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

[luzula]

Short reply, because I'm going to bed now. (I was actually planning to write porn tonight, but ended up doing math-related stuff instead. Funny, it's more often the other way around. *g*)

I am not making models, so I don't actually know much about making these things work in applications--I'm doing pure math. Basically, in the stuff I do, we try to allow for as broad a class of boundaries as we can, so that it's applicable to as many types of problems as possible. In a specific problem, boundaries can be really concrete--like, they can be given by the walls of a room or something.

[meresy.livejournal.com]

Have a good sleep! (A healthy porn-to-math ratio is recommended by doctors! Or biochemists. Trufax.)

Wheee, pure math. That is a bogglement to me indeed, since at the end of the day biology is very much about concrete things. Molecules are on the tiny end, and you have to make up ways to represent them, but they are not abstract. *g*