## Beef apple, cumin crouton

Just a note to myself to remember two things from tonight’s dinner:

• If you fry apple slices in the greasy pan you cooked the hamburgers in, you get a very good hamburger topping.  They are also good plain.
• In a saucepan fry ripped-up bread in olive oil with a lot of garlic and cumin and some walnuts.  Add to chopped tomato and cucumber and it is salad.
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## We ask America and America tells us something unreasonably precise

The latest Wisconsin poll from We Ask America doesn’t tell us anything new.  Scott Walker remains unpopular, but not Rick Scott unpopular.  I bring it up the new polling only to object to the decision to report the results with two digits after the decimal point.  Nothing is gained by telling us that 45.15% of a 1300-person sample approves of Walker’s governorship.  Nor does it help matters to warn of the 2.72% margin of error.

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## Madison in the ’60s: reactionary politics and opposition to racial equality

I refer, of course, to the 1860s.  Stu Levitan’s superb Isthmus feature tells the story of Madison during the Civil War, when the city wasn’t exactly the beacon of progressive values it is today:

Lincoln carried Wisconsin both before and during the war, but he did not carry Madison. In 1860, the city narrowly voted for Democrat Steven Douglas over Lincoln, 783 to 747. In 1864, when the choice was between Lincoln, a president committed to prosecuting the war until abolition, and Democratic candidate Gen. George McClellan, who would have allowed slavery to continue, Madison went for McClellan, 794-705.

Madison also twice rejected Republican governor Alexander P. Randall, an early and forceful foe of the Southern secessionists. While Wisconsin handily reelected the abolitionist Randall in 1859, he lost Madison, 956-701.

That same year, Madison reelected as its mayor George B. Smith, who in a 1862 diary entry said Lincoln was “responsible for the miserable state of things, and for this and many special and arbitrary acts which he has committed and authorized, I solemnly believe that he ought to be impeached and legally and constitutionally deposed from the high office of President.”

Levitan’s story of Madison in the war years serves as a useful guide to our early city fathers — the guys whose surnames are on the street signs, like Fairchild, Keyes, Randall, and Vilas.  This is one of the reasons we need local newspapers.  Where else are you going to read this?

Not everything about Madison was different 150 years ago, by the way.

A report from the city finance committee in 1860 was blunt: “The truth is that as a city, we have been swindled and robbed and the state prison for life would be none too good for the men who have done this. We are bankrupt for the present, bankrupt for the future, bankrupt forever, unless we can effect a reasonable compromise.”

A compromise would be reached and disaster averted. But this took three years to work out, and it left the city paying its debts at about 60 cents on the dollar.

Even incoming Mayor Vilas, a successful politician in his native Vermont before moving to Madison in 1850, sounded like an angry tea partier. Probably the richest man in Madison, he unloaded at his mayoral inaugural in April 1861, declaring that citizens “had been swindled” and that “taxes have become odious.”

Vilas bemoaned “the corruptions of men in office, and the licentious practices that prevail upon the people in selecting them.” He warned that “the imbecility and disregard of justice and right manifested in the discharge of official trusts” could result in revolution and a new government “subversive of the rights and liberties of the people.”

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## Prickly live on WMBR, 1995 and 1997

Matthew White, formerly of Prickly, now of Chores, sent me these files long ago and I never got around to posting them — two live-in-studio performances by Prickly on WMBR.  The shows cover the majority of their catalogue, with drastically better sound quality than the ancient demo tape I posted previously.  Plus, between-song banter!

Prickly Live on WMBR May 1995

Prickly Live on WMBR Jan 1997

Update:  Above links are long dead but I’ve now put this stuff and more on Google Drive.

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## John Tierney can have my rubberized playground surface when he pries it from my cold, dead hands

In the NYT, John Tierney unloads this week on playground equipment, which in his view is not high enough or dangerous enough and is contributing to the weak moral fiber of These Kids Today.  Kids need to break their arms more, because breaking your arm and getting over it is part of growing up.

Even if children do suffer fewer physical injuries — and the evidence for that is debatable — the critics say that these playgrounds may stunt emotional development, leaving children with anxieties and fears that are ultimately worse than a broken bone…. Sometimes, of course, their mastery fails, and falls are the common form of playground injury. But these rarely cause permanent damage, either physically or emotionally. While some psychologists — and many parents — have worried that a child who suffered a bad fall would develop a fear of heights, studies have shown the opposite pattern: A child who’s hurt in a fall before the age of 9 is less likely as a teenager to have a fear of heights.

