Tag Archives: mathematics

Linkage

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Cosma reviewed Networks, Crowds, and Markets by Easley and Kleinberg for the American Scientist. I have had the book for a while and just haven’t gotten around to reading it yet, but I should. Its a weighty tome, perhaps a bit too weighty to take to the beach (or on a plane, or…). Alex Dimakis said he reads a little bit before going to bed at night. That’s a heavy glass of warm milk. A fun quote from the review:

What game theorists somewhat disturbingly call rationality is assumed throughout—in other words, game players are assumed to be hedonistic yet infinitely calculating sociopaths endowed with supernatural computing abilities.

Ah, game theory. I anticipate experiencing the unease Cosma feels about the “realities behind the mathematics.”

MIT wants to teach math writing. I thought I learned how to write math by having my Phase II paper draft doused liberally in red ink by Prof. Kleiman. But this is something else entirely. I think a more important thing is to help those who work in mathematical fields or who use mathematics. Perhaps this will be a resource that engineering graduate students can use to improve their own writing.

The Connected States of America, including an interactive map showing how much people in place A talk to people in place B. Via MeFi.

Since I am moving to Chicago this fall, it’s time to get familiar with the L.

Our Paperwork Explosion – an add for IBM. Very weird. Also, vaguely menacing. I love the music though! AVia MeFi.

A mountain-climber’s axe! A mountain-climber’s axe! CAN’T YOU GET THAT THROUGH YOUR SKULL? (Trotsky dies. Bell.)

Linkage

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I never knew Jell-O could be so graceful.

I kind of like this version of Take Five from Sachal Music.

Sometimes the Library of Congress does awesome things. This jukebox is up there.

I wouldn’t have believed before that there is money in a bannass stand, but I could be wrong.

The clarity in this press nugget leaves a lot to be desired. The statement “the trio has found a way to determine the smallest number of nodes that must be externally controlled to force a given network from any initial state to any desired final state,” is so vague! The full article is here. It turns out they are looking at a linear control problem d\mathbf{x}/dt = A \mathbf{x}(t) + B \mathbf{u}(t) where the different elements of the state are related via a graph matched to A and you want the input \mathbf{u}(t) to only be nonzero on a subset of the nodes. Thanks to Ann Wehman for the pointer.

Linkage

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Yes yes yes, all my posts are link posts now. I swear, I’ll get back to something more interesting soon, but I always promise that.

People post funny things to ArXiV.

Razib discusses new studies of the genetic origin of Indians.

Tips for food photography. I seem to know several food bloggers now.

A new study about bullying.

The University of Michigan is allowing longer tenure processes. This is in part to address the pressures of getting tenure and starting a family at the same time, but also particularly the culture in the medical school, where “very few faculty in medical schools actually take advantage of such policies [to halt the tenure clock].” The academic Senate Assembly was opposed to the change.

Linkage

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I’m blogging from Chicago, where it is a balmy 42 degrees but sunny. Whither spring, I ask! Actually, I’m not blogging so much as linking to a bunch of stuff.

For San Diegans, the SD Asian Film Festival Spring Showcase is going on. It looks like I’ll miss a lot of it but I might try to catch something at the end of the week.

Less Pretentious & More Accurate Titles For Literary Masterworks — funny but possibly NSFW.

This home-scanning program seems creepy, regardless of the constitutionality issues.

Unfortunate headlines strike again.

I really like scallion pancakes. I’ll have to try this out when I get back to San Diego.

I agree that this video is awesome. Yo-Yo Ma and Lil Buck. I think that dude is made of rubber. And steel.

Tom Waits was induced into the Rock and Roll Hall of Fame. I just hope I get to see him live some day.

Some things to skim or read from ArXiV when I get the chance:
Sequential Analysis in High Dimensional Multiple Testing and Sparse Recovery (Matt Malloy, Robert Nowak)
Differential Privacy: on the trade-off between Utility and Information Leakage (Mário S. Alvim, Miguel E. Andrés, Konstantinos Chatzikokolakis, Pierpaolo Degano, Catuscia Palamidessi)
Capacity of Byzantine Consensus with Capacity-Limited Point-to-Point Links (Guanfeng Liang, Nitin Vaidya)
Settling the feasibility of interference alignment for the MIMO interference channel: the symmetric square case (Guy Bresler, Dustin Cartwright, David Tse)
Decentralized Online Learning Algorithms for Opportunistic Spectrum Access (Yi Gai, Bhaskar Krishnamachari)
Online and Batch Learning Algorithms for Data with Missing Features (Afshin Rostamizadeh, Alekh Agarwal, Peter Bartlett)
Nonuniform Coverage Control on the Line (Naomi Ehrich Leonard, Alex Olshevsky)
Degree Fluctuations and the Convergence Time of Consensus Algorithms (Alex Olshevsky, John Tsitsiklis)

Linkage (and a puzzle)

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I saw Scott’s talk today on some complexity results related to his and Alex Arkhpov’s work on linear optics. I missed the main seminar but I saw the theory talk, which was on how hard it is to approximate the permanent of a matrix X whose entries (X_{ij}) are drawn iid complex circularly-symmetric Gaussian \mathcal{CN}(0,1). In the course of computing the expected value of the 4th moment of the permanent, he gave the following cute result as a puzzle. Given a permutation \sigma of length n, let c(\sigma) be the number of cycles in \sigma. Suppose \sigma is drawn uniformly from the set of all permutations. Show that

\mathbb{E}[ 2^{c(\sigma)}] = n + 1.

At least I think that’s the statement.

In other news…

  • Ken Ono has announced (with others) an algebraic formula for partition numbers. Very exciting!
  • Cosma thinks that New Yorker article is risible, but after talking to a number of people about it, I realized that the writing is pretty risible (and that I had, at first pass, skimmed to the part which I thought was good to report in the popular (or elitist) press, namely the bias towards positive results. Andrew Gelman points out that he has written about this before, but I think the venue was the more interesting part here. What was risible about the writing is that it starts out in this “ZOMG OUR SCIENCE POWERZ ARE FAAAAAAADINNNNNNGGGGGGG,” and then goes on to say slightly more reasonable things. It’s worthy of the worst of Malcolm Gladwell.
  • Tofu is complicated.
  • The 90-second Newbery contest.

A nice formula for the volume of an L_p ball

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I recently came across this paper:

Volumes of Generalized Unit Balls
Xianfu Wang
Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 390-395

which has nice formula for a “generalized unit ball” in \mathbb{R}^n:

\mathbb{B}_{p_1,p_2,\ldots,p_n} = \{ \mathbf{x} = (x_1, x_2, \ldots, x_n) : |x_1|^{p_1} + |x_2|^{p_2} + \cdots + |x_n|^{p_n} \le 1 \}

These balls can look pretty crazy (as some pictures in the paper show).

The main result is that for p_1, \ldots, p_n > 0, the volume is equal to

\mathrm{Vol}(\mathbb{B}_{p_1,\ldots,p_n}) = 2^n \frac{ \Gamma(1 + 1/p_1) \cdots \Gamma(1 + 1/p_n) }{ \Gamma(1/p_1 + 1/p_2 + \cdots + 1/p_n + 1) }

The formula for the volume of the n-sphere in the L_2 norm is well known, but this formula lets us calculate all sorts of volumes. For example, for the unit L_1 ball we get the rather clean and beautiful formula

2^n \frac{\Gamma(2)^n}{\Gamma(n + 1)} = \frac{2^n}{n!}

The proof given in the note is by induction, and a remark at the end points to several other proofs based on Laplace transforms.