What my research is about

[luzula]

sionnain asked what my research is about. My first answer was that I study partial differential equations, and the kind of things I ask are: If an operator has a particular regularity, what regularity will the solution have? How does the solution behave at the boundary?

At which point sionnain asked what those terms mean. *g*

First, you need to remember what an equation is. This is an example:

x + 1 = 2

The important thing with an equation is that there's something that's unknown, and your goal is to find out what the unknown thing is equal to. In this case the unknown thing is the "x", and it's easy to see that it must be equal to 1. So x=1 is the solution of the equation.

In differential equations, which is what I do, the unknown thing isn't a number any more, but a function. A function is something where you enter one thing (often a number, which is called the variable), and the function does something with the variable according to a rule, and gives you a result. For example, the function

f(x) = 2x

takes the number x and doubles it. More generally, you can think of a function kind of as a recipe--you enter in all the ingredients, the recipe tells you what to do with them, and then you get a cake as the output.

Often a function isn't defined everywhere, that is, there's a rule that says we can only enter certain numbers. For example, we could decide that we only allow numbers between 0 and 1. In this example, the boundary of the area where the function is defined is the 0 and the 1, because they are at the "edges" of the area.

The regularity of a function has to do with how much the output changes if you only change the input a little bit. If the output jumps around a lot when you change the input just a little bit, it has low regularity. This is generally undesirable in applications.

An operator is sort of like a meta-function. Ordinary functions take a number as input and give you another number as output. Instead, an operator takes a function as input and gives you another function as output. Operators are often used to construct differential equations (also, they often involve taking the derivative of the function, if you know what that is). You get an equation like this: if A is an operator, f(x) is an unknown function and g(x) is a known function, then

A(f(x)) = g(x)

is an equation. Often, one also specifies the values of f(x) at the boundary. We want to find the solution, i e figure out what kind of function f(x) is.

Hopefully these sentences will make more sense now: If an operator has a particular regularity, what regularity will the solution have? How does the solution behave at the boundary?

So what do you use these things for? Well, partial differential equations have lots of applications. For example, you can think of heat as a function of three space variables and one time variable. We use a particular point in space and time as the input, and the heat function tells us the temperature at that point. We can specify the temperature in a room at a specific time, and also specify what the walls do (maybe they are insulating?). This is equivalent to specifying the values of the heat function at the boundary. Then, solving the heat equation is the same as figuring out how the temperature inside the room changes as time passes.

Comments

[kill-claudio.livejournal.com]

Oh, this is so helpful! I actually understood most of that - I got lost a bit around operators. So, an operator works on a function in the same way a function works on a number? It helps you to find the function that you don't know?

I had to take a couple of statistics classes at uni; the sheer volume of information you get out of an archaeological dig sort of makes some kind of statistical analysis necessary, otherwise it's pure guess work. We did ... analysis of variance, I think? And the Mann-Whitney test. I'm afraid I've forgotten most of it already. :-/

[luzula]

So, an operator works on a function in the same way a function works on a number? It helps you to find the function that you don't know?

Yes, that's it!

And yeah, one tends to forget the stuff that one is not actively using. I took maybe a semester of probability and statistics, but I've forgotten most of it, too, because I never use it.

I suppose my question to an archaeologist would be "do you get to do fun fieldwork and actually dig up cool stuff, or is it mostly indoors lab work?" Uh, I suppose this mostly reveals my opinion that outdoors fieldwork = fun, and lab work = boring. *g*

[kill-claudio.livejournal.com]

I suppose my question to an archaeologist would be "do you get to do fun fieldwork and actually dig up cool stuff, or is it mostly indoors lab work?" Uh, I suppose this mostly reveals my opinion that outdoors fieldwork = fun, and lab work = boring. *g*

Well, that's often true! Although if it's pouring with raining then sometimes you can be pretty glad to be in the lab.

Most archaeologists specialise in a particular kind of evidence; ceramics, animal remains, human remains, metallurgy, plant remains, worked flint, you get the idea. So they might go out on site if their expertise is needed, or they might get the material sent to the lab to do research on it as a follow up to the dig, or they might have a research project going re-analysing old materials. New theories have to be tested on the evidence that's already on record as well as new stuff coming out of the ground.

This is where museums are such a necessary part of archaeology, because everything that's dug has to be properly curated by someone who knows what they're doing. Being able to display it for the public is a side benefit to archaeologists; although naturally, that's where the money comes from.

[luzula]

Hmm, interesting. So what do you specialize in?

[kill-claudio.livejournal.com]

Human remains. I'm mostly a burial archaeologist; I've worked in a few cemeteries, including a pretty big Anglo-Saxon one in Norfolk. Most archaeologists have a period specialisation as well, and mine is the early Palaeolithic, for which most of the evidence is bones and maybe a bit of flint, so it was kind of inevitable.