Recent content by I<3Gauss

Just a quick question. If Q is a dense set of a metric space X, and P is a dense set of a metric space Y, then is Q x P a dense set of X x Y? I am fairly sure this is the case. If this is true, then I want to use this statement to show that the open sets of the product of finite number of...

After reading through this thread, I just want to say that it is definitely possible to do research mathematics without a graduate or undergraduate degree. However, so is getting hit by lightning, or being mauled by a cow. The odds are not in your favor, and you will be at a disadvantage...

If it makes you feel any better, I was in a similar situation as you when I decided I wanted to do graduate level mathematics near the end of my junior year of college (2009). I was also an economics major, and at that point, I only had Calc I-IV, and ODE's with a basic intro to proofs course...

Whovian, What gave you the idea to make such a substitution? What technique are you using?

Does anyone know how to prove the following statement? I haven't messed with integrals for awhile and I have to say that I am kind of rusty on this. From initial attempts, it seems the integral on the left is not something you can integrate directly... Maybe Taylor Expansion of cos^2(x) would...

Thanks for the really simple example!

While reading a proof on the closure of the span of finite number vectors in a hilbert space with respect to the norm induced topology, I became stumped on a particular step of the proof using the Bolzano Weierstrass theorem. For finite dimensional vector spaces, Bolzano Weierstrass states...

thanks for the tip, i think i kind of see why this is now. I guess if a poset P was the join irreducible set of some Lattice L, and this particular poset P is isomorphic to the join-irreducibles of L(P), which is the set of all Ix, then L would be isomorphic to L(P)? The tip that you gave...

That's an interesting idea but however, i also do not have the guts yet or the know how to make such an assumption.

The poset on the set of order ideals of a poset p, denoted L(p), is a distributive lattice, and it is pretty clear why this is since the supremum of two order ideals and the infimum of 2 order ideals are just union and intersection respectively, and we know that union and intersection are...

oh whoops again, typo, thanks for that correction!

oh whoops, i c omg, can't factor... (x^2 + x^4 + x^6 + x^8)^k = [ (x^2) (1 + x^2 + x^3 + x^4 +...) ]^k but 1 + x^2 + x^3 + x^4 + ... does not equal 1/(1-x) cuz 1/(1-x) = 1 + x + x^2 + x^3 + x^4 Thanks guys, I now know i can do enumerative combinatorics but can't do algebra or...

Okay, here goes (x^2 + x^4 + x^6 + ...)^k = (x^2k) (1 + x + x^2 + x^3 + ...)^k = (x^2k) [(1-x)^-k] By the general binomial theorem, we now have = (x^2k) [sum(n=o to infinity) (-k choose n) (-x)^n] well, we know that (-k choose n) is equal to (-1)^n (k+n-1 choose n) plugging...

I was reading Stanley's first volume on Enumerative Combinatorics, and I am seemingly stuck on a basic question regarding compositions. It may be that my algebra skills are rusty, but I just cannot get the correct formula for the number of compositions of n into even numbers of even valued...