X-Y correspondence

Google hits for "Pascal-Fermat correspondence": 3,230. For "Fermat-Pascal correspondence": 206.

Does anyone have any idea why? In particular it seems like one would sort either on importance (but which of these two is more important?) or alphabetically (but that gives the wrong result).

And given two individuals X and Y, how can we predict whether "X-Y correspondence" or "Y-X correspondence" is more common? I'm sure there are no hard and fast rules here but there must at least be some trends, and would expect a situation similar to how gender is assigned to words in languages, foreign to me, where words have gender.

Which comes first, the nominal or real record high?

A quick puzzle: say that gas prices hit a record high in nominal dollars. And say they also, around the same time, hit a record high in real (inflation-adjusted) dollars. Which comes first?

Say the nominal price is f(t) and the real price is h(t) = f(t)g(t), where g is monotone decreasing. Then h'(t) = f'(t) g(t) + f(t) g'(t). So if the nominal price is at a maximum at time T, then f'(T) = 0 and so h'(T) = f(T) g'(T). f(T) is positive because it's a price and g'(T) is negative by assumption. So h'(T) is negative and at the time of the nominal high, the real price is already decreasing. The real high comes first, and there's a short period in which the real price is decreasing but the nominal price is still increasing. This makes sense - a nominally constant price is decreasing in real dollars.

This is brought to you by an example from Geoff Nunberg's book Talking Right, on how the Right has distorted political language in the United States in an effort to marginalize the Left. At the particular point I'm commenting on, argues that the right likes to proclaim "record highs" in gas prices which are basically always in nominal dollars and therefore make the gas price rise look worse than it is. My argument, however, says nothing about that - taking derivatives makes this a local problem, and "record highs" (nominal or real) are global maxima.

"Orthogonal" at the Supreme Court

Orin Kerr at the Volokh Conspiracy points to arguments in a U. S. Supreme Court case yesterday which used the word "orthogonal" in the technical-jargon sense defined, say, at the Jargon File. (See page 24 of the original transcript.) There's a follow-up here by Eugene Volokh, basically saying that there's no point in using big words if your audience doesn't understand them. (And the justices did stop to ask what the word meant.)

What is "classical"?

John Cook quotes a definition of "classical", due to Ward Cheney and Will Light in the introduction to their book on approximation theory. Basically, something is "classical" if it was known when you were a student.

The problem with this definition is that it depends on the speaker, which is really not a good property for a definition!

Best bad math joke ever

One of my favorite bad math jokes ever is now in Wikipedia, and no, I didn't add it.

Namely, exercise 6.24 of Richard Stanley's Enumerative Combinatorics, Volume 2 asks the reader to

"Explain the significance of the following sequence: un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu..."

The answer is that these are the "Catalan numbers", i. e. the numbers in the Catalan language. If this seems random, note that exercise 6.21 is the famous exercise in 66 parts (169 in the extended online version, labelled (a) through (m⁷)), which asks the reader to prove that 66 (or 169) different sets are counted by the Catalan numbers.

I'm telling you about this joke because the Wikipedia article on Catalan numbers begins with a link to the list of numbers in various languages.

An alternative version of this joke (American Mathematical Monthly, vol. 103 (1996), pages 538 and 577) asks you to identify the sequence "una, dues, cinc, catorze, quaranta-dues, cent trenta-dues, quatre-cent vint-i-nou,...", which are the Catalan numbers 1, 2, 5, 14, 42, 132, 429... in the Catalan language. (I'm reporting the spellings as I found them in my sources; the first series is in the masculine and the second is in the feminine, as Juan Miguel pointed out in the comments.)

Why isn't it expnormal?

We say that a random variable X has a lognormal distribution if its logarithm, Y = log X, is normally distributed. The normal distribution often occurs when a random variable comes about by combining a bunch of small independent contributions, but those contributions combine additively; when the combination is multiplicative instead, lognormals occur. For example, lognormal distributions often occur in models of financial markets.

