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PROFESSOR: Having looked at the quizzes in a preliminary fashion, I come to the conclusion that some notes on tensors would be useful for the remainder of the term. So at great pain and personal sacrifice I will endeavor to [INAUDIBLE] All right, let me remind you of where we were a week ago-- before a slight unpleasantness intervened-- and we had just begun to look at the properties of a very useful surface, the representation quadric. And we said that to define it we would take the elements of a second rank tensor, something of the form A11, A12, A13, A21, A22, A23, A31, A32, A33. And we'd use those elements as coefficients in a second rank, a quadratic equation, of the form Aij Xi Xj equals 1. OK, so the Xi and Xj are the coordinates in our three-dimensional space. And the set of coordinates that satisfy this equation-- setting the right hand side equal to a constant-- will define some surface in space. And it's going to be a surface that is a quadratic form, s...

ALBERT MEYER: We've looked at independence for two events. What about when we have a bunch of events? Well, in that case we want to look at the idea of mutual independence. So let's check that out. I'll say that if I have n different events, I'll say that they're mutually independent, intuitively. If the probability that one of them occurs is unchanged by which other ones happen to have occurred. So expressed in conditional probability, which is the way to make it precise, what we're really saying is that events A1 through An are mutually independent when the probability of Ai is equal to the probability of Ai, given the intersection of any of the other As as long as i is not one of them. So take A1, A2, or A1, A2, A3, and so on. And A5 is going to be independent of all of those other intersections. If we shift over to the other definition of independence that we used for two sets, in terms of products, you could say that n sets are mutually indepe...

PROFESSOR: We've briefly looked at graph isomorphism in the context of digraphs. And it comes up in even more fundamental way really for simple graphs where the definition is a bit simpler. So let's just look at this graph abstraction idea and how isomorphism connects with it. This is an example of two different ways of drawing the same graph. That is here's a 257, and there's 257. It's connected directly to 122, as here. And also 257 is connected to 99, as here. And if you check, it's exactly the same six vertices and exactly the same eight edges. But they're just drawn differently. So we don't want to confuse a drawing of a graph, like these two, with the graph itself. The graph itself consists of just the set of nodes and the set of edges. And if you extracted that from these two diagrams, you would get the same set of nodes and the same set of edges. So same graph, different layouts. But here's a case where it's really the same ...