Languagehat.com on the word "normal".
I didn't know that the original meaning of the Latin word norma had to do with right angles (which is probably the most common technical meaning, though there are lots of others); the meaning of "following a rule" is a metaphorical extension of that, coming from the square that was used to make right angles.
04 January 2008
The math of Futurama
Dr. Sarah's Futurama Math, from Sarah Greenwald. Apparently a new Futurama DVD was just recently released, if you care about that sort of thing. (Personally, I like the show but not enough to go out of my way to watch it.) The DVD includes a lecture on the math of Futurama. I didn't know that a lot of the writers of the show had serious mathematical training, but it doesn't surprise me at all.
Also, simpsonsmath.com from Greenwald and Andrew Nestler. I like this one more, because I can get The Simpsons but not Futurama on my dirt-cheap cable package, so the Simpsons references are more current to me. I linked to this one a long time ago, but you probably weren't reading this blog then, because at the time I had maybe one percent of the readers I have now.
(In a not-all-that-strange coincidence, I'm reading William Poundstone's biography of Carl Sagan. Sagan was born in 1934, and often cited the "real" Futurama, a pavillion at the 1939 New York World's Fair, as one of the first things that pushed him towards being a scientist.)
Also, simpsonsmath.com from Greenwald and Andrew Nestler. I like this one more, because I can get The Simpsons but not Futurama on my dirt-cheap cable package, so the Simpsons references are more current to me. I linked to this one a long time ago, but you probably weren't reading this blog then, because at the time I had maybe one percent of the readers I have now.
(In a not-all-that-strange coincidence, I'm reading William Poundstone's biography of Carl Sagan. Sagan was born in 1934, and often cited the "real" Futurama, a pavillion at the 1939 New York World's Fair, as one of the first things that pushed him towards being a scientist.)
03 January 2008
How do people afford Manhattan?
Apartment Prices in Manhattan Defy National Real Estate Slide.
Assuming that that "$828,0000" is actually $828,000, this makes sense; the distribution of housing prices is skewed, just like the distribution of incomes. It's rare, though, to see an article that mentions both the mean and the median of a distribution.
The average price for an apartment reached $1.4 million in the last quarter of 2007.... The reports noted, however, that average prices were being pushed to record levels because of the increasing number of apartments selling at the top end of the market, above $10 million... Still, real estate brokers and economists point out that the overall market has performed just as well as the luxury market. In fact, half of the sales in Manhattan involved apartments that sold for less than $828,0000 [sic], according to data tracked by Halstead Property and Brown Harris Stevens.
Assuming that that "$828,0000" is actually $828,000, this makes sense; the distribution of housing prices is skewed, just like the distribution of incomes. It's rare, though, to see an article that mentions both the mean and the median of a distribution.
How many cold nights are there?
Nineteen degrees Fahrenheit (-7 Celsius) in Philadelphia this morning.
A photographer that I know went out and took some beautiful pictures of downtown Philadelphia overnight; then he complained that he picked the "coldest day of the year" to do it. Of course, he meant the coldest day of the year so far.
But I suspect it won't be the coldest day of the year, for the following reasons:
But that raises an interesting question: what can we expect the coldest temperature all year to be?
The normal low temperature in Philadelphia on any given night in late January is 25 degrees -- that's when the normal low is at its lowest -- but only an idiot would say that this means the average lowest temperature all year is 25 degrees. (In fact, even without looking at the records, I'd be shocked to learn that there has been a year in recorded history in which Philadelphia didn't go below 25 at some point.) One tempting thing to do is to make the following assumptions:
That turns out to be roughly 2 standard deviations below the mean. Unfortunately, I don't know what the standard deviation is!
And there's a much bigger problem here -- I've implicitly assumed that the annual minimum always falls in a certain cold period, and that the temperature on each day is independent of each other day! The second assumption is spectacularly bad. If it's colder than average today, it'll probably be colder than average tomorrow.