But of course X = exp Y, so the variable we care about is the exponential of a normal. Why isn't it called expnormal?

Complices are made up of simplexes, or something like that

How come the plural of "simplex", in standard mathematical usage, is "simplices", but the plural of "complex" isn't "complices"?

My first thought is that it's because "complex" is also a noun in standard English, so it pluralizes like the English noun, while "simplex" isn't.

(As you may have guessed, I'm reading something that mentioned simplicial complexes.)

The Arbesman limit

Samuel Arbesman talks how to get something named after yourself. Of course, he names something after himself -- the "Arbesman limit", which is the number of things that one person can have named after themselves. (Gauss, Euler, etc. provide a lower bound for this limit.)

Supposedly Banach originally named his spaces "spaces of type B" or something like that, figuring that people would see the B, assume it standed for Banach, and start calling them Banach spaces. If that's true, it worked.

Permutatinos

One of my most common typos is "permutatinos" for "permutations". A friend points out that this is quite apt.

How do you pronounce ≤ and ≥?

I'm taking a break from proofreading a paper. I'm reading it out loud, because I find this is the best way to catch mistakes; it forces me to look at every word.

There are inequalities in this paper, so the signs ≤ and ≥ come up a lot. How do you pronounce these? When I was in college I pronounced them "less than or equal to" and "greater than or equal to". But sometime around the first year of graduate school I seem to have shifted to "at most" and "at least", which have the obvious advantage of being shorter.

Edit (11:15 pm): It appears I've mentioned this before.

Etymology of "theorem"

Is there some connection between the etymology of "theorem" and words like "theology" or "theist"?

For "theorem" the OED says: theorem, from the late Latin theorema, from the Greek θεωρημα, spectacle, speculation, theory, (in Euclid) a proposition to be proved, from θεωρειν to be a spectator (θεωροσ), to look at, inspect. (This isn't an exact quote; I've expanded some of the abbreviations, and suppressed some of the accent marks. But if you're the sort of person who could actually answer my question you probably already knew that.)

But for the words where "the-" or "theo-" is god-related, of which there are a lot, it just says things like "from Greek θεοσ, God" and doesn't go any further. And maybe you could imagine that people "look at" or "inspect" God. Of course I recognize that the OED is not the best possible source for these things -- but I'm suspecting that someone in my audience has also noticed this apparent coincidence of words and knows the answer.

(I just want to reiterate that the title "God Plays Dice" is not a religious thing; it's alluding to the quote of Einstein, as I've written before.)

A lexicographical copout

Definitions of "set" (as a noun) in the OXford English Dictionary include:

Math. Used variously, as defined by the individual author. Obs.

(This is set, n.², definition 10b. Definition 10c is "An assemblage of distinct entities, either individually specified or which satisfy certain specified conditions", which is what mathematicians usually mean by "set". set, n.² basically is the definitions of "set" that mean "bunch of things", while set, n.¹ has to do with things hardening, hanging, etc.

I was led to this definition by a talk about the modern dictionary from the TED conference, in which Erin McKean, an editor for the OED, talks about how in the future dictionaries won't be like they are now. (A transcript of the talk is here.) She mentions that set has many definitions, one of which is "miscellaneous technical senses". I was actually hoping that this would include the mathematical sense, and I could write a post about how the compilers of the OED hate mathematicians. The numbered definition she was talking about, set, n.¹ definition #30, has various lettered sub-definitions; usually the sub-definitions under a number are related, but this one isn't. McKean's theory is that it was Friday afternoon and the people writing it wanted to go down to the pub, and she called it a lexicographical copout.

But McKean also makes a more substantial point in the talk; it's not her job to enforce rules about the language, but rather to describe it. And nowhere is this more true in mathematics, where we basically just define words, in our papers, to mean whatever we want them to mean. The definition I gave above may be obsolete -- we've pretty much converged on a meaning for set -- but the spirit there lives on.

