Archive for Boston

not for the faint-hearted!

Posted in Books, Kids, pictures, Travel with tags , , , , , on March 21, 2016 by xi'an

While flying over to Boston yesterday, I had a look at The Martian on my seat screen but this proved too much of a hardship: after watching the early self-surgery scene, which is definitely realistic and somewhat gory, I just fainted. Really and truly fainted, which means I came back to my senses being dragged on the plane floor by two Air France flight attendants!, hearing and seeing them but being unable to react for a dozen seconds. There was a doctor in the plane who checked upon me while I was coming back to my senses and his final advice was to stop watching this “kind of movies”, as if I knew I was going to faint from watching a  PG-13 movie… (It actually happened to me once earlier, in that I came close to fainting from watching The Last Temptation of Christ in Ithaca in the 80’s, while protesters were demonstrating outside the cinema.) Quite an embarrassment, frankly! And I did not even watch the rest of the movie…

glorious Boston sunrise

Posted in pictures, Travel, University life with tags , , , , , on March 21, 2016 by xi'an

snapshot from Boston [guest shot]

Posted in pictures, Travel with tags , , , , on July 4, 2015 by xi'an

boston

Cauchy Distribution: Evil or Angel?

Posted in Books, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on May 19, 2015 by xi'an

Mystic2Natesh Pillai and Xiao-Li Meng just arXived a short paper that solves the Cauchy conjecture of Drton and Xiao [I mentioned last year at JSM], namely that, when considering two normal vectors with generic variance matrix S, a weighted average of the ratios X/Y remains Cauchy(0,1), just as in the iid S=I case. Even when the weights are random. The fascinating side of this now resolved (!) conjecture is that the correlation between the terms does not seem to matter. Pushing the correlation to one [assuming it is meaningful, which is a suspension of belief!, since there is no standard correlation for Cauchy variates] leads to a paradox: all terms are equal and yet… it works: we recover a single term, which again is Cauchy(0,1). All that remains thus to prove is that it stays Cauchy(0,1) between those two extremes, a weird kind of intermediary values theorem!

Actually, Natesh and XL further prove an inverse χ² theorem: the inverse of the normal vector, renormalised into a quadratic form is an inverse χ² no matter what its covariance matrix. The proof of this amazing theorem relies on a spherical representation of the bivariate Gaussian (also underlying the Box-Müller algorithm). The angles are then jointly distributed as

\exp\{-\sum_{i,j}\alpha_{ij}\cos(\theta_i-\theta_j)\}

and from there follows the argument that conditional on the differences between the θ’s, all ratios are Cauchy distributed. Hence the conclusion!

A question that stems from reading this version of the paper is whether this property extends to other formats of non-independent Cauchy variates. Somewhat connected to my recent post about generating correlated variates from arbitrary distributions: using the inverse cdf transform of a Gaussian copula shows this is possibly the case: the following code is meaningless in that the empirical correlation has no connection with a “true” correlation, but nonetheless the experiment seems of interest…

> ro=.999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> cor(x[,1]/x[,2],y[,1]/y[,2])
[1] -0.1351967
> ro=.99999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> cor(x[,1]/x[,2],y[,1]/y[,2])
[1] 0.8622714
> ro=1-1e-5;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y)))
> cor(x=z,y=w)
[1] 0.9999732
> ks.test((z+w)/2,"pcauchy")

        One-sample Kolmogorov-Smirnov test

data:  (z + w)/2
D = 0.0068, p-value = 0.3203
alternative hypothesis: two-sided
> ro=1-1e-3;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y)))
> cor(x=z,y=w)
[1] 0.9920858
> ks.test((z+w)/2,"pcauchy")

        One-sample Kolmogorov-Smirnov test

data:  (z + w)/2
D = 0.0036, p-value = 0.9574
alternative hypothesis: two-sided

The Unimaginable Mathematics of Borges’ Library of Babel [book review]

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , on September 30, 2014 by xi'an

This is a book I carried away from JSM in Boston as the Oxford University Press representative kindly provided my with a copy at the end of the meeting. After I asked for it, as I was quite excited to see a book linking Jorge Luis Borges’ great Library of Babel short story with mathematical concepts. Even though many other short stories by Borges have a mathematical flavour and are bound to fascinate mathematicians, the Library of Babel is particularly prone to mathemati-sation as it deals with the notions of infinite, periodicity, permutation, randomness… As it happens, William Goldbloom Bloch [a patronym that would surely have inspired Borges!], professor of mathematics at Wheaton College, Mass., published the unimaginable mathematics of Borges’ Library of Babel in 2008, so this is not a recent publication. But I had managed to miss through the several conferences where I stopped at OUP exhibit booth. (Interestingly William Bloch has also published a mathematical paper on Neil Stephenson’s Cryptonomicon.)

