Recent content by Rasalhague

I guess he must have meant "connected" rather than "simply connected" when he writes that Dynkin diagrams classify "a bunch of things" including: "simply connected complex simple Lie group" and "compact simply connected simple Lie groups".

SO(2,C) is connected, but not simply connected (Fulton & Harris: Representation Theory, Prop. 23.1). https://en.wikipedia.org/wiki/Orthogonal_group#Over_the_complex_number_field

I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex...

Got it, thanks!

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...

Thanks to everyone who responded to this thread. I sent out a bunch of queries. Most did say I wouldn't be eligible. A couple said I would. Someone also suggested to me the IMMERSE program for pre-grads at Nebraska: http://www.math.unl.edu/programs/mctp/immerse

Is it possible to do an REU without being currently enrolled in an undergraduate degree program? Some REU descriptions do state this is a requirement. Others seem to expect it, but don't explicitly rule out other possibilities. Does anyone have experience with this? I have a bachelor's degree...

It really depends on the course: the instructor(s) and how well organized it is, whether they have enough assistance. I've taken some great ones. I've learned a lot. Most have had few issues, but they were still very useful and rewarding. I don't think I ever dropped a Coursera course because of...

Great, thanks, Bill!

If I define a function, such as f[x_, y_] := {-2 x + 2 x^2, -3 x + y + 3 x^2} I can compute the derivative with D[f[x, y], {{x, y}}] but what is the syntax for evaluating this derivative at a point?

I've taken online courses in the past at Coursera and the Saylor Foundation and had a mostly good experience with them. I've taken them firstly to learn and secondly on the off-chance that they may provide some (albeit tenuous) evidence of achievement, or at least of intent. They also provide...

Ah, reading further on that Mathworld article, it seems one definition concerns real-valued random variables from a finite sample space, another definition concerns tuples of such random variables. But still, there appear to be a variety of concepts here to which the name covariance is attached...

I'm not sure how to reconcile Serfling's formula (1) with the way Wolfram Mathworld writes it out explicitly for the case where N = M: http://mathworld.wolfram.com/Covariance.html Are there two somewhat different concepts each called covariance, each corresponding to its own way of...

The formula defines covariance for discrete variables in Simon & Blume (1994): Mathematics for Economists, end of section A5.4, and in Robert J. Serfling's online intro 'Covariance and Correlation', formula (1) which he identifies with E[(X-EX)(Y-EY)P(X,Y)] in the formula which follows that...

Var(X) = Cov(X,X) ?? Var(X)=\sum_{i=1}^N P(X_i)(X_i-EX)^2. Cov(X,Y) = \sum_{i=1}^N\sum_{j=1}^M P(X_i,Y_j)(X_i - EX)(Y_j - EY). If, for instance, P(X_i) = 1/N and X = Y = (1,2,3), then Var(X) = \frac{1}{3} ((1-2)^2 + (2-2)^2 + (3-2)^2) = \frac{2}{3}, but Cov(X,X) = \sum_{i=1}^3 \sum_{j=1}^3...