Open course ware

First of all, the way a nonlinear autonomous system looks, you have had some practice with it by now. This is nonlinear. The right-hand side are no longer simple combinations ax plus by. Nonlinear and autonomous, these are function just of x and y. There is no t on the right-hand side. Now, most of today will be geometric. The way to get a geometric picture of that is first by constructing the velocity field whose components are the functions f and g. This is a velocity field that gives a picture of the system and has solutions. The solutions to the system, from the point of view of functions, they would look like pairs of functions, x of t, y of t. But, from the point of view of geometry, when you plot them as parametric equations, they are called trajectories of the field F, which simply means that they are curves everywhere having the right velocity. So a typical curve would look like -- There is a trajectory. And we know it is a trajectory because at each point the ve...

PROFESSOR: A new way of thinking about this is based on integral equations. So it's a nice method. It's less complicated than it seems, suddenly leads to some expressions for the scattering amplitude. We'll find another formula for this quantity, f of k of theta, when we cannot calculate it with phase shifts. But phase shifts is very powerful if your problem has spherical symmetry. It's probably the first thing you try, because there's nothing to prevent you from finding a solution, in that case. OK, so integral equations, so integral scattering equation, OK, integral scattering equation. So what are we solving? We're solving minus h squared over 2m Laplacian, plus v of r of potential, that is not necessarily spherical, equals or acts on a psi of r, to give you h squared, k squared over 2m, psi of r. So it's an equation for eigenstates, the Schrodinger equation. It's good. It's nice. And this is what we've been solving. We've be...

PROFESSOR: Another way to talk about congruence and remainder arithmetic is to work strictly with remainders, which makes things a little simpler because you don't have to worry about the fact that the product of two remainders may, for example, be too big to be a remainder. To knock it back in range, you have to take the remainder again. And that's what this abstract idea of the ring of integers modulo n, the ring Z sub n, captures in a quite elegant way. So it's going to allow us to talk strictly about equality instead of congruence. And let's remind ourselves that the basic idea behind working with a remainder arithmetic was that every time we got a number that was too big to be a remainder, we just hit it with the remainder operation again to bring it back in range. And so the operations in Zn work exactly that way. The elements of Zn are the remainders. That is, the numbers from 0, including 0, up to n, but not including n. So there are n of them from...

So is there any way to prove that pidgins never existed, or can they only not be proven to exist? So actually, I think this question hinges on the way you def-- on what you mean by pidgins, what is meant by pidgins, right? So at least wanted define pidgins, which is to say that there are these systems of communication which are created when people come from different language background and they have to communicate, they have speak to each other. So like in Africa, in Asia-- in fact, this term pidgin arose in the coast of China, and it referred to this business language that people used to cross linguistic barriers. So from that perspective, pidgins do exist, right, if you take it as that definition. Pidgins have been documented over and over again. They are non-native languages, and typically, they indeed reduce in structure. But as people use them more and more, they can develop structure. So-- go ahead. OK. AUDIENCE: I guess my question be more like, if-- you were sayi...