequalities which together give both a lower and an upper bound for a certain quantity?  French has encadrement for things like this, as in

Pour tout réel x, on a l’encadrement

 x-\frac{x^2}{2}\leq \cos(x)\leq x-\frac{x^2}{2}+\frac{x^4}{24},

This is often quite convenient — of course, one can say “we have the following inequalities…”, but the extra information is useful to have, and it helps avoiding too many repetitions.

Note: The OED only lists encadré as an English word, as a technical term in crystallography:

A crystal is named encadré, when it has facets which form kinds of squares around the planes of a more simple form already existing in the same species, R. Jameson, A treatise on the external characters of minerals, 1805.

Science and mathematics

Quite by chance, I’ve stumbled in the archive of Nature (alas, not freely available) on a paper by J. Sylvester (dated December 30, 1869) concerning, roughly, the status of mathematics among sciences. He says his text was a reaction to earlier talks and articles by Huxley (the biologist, not the limericks writer…). His esteemed opponent having stated

Mathematics “is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation”, but knows only deduction,

Sylvester argues strongly in the opposite direction:

I think no statement could have been made more opposite to the fact,

and goes on to give examples from his own work, in particular, where conclusions were reached, and entire theories were constructed, based on simple apparently accidental remarks, by processes of observation, induction and imagination.

Besides this discussion, reading this paper is quite fascinating. Mostly, it must be said, because it is rather incredibly hard to read. Not only physically (the font size is small, and the footnotes even smaller, and printed 2-up, it really exercises your eyesight), but also because of the language, which I believe should cause many a lover of the English language to either faint or burst out laughing; the Gothic Victorian style (cleverly ridiculed by Jane Austen in “Northanger abbey”) is here put into overdrive for the purpose of scientific discussion. There are rather frightening mathematical terms which, presumably, a few living readers can still interpret,

canonisant, octodecadic skew invariant, invariantive criteria, amphigenous surface, a catena of morphological processes

and there are Latin, Greek and French quotations, untranslated, and a German one (which, strangely, Sylvester feels to be in need of translation). The following passage is quite typical:

Now this gigantic outcome of modern analytical thought, itself, only the precursor and progenitor of a future still more heaven-reaching theory, which will comprise a complete study of the interoperation, the actions and reactions, of algebraic forms (Analytical Morphology in its absolute sense), how did this originate? In the accidental observation by Eisenstein, some twenty or more years ago, of a single invariant (the Quadrinvariant of a Binary Quartic) which he met with in the course of certain researches just as accidentally and unexpectedly as M. Du Chaillu might meet a Gorilla in the country of the Fantees, or any one of us in London a White Polar Bear escaped from the Zoological Gardens. Fortunately, he pounced upon his prey and preserved it for the contemplation and study of future mathematicians…

But there are also interesting things, like a discussion of the status of higher-dimensional geometry, and indeed a forecast of Flatland (the book of that title was only published 15 years later):

for as we can conceive beings (like infinitely attenuated book-worms in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions…

The follow-up paper is much in the same style (with beauties such as “the Eikosi-heptagram“, and flights of fancy like “my own latest researches in a field where Geometry, Algebra and the Theory of Numbers melt in a surprising manner into one another, like sunset tints or the colours of the dying dolphin” – this theory is that of “the Reducible Cyclodes”), and also quite insightful sometimes. For instance, there is en passant, the following very convincing footnote:

Is it not the same disregard of principles, the indifference to truth for its own sake, which prompts the question “Where’s the good of it?” in reference to speculative science, and “Where’s the harm of it?” in reference to white lies and pious frauds? In my own experience I have found that the very same people who delight to put the first question are in the habit of acting upon the denial implied in the second. Abit in mores incuria.

(Sylvester writes the “i” in the word “in” in the last quotation as a dotless i; I doubt it’s a typographical error, but I can’t find an indication that this is proper Latin grammar; does any reader here have an insight on this?)

a long the… riverrun

If you had to write a circular novel, that can be started at any point of the narrative and that would circle back to itself, how long would it be? Probably, most mathematicians would say that either 63, 628, or 6283, pages would be the most appropriate.

And so it’s probably not surprising that the one circular novel I know, Joyce’s “Finnegans wake“, is indeed (in all editions I’ve seen) 628 pages long. Amusingly, this is not mentioned in the introduction to the one I have (which is, also, about the only thing I’ve read of the book, with the exception of the first few and last paragraphs, and isolated snatches here and there).

The reason for this post is really that today is Bloomsday, the day when, fictionally, the action of the earlier “Ulysses” evolves, and that Zürich is well-connected with Joyce. Not only is he buried there, very close to the zoo, so I can use one appropriate quotation

As the lion in our teargarten remembers the nenuphars of his Nile…

from “Finnegans wake” (one of those few snatches I have actually read), but he also wrote parts of “Ulysses” in a house on Universitätstrasse, quite close to the main building of ETH Zürich where I work, as memorialized by the plaque below:

Joyce plaque in Zurich

(I’ve heard that there are a few other such places in Zürich, the reason being that, unable to pay the rent, Joyce had to move frequently…)

Équidistribution, or équirépartition ?

For years, I have been convinced that the proper French translation of “equidistribution” was not the faux ami (false friend) “équidistribution”, but rather the word “équirépartition”. The latter is for instance used by Serre (and Bourbaki).

But then  I realized recently that Deligne uses “équidistribution” in his great paper containing his second proof of the Riemann Hypothesis over finite fields, which contains in particular his famous equidistribution theorem (see Section 3.5, entitled “Application: théorèmes d’équidistribution”).

Since, in fact, neither word appears in the French dictionaries I have available (unsurprisingly: “equidistribution” is not in the OED), and since moreover “distribution” and “répartition” do appear and are identified as synonyms, it seems now that in fact both words should be acceptable…

Peano paragraphing

Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the “Peano paragraphing method”. This is a numbering system for sections/subsections/etc in books where the different items are identified by a decimal number (e.g., 9.132), where the integral part is the chapter number, and the decimal part is arranged in increasing order within each chapter. So for instance 9.301 is a subsubsection lying between 9.3 and 9.31.

I had noticed this system in Titchmarsh’s book “The theory of functions”, from 1932, without understanding it (it is not explained, nor attributed to Peano). Then I saw it again just recently as I was looking up a reference in Whittaker and Watson’s “A course of modern analysis” from 1927, where the explanation and attribution are given in a remark at the beginning. This greatly clarified my previous perplexity in navigating the book of Titchmarsh, which I had found extremely confusing; for instance in Chapter 9, we have

9.1, 9.11 up to 9.15, 9.2, 9.3, 9.31, 9.32, 9.4, 9.41 to 9.45, 9.5, 9.51 to 9.55, 9.6, 9.61, 9.62, 9.621 to 9.623, 9.7…

Looking into other classical books, I can see this system in Watson’s treatise on Bessel functions, but it is not used in either Hardy and Wright’s “Introduction to the theory of numbers”, nor in Titchmarsh’s “The theory of the Riemann zeta function”. It is also absent from Zygmund’s “Trigonometric series” (which, on the other hand, uses a continuous numbering scheme X.Y (Chapter.Item) both for equations, theorems, etc), and from Hardy and Rogosinsky’s “Fourier series”.

Note finally that it seems rather euphemistic to say that this is “lesser known”: neither Google nor Wikipedia seem to be able to give a reference or explanation!