This article is more annoying than Tierney’s usual schtick, because this time I agree with his overall psychological stance.  I’m on board with the CBT model of phobia treatment, in which you attenuate a fear by graduated exposure.  I watch CJ struggle with things that scare him all the time, and I share his pride when he handles them.

That said, I think it’s premature to worry that you’re letting your kids grow up underfractured.  Tierney declines to say what “studies” he’s referring to above, but I’m pretty sure it’s Evidence for a non-associative model of the acquisition of a fear of heights, a 1998 paper in Behavior Research and Therapy by R. Poulton et al.  It’s a good paper!  But let’s look at what it really says. They used data from a longitudinal study to see what relation, if any, there was between severe falls early in life and fear of heights later.  What they found was this.  A fall before age 5 didn’t significantly affect fear of heights at age 11.  A fall before age 5 also didn’t significantly affect fear of heights at age 18.  Also, a fall between the ages of 5 and 9 didn’t significantly affect fear of heights at age 11.  But there was a significant negative association between falls between the ages of 5 and 9 and fear of heights at age 18.

That’s pretty far from “safe playgrounds stunt kids’ growth.”  All the more so when you stop to think that there might be other reasons that kids who were fearless about heights at 18 might have broken their arms more as kids.  Maybe they were fearless about heights to start with!  The authors of the study explicitly raise this possibility.  Tierney does not — strangely, considering how much he digs innate biological explanations when it’s time to explain where all the women math professors are.

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## Math is like Earthball, not like ARML

Cathy’s post touched off a lot of discussion of math contests, and whether they do or do not, in her formulation, suck.  My thoughts on this are pretty simple.

Big good thing about math contests:  They reveal that math is more than what’s taught in school, and that there’s a whole community of kids around the world who are passionate about math.

Big bad thing about math contests:  They help promote the idea that the most important thing about math is whether you’re the best at it.

Of course you can design your contest to provide more of the big good and less of the big bad.  The question isn’t so much whether the good minus the bad is positive; it’s whether there are other ways of getting at the good that avoid the bad.  I think programs like Hampshire and MathCamp and PROMYS and Ross are like this.  But they clearly don’t scale to the size of an AMC.  Mary O’Keeffe‘s comments on Cathy’s blog were particularly interesting, since they give a good sense of what math contests are like in 2011 for those of us whose direct experience is substantially less recent.

Some people, I think, don’t think my big bad is so bad.  I disagree.  Somebody on my Facebook feed  recently linked to this letter from algebraist Donald Weidman to Science, headed “Emotional Perils of Mathematics.”  Weidman numbers among these perils the following frustration:

“The history of mathematics makes plain that all the general outlines and most of the major results have been obtained by a few geniuses who are not the ordinary run of mathematicians.  These few big men make the long strides forward, then the lesser lights come scurrying in to fill in the chinks, make generalizations, and find some new applications; meanwhile the giants are making further strides.”

This is so profoundly wrong it makes my teeth hurt.  Mathematics is like Earthball.  The weight of our ignorance is tremendous and all of us push together to move it a bit to one side over the course of our lifetimes.  We may vary in strength but what we have in common is that there’s very little we can do alone.

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## Should you be surprised by the diameter of the nxnxn Rubik’s group?

A press release from MIT reports on a new theorem of Erik Demaine, Martin Demaine, Sarah Eisenstat, Anna Lubiw, and Andrew Winslow:  the group of transformations of the n x n x n Rubik’s cube has diameter on order n^2 / log n in the standard generators.  The press release quotes (Erik) Demaine as saying

“That that’s the answer, and not N2, is a surprising thing,”

That they were able to prove this is surprising, and impressive, indeed!  Upper bounds on diameters of Cayley graphs are hard; you use a big spectral hammer if you have one at hand, and otherwise you really have to get dirty and explain algorithmically how you’re going to get all the group elements from short words.  And in general, the best you can hope for is an asymptotic result like the one obtained by Demaine et al; the precise diameter of the 3x3x3 Rubik’s group was only just determined to be 20 with the help of 35 CPU-years donated by Google.

But the truth of the theorem is much less surprising, in my opinion.  Let’s start with the lower bound, which is easy, as Demaine recounts.

“In the first hour, we saw that it had to be at least N2/log N,” Demaine says.

Why was this so fast?  Because the number of generators is just 6 N; for each of the 3 coordinate axes, there are N clockwise quarter-turns perpendicular to the axis and N counterclockwise quarter-turns.  So the total number of words of length k in those generators is on order N^k; for this to cover the whole group, whose size is known to be exponential in N^2, you need k to be at least N^2 / log N.