"Torsion" as an adjective

I had an algebra professor my first year of grad school who would describe groups in which all elements had finite order as "torsion". For example, he'd say something like "since S₄ is torsion"...

Today I was browsing through Tao and Vu's additive combinatorics. They give the following definition:

Definition 3.1 (Torsion) If Z is an additive group and x ∈ Z, we let ord(x) be the least integer n ≥ 1 such that n · x = 0, or ord(x) = +∞ if no such integer exists. We say that Z is a torsion group if ord(x) is finite for all x ∈ Z, and we say that it is an r-torsion group for some r ≥ 1 if ord(x) divides r for all x ∈ Z. We say that Z is torsion-free if ord(x) = +∞ for all x ∈ Z

In the ensuing sections they often refer to "a torsion group". But they never use "torsion" alone, or if they do I didn't see it. This seems to me to be good evidence that the use of bare "torsion" isn't universal. (It also sounds weird to my ears, but a lot of things this particular professor said sounded weird to my ears and turned out to be standard usage among mathematicians. Since I was a first-year I had some learning to do.)

And the way that the word "torsion" is used here seems different than, say, an "open set". You can start a proof by saying "Let U be open." But "Let G be torsion." seems lazy. Perhaps mathematical English has two sorts of adjectives -- adjectives of the second kind like "open", which can be used without the implied noun they modify, and adjectives of the first kind like "torsion", which can only be used with the implied noun.

(The swapping of "second" and "first" is deliberate; it's like the Stirling numbers. I can never remember which is which.)

When Obama wins...

When Obama wins is Jason Kottke's collection of people's postings to Twitter that begin "when Obama wins".

It appears to have started with When Obama wins... unicorns will crap ice cream and pastries.

Of course this is vacuously true! Or at least it would be given the nonexistence of unicorns, which I think we can take as an axiom. To disprove it one would need to produce a counterexample, namely a unicorn that does not crap ice cream and pastries. But we can't do that, since there are no unicorns!

Now, if it said "When Obama wins... everybody will have their own unicorn that craps ice cream and pastries", that would be falsifiable. (And almost certainly false. Let's face it, it's hard to make sure everybody gets a unicorn.)

Discreet mathematics

What is discreet mathematics? Urbandictionary.com tells us: "Discreet mathematics refers to the subtle study of mathematics. Discreet mathematics is characterised by furtive looks in maths textbooks disguised as pr0n and by secret maths lectures held in abandoned warehouses at midnight..."

Of course, discrete mathematics is an entirely different beast.

edit, 6:31 pm: A poster on a cryptography mailing list pointed out that cryptography is "discreet" mathematics.

An "algorithmic curve"?

Steampunk Moves Between Two Worlds, from today's Monkey Writes.

“There seems to be this sort of perfect storm of interest in steampunk right now,” Mr. von Slatt said. “If you go to Google Trends and track the number of times it is mentioned, the curve is almost algorithmic from a year and a half ago.” (At this writing, Google cites 1.9 million references.)

Here's the curve at Google Trends. It looks linear to me, although with a lot of noise. I suspect, though, that the speaker meant to say it was exponential (because exponentials grow really fast, so this would be in keeping with the rest of the article), then confused that with "logarithmic" (which is a pretty common mistake), and then rearranged a few letters to get "algorithmic".

And a random Google Trends fact: the number of people searching for mathematics has declined steadily over the past four years. Math has been basically flat over that time period, but shows a very large dip during the summers. Presumably the people searching for "math" are students. Maths displays what appears to be a more complicated seasonal pattern; can this be explained by the UK academic calendar. As for "mathematics" declining, I suspect people find the word too long.

