One dimension ({0,1} n*n matrix with exactly one 1 per row or column): The number of permutations satisfies Stirling's formula: it's [(1+o(1)(n/e)]^n
Two dimensions ({0,1} n*n*n matrix with exactly one 1 per row, column or shaft): The number of Latin squares, as seen from Van Lint and Wilson's book, is [(1+o(1))(n/e^2)]^{n^2}.
Conjecture: in d dimensions, it's [(1+o(1))(n/e^d)]^{n^d}
Linial and Zur Luria proved the "less than or equal to" side. The "greater than or equal to" is still open.
Inside and outside plans. Infer the internal organization of a building from the outside setup of doors and windows. Where is the staircase? How many apartments are there? Where are the separations between apartments? Much of the building plan can be guessed from examining the disposition of the windows.
Ceiling heights. It happens that adjacent buildings have the same height yet a different number of floors: one has 7 floors, the one next to it has 8 floors, yet their roofs are lines up. Why? Because the floors have different heights. Correlate floor height to the age of the buildings.
Symmetry and near-symmetry. Find perfectly symmetrical patterns. Find patterns that are not symmetrical. Why not? Are those departures from perfect symmetry accidental or voluntary? Example: The central column of Notre Dame's facade, the tip of the roof behind it, and the tall steeple in the back, are not quite lined up, thus missing an opportunity for pleasing symmetry.
Yesterday I heard a microphone in the cloister-like courtyard outside my office. There was a small crowd around a man who was saying slowly, articulating carefully: "... Third: David Smith. [Sound of hands clapping]. Fourth: Jane Doe. [Sound of hands clapping]...": he was publicly proclaiming the results (in the humanities) of the entrance exam of Ecole Normale Superieure!
A few minutes later, I walked across the yard and saw people scattered about, some with a big smile on their face, others crying. Many people were on the phone.
I wonder: Results and rankings are public in France: they are published in the newspapers, and grades are posted on bulletin boards outside schools. But this ceremonial in the historical courtyard goes one more step towards delivering praise and humiliation, and maximizes anxiety among candidates. Could this setup be considered a form of hazing?
Day after the conference deadline: goofing off.
Day after the day after the conference deadline: cleaning up. Taking stock of all the urgent tasks that got shoved aside to make room for the conference submission, apologizing for things that were not done and people who were ignored, and working at getting things back under control.
Almost every computer scientist I know works in that way. Almost none of the mathematicians or physicists I know work in that way. Why have we chosen such a stressful and unpleasant way of organizing our work? More importantly, why is it so hard to break the habit? Part of the reason is that the minority rules in that case: in a group of collaborators, it is enough that one person works in that mode to force all participants to adopt it. In the DAG of tasks that need to be done, as long as a few critical tasks belong to a person of that culture, all members of the project have to also follow the same pattern of behavior, I think.
So what can be done to change behavior? Getting rid of hard deadlines would change the way the field works. One possible direction is the possibility of rolling deadlines.
There are two kinds of homeless people in Paris. There are some men in their fifties surrounded by bags of stuff, and who look like out of control alcoholics. They have been around for as far as I remember. There are also some people who don't speak much French, who don't have stuff but often are accompanied by something to inspire compassion: women with young children, men with dog and puppies, person kneeling in an attitude of prayerful meditation. I am told that they are "roms" (some kind of gypsies?) and that they form a network similar to Fagin's beggar group.
Yesterday I saw someone different, an older woman with bags of stuff, who was begging without much energy. It was around midnight, on a pedestrian street, and I wondered where she was going to spend the night. I asked if she was going to sleep there. She said, no, of course not, because for a woman there was too much of a risk of being getting into trouble. Instead, she goes to a safe spot, a place right next to the police station. She gets together with two or three other women, and they all sleep near one another, so that they can help if one of them is attacked. Before going to sleep, she also waits until the men have gone to bed, so that they don't notice that there is a woman nearby. Finally, she takes care to wear gender-neutral clothes: a black wool hat and some baggy pants, so that, seen from the back, in the dark one might think that she is a man. Finally, she added that the place near the police station was protected from the rain, and gave me the address. I wished her good night and left.
The thought crossed my mind to offer her a bed for the night in my guest room, but it did not linger. The weather was nice, and sleeping outside did not seem like the worst thing in the world that evening.
Now I wonder if she thought I might be looking for a place where to crash. I was wearing old jeans and a slightly frayed T-shirt, and carrying a backpack. Perhaps she thought I had nowhere to go to, and was giving me advice? How considerate of her!
Today my friends in the US are expressing their joy: the project of reform of health insurance was not struck down by the Supreme Court.
To each their own priorities.
Yesterday, walking in Paris I saw four perfectly shaped plastic mannequins displayed on the street just outside a store, facing away from me. Next to them was a large woman, also facing away from me. They had a similar attitude, almost a pose, as if all five of them were idly waiting. I took an imaginary picture of the scene with the camera of my mind, and gazed at it with interest, trying to come up with an appropriate caption: "Find the intruder", perhaps? "Life is not fair"?
Then the woman, who had been talking with another woman, turned around and they both looked at me. Embarrassed, I realized that they had caught me staring. Worse, I suddenly understood that they were prostitutes and might be wondering if I was interested in their services. I hastily withdrew into my best "I'm not really here" look and passed them without another glance.
MoraleLesson: after a red eye, take a day off.