Now, what is unimaginable in the maths behind Borges’ great Library of Babel??? The obvious line of entry to the mathematical aspects of the book is combinatorics: how many different books are there in total? [Ans. 10¹⁸³⁴⁰⁹⁷…] how many hexagons are needed to shelf that many books? [Ans. 10⁶⁸¹⁵³¹…] how long would it take to visit all those hexagons? how many librarians are needed for a Library containing all volumes once and only once? how many different libraries are there [Ans. 1010⁶…] Then the book embarks upon some cohomology, Cavalieri’s infinitesimals (mentioned by Borges in a footnote), Zeno’s paradox, topology (with Klein’s bottle), graph theory (and the important question as to whether or not each hexagon has one or two stairs), information theory, Turing’s machine. The concluding chapters are comments about other mathematical analysis of Borges’ Grand Œuvre and a discussion on how much maths Borges knew.

So a nice escapade through some mathematical landscapes with more or less connection with the original masterpiece. I am not convinced it brings any further dimension or insight about it, or even that one should try to dissect it that way, because it kills the poetry in the story, especially the play around the notion(s) of infinite. The fact that the short story is incomplete [and short on details] makes its beauty: if one starts wondering at the possibility of the Library or at the daily life of the librarians [like, what do they eat? why are they there? where are the readers? what happens when they die? &tc.] the intrusion of realism closes the enchantment! Nonetheless, the unimaginable mathematics of Borges’ Library of Babel provides a pleasant entry into some mathematical concepts and as such may initiate a layperson not too shy of maths formulas to the beauty of mathematics.

talk in Linz [first slide]

Posted in Mountains, pictures, Running, University life with tags , , , , , , , , , on September 17, 2014 by xi'an

summer reads

Posted in Books, Travel with tags , , , , , , , , , , on August 23, 2014 by xi'an

wells3I had planned my summer read long in advance to have an Amazon shipment sent to my friend Natesh out of my Amazon associate slush funds. While in Boston and Maine, I read Richard Dawkins’ The God delusion, the fourth Kelly McCullough’s Fallen Blade novel, Blade reforged, the second Ancient Blades novel, unrelated to the above, A thief in the night, by David Chandler, and also the second Tad Williams’ Bobby Dollar novel, Happy Hour in HellThe God delusion is commented on another post.

Blade reforged is not a major novel, unsurprisingly for a fourth entry, but pleasant nonetheless, especially when reading in the shade of a pavilion on Revere Beach! The characters are mostly the same as previously and it could be that the story has (hopefully) come to an end, with (spoilers!) the evil ruler replaced by the hero’s significant other and his mystical weapons returned to him. A few loose ends and a central sword fight with a more than surprising victory, but a good summer read. Checking on Kelly McCullough’s website, I notice that two more novels are in the making….

Tad Williams’ second novel Happy Hour in Hell is much less enjoyable as the author was unable to keep up with the pace and tone of the highly imaginative first novel, full of witty and hard-boiled exchanges. The first novel introduced the (after-)life of a guardian angel in California, Doloriel (a.k.a. Bobby Dollar), with enough levels of political intrigue between Heaven and Hell and Earth and plots, pursuits, assassination attempts, etc., to make it a page-turner. This second novel sends Doloriel on a suicide mission to Hell… and the reader to a Hell of sorts where the damnation is one of eternal boredom! What made the first novel so original, namely the juxtaposition of the purpose of a guardian with his every-day terrestrial life, is lost. All we have there is a fantastic creature (from Heaven) transposed in another fantastic environment (Hell) and trying to survive without a proper guide book. The representation of Hell is not particularly enticing (!), even with acknowledged copies from Dante’s Inferno and Hieronymus Bosch’s paintings. There is a very low tolerance level to my reading of damned souls being tortured, dismembered, eaten or resuscitated, even when it gets to the hero’s turn. Add to that a continuation of the first book’s search for a particular feather. And an amazing amount of space dedicated to the characters’ meals. This makes for a very boring book. Even for a rainy day on a Maine lake! The depiction of the levels and inhabitants of Hell reminded me of another endless book by Tad Williams, Shadowmarch, where some characters end up in a subterranean semi-industrial structure, with a horde of demon-like creatures and no fun [for the reader!]. Ironically, the funniest part of reading Happy Hour in Hell was to do it after Dawkins’ as some reflections of the angel about the roles of Heaven and Hell (and religion) could have fitted well into The God delusion! (Too bad my Maine rental had Monty Python’s Holy Grail instead of The Life of Brian, as it would have made a perfect trilogy!)

Most sadly, David Chandler’s A thief in the night had exactly the same shortcomings as another book  I had previously read and maybe reviewed, even though I cannot trace the review or even remember the title of the book (!), and somewhat those of Tad Williams’ Happy Hour in Hell as well, that is, once again a subterranean adventure in a deserted mythical mega-structure that ends up being not deserted at all and even less plausible. I really had to be stuck on a beach or in an airport lounge to finish it! The points noted about Den of Thieves apply even more forcibly here, that is, very charicaturesque characters and a weak and predictable plot. With the addition of the unbearable underground hidden world… I think I should have re-read my own review before ordering this book.