Also, I've assumed that the low temperatures on a given day of the year are normally distributed, which probably isn't true...
However, one could use a method like this to estimate, say, the mean number of days below, say, fifteen degrees in a given year. If we know the distribution then we can compute, say, that the probability of it being below fifteen degrees on the night of January 3 is ten percent; adding up these probabilities for every night of the year gives an expected mean number of cold nights; call this μ. But the variance is important here, as well. All I can say instinctively is that the variance is probably greater than that of a Poisson distribution with mean μ (which is what you'd usually use to model "rare events"), since if it's cold today it'll probably be cold tomorrow, and it probably doesn't tell us much about whether next week will be cold; in particular I suspect that within the winter season, there aren't pairs of days for which being cold is inversely correlated.
(Of course, this is all testable, but I don't care that much.)
A photographer that I know went out and took some beautiful pictures of downtown Philadelphia overnight; then he complained that he picked the "coldest day of the year" to do it. Of course, he meant the coldest day of the year so far.
But I suspect it won't be the coldest day of the year, for the following reasons:
- the coldest weather in Philadelphia usually comes in late January. (But this is a bit specious; if this morning's low had been 5oF (-15o C) I wouldn't be saying that.)
- In the last eleven years, it has been 19 degrees or below on some day after January 3 every year. (There's more complete climate data available out there, but I don't feel like looking at it.)
- Forecasters are forecasting lows around 16 degrees F (-9 C) for tonight.
But that raises an interesting question: what can we expect the coldest temperature all year to be?
The normal low temperature in Philadelphia on any given night in late January is 25 degrees -- that's when the normal low is at its lowest -- but only an idiot would say that this means the average lowest temperature all year is 25 degrees. (In fact, even without looking at the records, I'd be shocked to learn that there has been a year in recorded history in which Philadelphia didn't go below 25 at some point.) One tempting thing to do is to make the following assumptions:
- the coldest day of the year will always fall between, say, January 5 and February 10, a period of 37 days. Call this number t;
- the low temperature on any day in that interval is normally distributed with mean 25 degrees and standard deviation σ
- thus the annual minimum temperature should be at the 1/(t+1) quantile of that normal distribution.
That turns out to be roughly 2 standard deviations below the mean. Unfortunately, I don't know what the standard deviation is!
And there's a much bigger problem here -- I've implicitly assumed that the annual minimum always falls in a certain cold period, and that the temperature on each day is independent of each other day! The second assumption is spectacularly bad. If it's colder than average today, it'll probably be colder than average tomorrow.
Also, I've assumed that the low temperatures on a given day of the year are normally distributed, which probably isn't true...
However, one could use a method like this to estimate, say, the mean number of days below, say, fifteen degrees in a given year. If we know the distribution then we can compute, say, that the probability of it being below fifteen degrees on the night of January 3 is ten percent; adding up these probabilities for every night of the year gives an expected mean number of cold nights; call this μ. But the variance is important here, as well. All I can say instinctively is that the variance is probably greater than that of a Poisson distribution with mean μ (which is what you'd usually use to model "rare events"), since if it's cold today it'll probably be cold tomorrow, and it probably doesn't tell us much about whether next week will be cold; in particular I suspect that within the winter season, there aren't pairs of days for which being cold is inversely correlated.
(Of course, this is all testable, but I don't care that much.)
02 January 2008
Stop-motion Tetris
Stop motion tetris is made out of people! Some people got together and played the part of the squares in the game of Tetris; a lecture hall with seats in a grid was the background.
What surprised me is how, at least in this particular game of Tetris, each individual piece got broken up pretty quickly, with only one or two pieces out of the original four remaining on the board. (The four people who made up each dropping piece were wearing the same color T-shirt, and there were enough different colors that for the most part one could assume that people wearing the same color and next to each other were part of the same piece.) This isn't obvious if you're used to, say, the NES version of Tetris (which is the one I played the most); the color scheme is different there. If you stop to think about it for a moment, though, it makes perfect sense that this should happen; it's very rare that four squares which fall at the same time all get eliminated at the same time.