The English language is not equipped for metric spaces

From Lydia Millet's novel Oh Pure And Radiant Heart, in which Robert Oppenheimer, Enrico Fermi and Leo Szilard, three of the chief minds behind the Manhattan Project, find themselves in contemporary America (p. 290):

Scientists at the Atomic Energy Commission took advantage of the testing in the Marshall Islands to study the effects of radiation on people.
In 1956, at an AEC meeting, one official admitted that Rongelap was the most contaminated place on earth. He said of the Marshall Islanders, reportedly without irony, "While it is true that these people do not live, I would say, the way Westerners do — civilized people — it is nevertheless true that they are more like us than mice.

To my ear, something about this last sentence is ambiguous -- and I suspect that a mathematician is more likely to spot this ambiguity than an average person. Let's assume that degrees of civilizedness fall on a scale from 0 to 1, with mice at 0 and Westerners at 1. Say that we have a magical civilizedness-measuring meter, and the Marshall Islanders fall at 1/3. Is the scientist's statement true?

If the scientist is saying that the distance in some abstract civilizedness-space (here caricatured by the unit interval) from Westerners to Marshall Islanders is less than the distance from Westerners to mice, then yes. The last clause could be rephrased as "it is nevertheless true that they are more like us than mice are like us." But if the scientist is saying that the distance from Westerners to Marshall Islanders is less than the distance from Marshall Islanders to mice, then it's not true if the islanders fall at 1/3; to force this interpretation, the original sentence could be rephrased as "it is nevertheless true that they are more like us than they are like mice." (I make no claim that these are the most elegant possible rephrasings, just that they clear up the ambiguity.)

Of course, in this particular case, I would claim the scientist intended the second interpretation; regardless of what one thinks about how civilized various groups of humans are, it is obvious that all such groups are more civilized than mice. (I apologize to fans of the Hitchhiker's Guide series.) So there is no need to even make the statement under the first interpretation! There is not much point in telling people something they already know.

Also, this shouldn't need saying, but the value 1/3 above is entirely hypothetical, and I do not mean to make any statements about the civilizedness of actual groups of life forms.

Cech check Czech

Did you know that Eduard Cech (whose cohomology is sometimes denoted by a "check") was Czech?

Open and closed?

At Language Log there's a post about how English-speakers use "open" and "closed", which are not grammatically the same sort of thing, in opposition to each other -- "open"/"close" or "opened"/"closed" would, on the surface, make more sense. (Compare French ouvert and fermé, which are both past participles.)

I won't try to summarize the linguistic content of the post; I'm not a linguist, although I did go through a phase where that seemed interesting.

But in mathematics-land, open and closed aren't even opposites, in the sense that open means not-closed and closed means not-open. Of course the complement of an open set is closed, and vice versa, but that's a more complicated relationship, because now we're talking about two sets, not one. This is one of about a zillion examples of how we take perfectly good natural-language words and give them specific meanings (group, ring, field, set, class, ...), which may or may not be preferable to making up entirely new words as some other fields (biology comes to mind) prefer.

Names of mathematicians in the MSC

Dave Rusin has a list of names of mathematicians which appear in the MSC scheme. (That's the "Mathematics Subject Classification." I almost called it the "MSC classification" but managed to stop myself.) For reference, here's the 2000 MSC. (Rusin's file treats the 1991 version.) The idea is that these must be Important Mathematicians because some subfield of mathematics has been named after them, for the most part.

Rusin claims that Abel is the only person usually given the honor of having his name lowercased ("abelian", not "Abelian"). I have a sense that "euclidean" also occurs, though I may be thinking of the French usage.

The most amusing one, to me, is that "Monte Carlo" is named after a person -- Charles III of Monaco -- who was as far as I know not a mathematician at all. Amerigo Vespucci is also on the list; 01A12 is the history of mathematics and mathematicians of indigenous cultures of the Americas. If I had been compiling the list I'd probably have missed that.

Rusin also calls Fibonacci a "rabbit-farmer". I'd laugh, except I'm pretty sure at least one of my students also thinks this.