The End 2016 Mathematics A To Z: The Fredholm Alternative


Some things are created with magnificent names. My essay today is about one of them. It’s one of my favorite terms and I get a strange little delight whenever it needs to be mentioned in a proof. It’s also the title I shall use for my 1970s Paranoid-Conspiracy Thriller.

The Fredholm Alternative.

So the Fredholm Alternative is about whether this supercomputer with the ability to monitor every commercial transaction in the country falls into the hands of the Parallax Corporation or whether — ahm. Sorry. Wrong one. OK.

The Fredholm Alternative comes from the world of functional analysis. In functional analysis we study sets of functions with tools from elsewhere in mathematics. Some you’d be surprised aren’t already in there. There’s adding functions together, multiplying them, the stuff of arithmetic. Some might be a bit surprising, like the stuff we draw from linear algebra. That’s ideas like functions having length, or being at angles to each other. Or that length and those angles changing when we take a function of those functions. This may sound baffling. But a mathematics student who’s got into functional analysis usually has a happy surprise waiting. She discovers the subject is easy. At least, it relies on a lot of stuff she’s learned already, applied to stuff that’s less difficult to work with than, like, numbers.

(This may be a personal bias. I found functional analysis a thoroughgoing delight, even though I didn’t specialize in it. But I got the impression from other grad students that functional analysis was well-liked. Maybe we just got the right instructor for it.)

I’ve mentioned in passing “operators”. These are functions that have a domain that’s a set of functions and a range that’s another set of functions. Suppose you come up to me with some function, let’s say f(x) = x^2 . I give you back some other function — say, F(x) = \frac{1}{3}x^3 - 4 . Then I’m acting as an operator.

Why should I do such a thing? Many operators correspond to doing interesting stuff. Taking derivatives of functions, for example. Or undoing the work of taking a derivative. Describing how changing a condition changes what sorts of outcomes a process has. We do a lot of stuff with these. Trust me.

Let me use the name `T’ for some operator. I’m not going to say anything about what it does. The letter’s arbitrary. We like to use capital letters for operators because it makes the operators look extra important. And we don’t want to use `O’ because that just looks like zero and we don’t need that confusion.

Anyway. We need two functions. One of them will be called ‘f’ because we always call functions ‘f’. The other we’ll call ‘v’. In setting up the Fredholm Alternative we have this important thing: we know what ‘f’ is. We don’t know what ‘v’ is. We’re finding out something about what ‘v’ might be. The operator doing whatever it does to a function we write down as if it were multiplication, that is, like ‘Tv’. We get this notation from linear algebra. There we multiple matrices by vectors. Matrix-times-vector multiplication works like operator-on-a-function stuff. So much so that if we didn’t use the same notation young mathematics grad students would rise in rebellion. “This is absurd,” they would say, in unison. “The connotations of these processes are too alike not to use the same notation!” And the department chair would admit they have a point. So we write ‘Tv’.

If you skipped out on mathematics after high school you might guess we’d write ‘T(v)’ and that would make sense too. And, actually, we do sometimes. But by the time we’re doing a lot of functional analysis we don’t need the parentheses so much. They don’t clarify anything we’re confused about, and they require all the work of parenthesis-making. But I do see it sometimes, mostly in older books. This makes me think mathematicians started out with ‘T(v)’ and then wrote less as people got used to what they were doing.

I admit we might not literally know what ‘f’ is. I mean we know what ‘f’ is in the same way that, for a quadratic equation, “ax2 + bx + c = 0”, we “know” what ‘a’, ‘b’, and ‘c’ are. Similarly we don’t know what ‘v’ is in the same way we don’t know what ‘x’ there is. The Fredholm Alternative tells us exactly one of these two things has to be true:

For operators that meet some requirements I don’t feel like getting into, either:

  1. There’s one and only one ‘v’ which makes the equation Tv  = f true.
  2. Or else Tv = 0 for some ‘v’ that isn’t just zero everywhere.

That is, either there’s exactly one solution, or else there’s no solving this particular equation. We can rule out there being two solutions (the way quadratic equations often have), or ten solutions (the way some annoying problems will), or infinitely many solutions (oh, it happens).

It turns up often in boundary value problems. Often before we try solving one we spend some time working out whether there is a solution. You can imagine why it’s worth spending a little time working that out before committing to a big equation-solving project. But it comes up elsewhere. Very often we have problems that, at their core, are “does this operator match anything at all in the domain to a particular function in the range?” When we try to answer we stumble across Fredholm’s Alternative over and over.

Fredholm here was Ivar Fredholm, a Swedish mathematician of the late 19th and early 20th centuries. He worked for Uppsala University, and for the Swedish Social Insurance Agency, and as an actuary for the Skandia insurance company. Wikipedia tells me that his mathematical work was used to calculate buyback prices. I have no idea how.

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My Math Blog Statistics, October 2014


So now let me go over the mathematics blog statistics for October. I’ll get to listing countries; people like that.

It was a good month in terms of getting people to read: total number of pages viewed was 625, up from 558, and this is the fourth-highest month on record. The number of unique visitors was up too, from 286 in September to 323 in October, and that’s the third-highest since WordPress started giving me those statistics. The views per visitor barely changed, going from 1.95 to 1.93, which I’m comfortable supposing is a statistical tie. I reached 18,507 total page views by the end of October, and maybe I’ll reach that nice round-ish 19,000 by the end of November.

The countries sending me the most visitors were the usual set: the United States with 393, the United Kingdom with 35, and Austria with 23. Curiously, Argentina sent me 20 readers, while Canada plummeted down to a mere nine. Did I say something wrong, up there? On the bright side my Indian readership has grown to nine, which is the kind of trend I like. Sending just a single reader this past month were Albania, Brazil, Denmark, Estonia, Finland, Indonesia, Japan, the Netherlands, Nicaragua, Norway, Poland, Saint Kitts and Nevis, Serbia, Spain, Sweden, Taiwan, Turkey, and the United Arab Emirates. Brazil, Estonia, Finland, the Netherlands, and Sweden were single-reader countries last month, and Finland and Sweden also the month before. I feel embarrassed by the poor growth in my Scandinavian readership, but at least it isn’t dwindling.

The most popular posts in October got a little bit away from the comics posts; the ones most often read were:

There weren’t any really great bits of search term poetry this month, but there were still some evocative queries that brought people to me, among them:

  • where did negative numbers come from
  • show me how to make a comic stip for rationalnumbers
  • desert island logarithm
  • herb jamaal math ludwig
  • in the figure shown below, Δabc and Δdec are right triangles. if de = 6, ab = 20, and be = 21, what is the area of Δdec?
  • origin is the gateway to your entire gaming universe.

That “origin is the gateway” thing has come up before. I stil don’t know what it means. I’m a little scared by it.