Incidentally, the people writing that post say "Now, we're not mathologists". What is a mathologist? I would argue that mathologists are to mathematicians as musicologists (i. e. people who study music in a scholarly fashion, as opposed to those who produce it) are to musicians. However, there don't seem to be many mathologists in this sense; it's quite difficult to get the sort of deep appreciation for mathematics that one needs without actually doing mathematics, or so it seems to me.
By the way, Tetris is also a traditional Russian board game involving vodka. (Not really. And the amounts of vodka that this description invokes would kill even a Russian.)
What surprised me is how, at least in this particular game of Tetris, each individual piece got broken up pretty quickly, with only one or two pieces out of the original four remaining on the board. (The four people who made up each dropping piece were wearing the same color T-shirt, and there were enough different colors that for the most part one could assume that people wearing the same color and next to each other were part of the same piece.) This isn't obvious if you're used to, say, the NES version of Tetris (which is the one I played the most); the color scheme is different there. If you stop to think about it for a moment, though, it makes perfect sense that this should happen; it's very rare that four squares which fall at the same time all get eliminated at the same time.
Incidentally, the people writing that post say "Now, we're not mathologists". What is a mathologist? I would argue that mathologists are to mathematicians as musicologists (i. e. people who study music in a scholarly fashion, as opposed to those who produce it) are to musicians. However, there don't seem to be many mathologists in this sense; it's quite difficult to get the sort of deep appreciation for mathematics that one needs without actually doing mathematics, or so it seems to me.
By the way, Tetris is also a traditional Russian board game involving vodka. (Not really. And the amounts of vodka that this description invokes would kill even a Russian.)
01 January 2008
Writers' strike day -307
Here in the United States, the Writers Guild of America -- who write a lot of the television shows -- are on strike, and have been for two months. As a result, a lot of TV shows are in reruns, and the networks are starting to move into "reality" television which doesn't need writers.
Anyway, zap2it.com, a web page with TV listings, currently has a banner ad reading: "WRITERS' STRIKE DAY -307: Find out how the Hollywood writers' strike will affect you." The strike actually started on November 5, 2007; this is day 58.
Apparently whoever wrote the code which automatically generates these banners didn't consider the possibility that the strike might run into 2008. And 308 days from today (the putative "Day 1" if the count increments by one each day) is November 4, not November 5... 2008 is a leap year! If I had to guess, I'd say that the code incorporates the fact that the strike started on the 309th day of the year... which is November 5 in an ordinary year, but November 4 in a leap year.
And do people remember how occasionally, around eight years ago, you'd see web sites referring to "19100" for 2000, "19101" for 2001, and so on, since the code which was automatically generating the dates hadn't been fixed for Y2K? This error reminds me of that, although there's no mathematical similarity. But they're both "stupid calendar tricks".
(While I'm on the topic of calendars, check out Claus Tondering's Calendar FAQ.)
Anyway, zap2it.com, a web page with TV listings, currently has a banner ad reading: "WRITERS' STRIKE DAY -307: Find out how the Hollywood writers' strike will affect you." The strike actually started on November 5, 2007; this is day 58.
Apparently whoever wrote the code which automatically generates these banners didn't consider the possibility that the strike might run into 2008. And 308 days from today (the putative "Day 1" if the count increments by one each day) is November 4, not November 5... 2008 is a leap year! If I had to guess, I'd say that the code incorporates the fact that the strike started on the 309th day of the year... which is November 5 in an ordinary year, but November 4 in a leap year.
And do people remember how occasionally, around eight years ago, you'd see web sites referring to "19100" for 2000, "19101" for 2001, and so on, since the code which was automatically generating the dates hadn't been fixed for Y2K? This error reminds me of that, although there's no mathematical similarity. But they're both "stupid calendar tricks".
