E. Kowalski’s blog

While cleaning up a bit the files on my laptop recently, I “discovered” an old computer program that I had written some time in 2003 or so, to display the graphical evolution of the partial sums of exponential sums, and more precisely of Kloosterman sums. In other words, given a prime number p and integers a and b, this program plots the points in the complex plane corresponding to the values

for

and draws the line segments joining each successive S_k. (As usual, the bar over x indicates the inverse modulo p).

The resulting paths look fittingly psychedelic:

The rather badly written code (for moderately modern Linux systems) can be downloaded here; the compilation should be a straightforward

./configure ; make

the resulting executable kloostermania is then found in the src directory.

Alternatively, still for pretty recent Linux systems (at least, Fedora on Intel machines), you can get the executable here.

(I should probably state formally that the license is GPL, not that there’s much danger of a proprietary software company deciding to make a fortune and deprive mathematicians of much freedom to play with the code by selling derivatives of this program; of course, if anyone decides to add features or to make the program work, e.g., on Mac, this would be much appreciated…)

The program should be fairly easy to use; the File menu offers the possibility of saving the current picture as a PNG file or to print it to a PostScript file (the New and Open commands are just decoys). The Edit menu’s only interesting item is the Preference command, where the colors can be changed by clicking on the respective patches, and where the drawing of the axes can be disabled if desired. The View menu’s equally single item is used to enter the parameter for a new drawing; if the modulus p is not prime, the next prime will be selected (computed in a ridiculously inefficient way…) Finally, the current parameters and the final value of the sum are displayed at the bottom of the window…

Few words, the OED informs us, have received more etymological examination than average (see the sense “average, n.²“). Ample consideration of this issue, we read, was given by eminent linguists, among whom are listed “Diez, Dozy, Littré, Wedgwood, E. Müller, Skeat, etc”. (The third one, É. Littré, is well-known in France, for his own XIXth Century French dictionary).

It seems that the mathematical sense arose from the following meanings:

2. Any charge or expense over and above the freight incurred in the shipment of goods, and payable by their owner. (In this sense it still occurs in petty average, and the now inoperative phrase, average accustomed in Bills of Lading: see quotations 1540 and 1865.)

3. spec. The expense or loss to owners, arising from damage at sea to the ship or cargo.

4. a. The incidence of any such charge, expense, or loss; esp. the equitable distribution of expense or loss, when of general incidence, among all the parties interested, in proportion to their several interests.

In this sense, it seems that “average” is directly related to the French word avarie (which, roughly, means any damage suffered by a ship or its cargo), both coming ultimately from the old Italian avaria. The OED traces the first use to 1200 (though it’s in an Old French text apparently), with English uses as far back as 1502.

In the mathematical sense, the first recorded uses seem to only come around 1750. One thing which seems to be not quite clear is whether one should say on average, or on an average, or on the average, or at an average, or something else altogether.

I was yesterday at the Minnesota Science Museum; after admiring the nice probability machine

I found in the excellent museum store an even nicer coffee mug, which promises much refreshment and strenghtening of the mind in the future:

(These mugs are produced by the Unemployed Philosophers Guild).

Somewhat later than I had hoped, I have updated the script of my spectral theory course. The version currently found online is complete as far as the material I intended to put in is concerned, but there are a few places where I haven’t written down all details (in particular for the proof of the Weyl law for the Dirichlet Laplace operator in an open subset of Euclidean space). I am also aware of quite a few small problems in the last chapter on Quantum Mechanics, due partly to notation problems (for the Fourier transforms, and for “physical” versus mathematical normalizations). I will need to re-read the whole text carefully to correct this; on the other hand, thanks to lists of corrections that I have already received from a few students, the number of typos is much less than before… I will however continue updating the PDF file as I continue checking parts of the text.

What delayed this version for a long time was the write-up of the last section on “The interpretation of Quantum Mechanics”; of course it’s in some sense an extraneous part of the script, since spectral theory barely enters in it, but I found it important to at least try to connect the mathematical framework with the actual physics. (This partly explains all the reading I’ve done recently about these issues). It is equally obvious that I am not the most knowledgeable person for such a discussion, but after all, there are good authorities that claim that no one really understands this question anyway…

What I end up discussing contains however one little mathematical result, which is cute and interesting independently of its use in Quantum Mechanics; it is a theorem of S. Kochen and E.P. Specker which states the following:

There does not exist any map

where S² is the sphere in R³ with the property that, whenever

are pairwise orthogonal unit vectors, we have

or in other words, two of the three values are equal to 1, and the other is equal to 0.

How this result enters into discussions of the interpretation of Quantum Mechanics is described by M. Jammer in his book on the subject (not the same as his book on the development of Quantum Mechanis, but another one, equally evanescent as far as the internet is concerned); more recently, J. Conway and S. Kochen have combined it with the Einstein-Podolsky-Rosen argument (or paradox) to derive what they call the “Free Will Theorem”, which is an even stronger version of the unpredictability of properties of Spin 1 particles (those to which the Kochen-Specker argument applies). Conway has given lectures in Princeton on this result and its history and consequences, which are available as videos online.

Coming back to the result above, considered purely from the mathematical point of view, it is interesting to notice that both the original proof and the version used by Conway-Kochen (which is due to A. Peres) show that the hypothetical map does not exist even for some finite sets of points on the sphere. It is of some interest to get a smallest possible set of such points. The proof I gave in the script, however, which is taken from Jammer’s book (who attributes it to R. Friedberg) is maybe theoretically slightly more complicated, but it is also somewhat more conceptual in that one doesn’t have to be puzzled so much at the reason why one finite set of vectors or another is really fundamental.