The End 2016 Mathematics A To Z: Image


It’s another free-choice entry. I’ve got something that I can use to make my Friday easier.

Image.

So remember a while back I talked about what functions are? I described them the way modern mathematicians like. A function’s got three components to it. One is a set of things called the domain. Another is a set of things called the range. And there’s some rule linking things in the domain to things in the range. In shorthand we’ll write something like “f(x) = y”, where we know that x is in the domain and y is in the range. In a slightly more advanced mathematics class we’ll write f: x \rightarrow y . That maybe looks a little more computer-y. But I bet you can read that already: “f matches x to y”. Or maybe “f maps x to y”.

We have a couple ways to think about what ‘y’ is here. One is to say that ‘y’ is the image of ‘x’, under ‘f’. The language evokes camera trickery, or at least the way a trick lens might make us see something different. Pretend that the domain is something you could gaze at. If the domain is, say, some part of the real line, or a two-dimensional plane, or the like that’s not too hard to do. Then we can think of the rule part of ‘f’ as some distorting filter. When we look to where ‘x’ would be, we see the thing in the range we know as ‘y’.

At this point you probably imagine this is a pointless word to have. And that it’s backed up by a useless analogy. So it is. As far as I’ve gone this addresses a problem we don’t need to solve. If we want “the thing f matches x to” we can just say “f(x)”. Well, we write “f(x)”. We say “f of x”. Maybe “f at x”, or “f evaluated at x” if we want to emphasize ‘f’ more than ‘x’ or ‘f(x)’.

Where it gets useful is that we start looking at subsets. Bunches of points, not just one. Call ‘D’ some interesting-looking subset of the domain. What would it mean if we wrote the expression ‘f(D)’? Could we make that meaningful?

We do mean something by it. We mean what you might imagine by it. If you haven’t thought about what ‘f(D)’ might mean, take a moment — a short moment — and guess what it might. Don’t overthink it and you’ll have it right. I’ll put the answer just after this little bit so you can ponder.

Close up view of a Flemish Giant rabbit looking at you from the corner of his eye.
Our pet rabbit on the beach in Omena, Michigan back in July this year. Which is a small town on the Traverse Bay, which is just off Lake Michigan where … oh, you have Google Maps, you don’t need me. Anyway we wondered what he would make of vast expanses of water, considering he doesn’t like water what with being a rabbit and all that. And he watched it for a while and then shuffled his way in to where the waves come up and could wash over his front legs, making us wonder what kind of crazy rabbit he is, exactly.

So. ‘f(D)’ is a set. We make that set by taking, in turn, every single thing that’s in ‘D’. And find everything in the range that’s matched by ‘f’ to those things in ‘D’. Collect them all together. This set, ‘f(D)’, is “the image of D under f”.

We use images a lot when we’re studying how functions work. A function that maps a simple lump into a simple lump of about the same size is one thing. A function that maps a simple lump into a cloud of disparate particles is a very different thing. A function that describes how physical systems evolve will preserve the volume and some other properties of these lumps of space. But it might stretch out and twist around that space, which is how we discovered chaos.

Properly speaking, the range of a function ‘f’ is just the image of the whole domain under that ‘f’. But we’re not usually that careful about defining ranges. We’ll say something like ‘the domain and range are the sets of real numbers’ even though we only need the positive real numbers in the range. Well, it’s not like we’re paying for unnecessary range. Let me call the whole domain ‘X’, because I went and used ‘D’ earlier. Then the range, let me call that ‘Y’, would be ‘Y = f(X)’.

Images will turn up again. They’re a handy way to let us get at some useful ideas.

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.