(While I'm on the topic of calendars, check out Claus Tondering's Calendar FAQ.)
Notation again
I just came across this article: Florian Cajori, History of symbols for n-factorial. Isis, Vol. 3, No. 3 (Summer, 1921), pp. 414-418. Available from JSTOR, if you have access. Cajori was the author of A History of Mathematical Notations, which is the canonical source on the subject of the history of mathematical notations; I will confess I have never seen a copy of his book.
I didn't realize how many historical notations there have been for the factorial. n! is of course the most common one these days. Γ(n+1) is seen sometimes, although I personally find it a bit perverse to use this notation if you know that n is a positive integer.
Supposedly Gauss used Π(n). Someone named Henry Warburton used 1n|1, a special case of
an|1 = a(a+1)...(a+(n-1)). (This is a variant of the Pochhammer symbol. It's not clear to me what the 1 in the superscript means.) Other notations include a bar over the number and writing the number inside a half-box (with lines on the left and below). Augustus de Morgan is mildly famous for not using a symbol, and once said: "Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results." (I'm copying this from Earliest Uses of Symbols in Mathematics, although I learned it from somebody's office door. I found the Cajori article while looking for this quote just now.)
Apparently adopting any sort of symbol was resisted by some people, at least in their more elementary writings (textbooks for undergraduates and the like), because they didn't want to overload their students with symbols. I'm not sure if I agree with this for the factorial. A little thought experiment, though -- why don't we have a symbol for 1 + 2 + ... + n? (Although a bit of reflection convinces me that the reason is because we have an explicit formula for this, namely n(n + 1)/2.) But n! probably arises more often.
While I'm on the subject, you should read Knuth's Two notes on notation (Amer. Math. Monthly 99 (1992), no. 5, 403--422; arXiv:math/9205211v1), which suggests the notation [P] for "1 if P is true, 0 if P is false"; this turns out to be a quite useful generalization of the Kronecker delta. It also suggests notation for the Stirling cycle and subset numbers (those are, um, the Stirling numbers of the first and second kinds, respectively? or the second and first kinds? See, those names are better.)
Happy New Year!
I didn't realize how many historical notations there have been for the factorial. n! is of course the most common one these days. Γ(n+1) is seen sometimes, although I personally find it a bit perverse to use this notation if you know that n is a positive integer.
Supposedly Gauss used Π(n). Someone named Henry Warburton used 1n|1, a special case of
an|1 = a(a+1)...(a+(n-1)). (This is a variant of the Pochhammer symbol. It's not clear to me what the 1 in the superscript means.) Other notations include a bar over the number and writing the number inside a half-box (with lines on the left and below). Augustus de Morgan is mildly famous for not using a symbol, and once said: "Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results." (I'm copying this from Earliest Uses of Symbols in Mathematics, although I learned it from somebody's office door. I found the Cajori article while looking for this quote just now.)
Apparently adopting any sort of symbol was resisted by some people, at least in their more elementary writings (textbooks for undergraduates and the like), because they didn't want to overload their students with symbols. I'm not sure if I agree with this for the factorial. A little thought experiment, though -- why don't we have a symbol for 1 + 2 + ... + n? (Although a bit of reflection convinces me that the reason is because we have an explicit formula for this, namely n(n + 1)/2.) But n! probably arises more often.
While I'm on the subject, you should read Knuth's Two notes on notation (Amer. Math. Monthly 99 (1992), no. 5, 403--422; arXiv:math/9205211v1), which suggests the notation [P] for "1 if P is true, 0 if P is false"; this turns out to be a quite useful generalization of the Kronecker delta. It also suggests notation for the Stirling cycle and subset numbers (those are, um, the Stirling numbers of the first and second kinds, respectively? or the second and first kinds? See, those names are better.)
Happy New Year!
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