If all those words were distinct, then the diameter of the group would be N^2/log N.  But they’re not distinct!  The Rubik’s group has relations; it is not free. Far from it.

But how far from it, is the question.

To measure this, we write W(k) for the number of distinct words of length k, or “the volume of the word ball.”  If the group were free on the given generators, we’d have W(k) = (6N)^k, exponential in k.  If, on the other hand, the group were free abelian on the given generators, W(k) would be on order of k^(6N), polynomial in k.  Commutativity crushes words together a lot.  And notice that a lot of the Rubik generators commute with each other:  whenever two twists are perpendicular to the same axis, they commute.  Does that make W(k) grow a lot more slowly?

The Cartier-Foata theorem is a beautiful identity that addresses this question.  It provides an exact formula for the number of length-k words in a “partially commutative monoid,” where some pairs of generators commute, and others don’t.  Namely:  if we construct a generating function

$P(t) = \sum_{k=0}^\infty W(k) t^k,$

then

$P(t) = (\sum_C (-t)^{|C|})^{-1}$

where C ranges over all commutative cliques, i.e. mutually commutative subsets of the generators.  When there are no commutation relations, the commutative cliques are just the subsets of size 0 and 1, so you get P(t) = 1/(1-mt) where m is the number of generators, and W(k) grows like m^k.  When all generators commute, every subset is a commutative clique, and you get P(t) = (1-t)^(-m), whose coefficients have polynomial growth.  Julianna Tymoczko and I used the Cartier-Foata formula to give a lower bound for the diameter of unipotent subgroups of Lie groups of finite type.  In our case, the number of generators grew with the rank n of the group, but the generators commute to such a great extent that the diameter was on order log |G| instead of log |G| / n.

The Rubik’s group is not quite that commutative.  The commutative cliques are just the subsets of generators which are perpendicular to the same axis, so there are

$3 {2N \choose a}$

commutative cliques of size a for all positive a.  Cartier-Foata tells us that the monoid whose only relations are these commutations has

$P(t) = (3(1-t)^{2N} - 2)^{-1}$

whose smallest pole, if I did this right, is at around (log 3/2)(1/2N).  So the t^k coefficient, which is an upper bound for the W(k) of the Rubik’s group, is exponential on order of (2/log(3/2)N)^k.  In particular, this poses no obstacle to filling up the whole group with words of length O(N^2 / log N).

Now that doesn’t mean you can actually do it!  W(k) might be much smaller.  But in my (limited) experience, commutativity relations really do a lot of the work.  (For instance, in my paper with Tymoczko, we give an algorithmic way of writing group elements as short words that shows that the Cartier-Foata bound is correct up to a constant, just as in the Rubik’s case.  Also as in the Rubik’s case, that upper bound is the hard part.)  Of course it’s easy to construct examples where the Cartier-Foata estimate is way off  (e.g. a free k-nilpotent group on N generators.)  But if you had made me guess what the asymptotic behavior of the Rubik’s group is, I like to think I would have gotten it right.

## Sports dude dialectic

Sports dude on minor league baseball:  “This is what baseball is really all about.  No overpaid superstars, no lockouts, no steroids — just kids playing their hearts out for the love of the game.”

Sports dude on women’s basketball:  “Sure, I’d like to get into it, but it’s just not that interesting to watch players at a lower level of athleticism.”

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## In which I agree with Pushkin

“Imagination is as necessary in geometry as it is in poetry.”

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## But he ate the banana with such authenticity

From the NYTimesMag’s interview with Marc Andreessen, one of the founders of Netscape:

After hearing a story about Foursquare’s co-founder, Dennis Crowley, walking into a press event in athletic wear and eating a banana, I developed a theory that bubbles might be predicted by fashion: when tech founders can’t be bothered to appear businesslike, the power has shifted too much in their favor.

Believe it or not, this goes deep into the interior mentality of the engineer, which is very truth-oriented. When you’re dealing with machines or anything that you build, it either works or it doesn’t, no matter how good of a salesman you are. So engineers not only don’t care about the surface appearance, but they view attempts to kind of be fake on the surface as fundamentally dishonest.

I got a B+ in “Intro to Sociology,” and even I know that to appear in a business setting wearing sweats and polishing off lunch is as much of a performance, and as deeply concerned with “surface appearance,” as is showing up in a \$5000 suit.  Actually, sorry, a little bit more concerned with surface appearance.

Bonus points for the suggestion that success in the Internet industry has nothing to do with salesmanship.