Recent content by futurebird

I found this thread and I just wanted to say that this identity inspired me to find these identities: - \frac{\pi ^2}{3!} = \displaystyle \sum_{j_1=1}^{\infty} -j_1^{-2} \frac{\pi ^4}{5!} = \displaystyle \sum_{j_1,j_2=1 \atop j_1 \neq j_2}^{\infty} (j_1j_2)^{-2} - \frac{\pi ^6}{7!} =...

Thank you so much... I wish I'd seen that, it makes so much sense.

Homework Statement I am reading about integer partitions. I'm learning a proof and I don't understand what would seem to be a simple step... as the book presents it without comment: \prod_{n=1}^{\infty} \frac{1-q^{2n}}{1-q^{n}}=\prod_{n=1}^{\infty} \frac{1}{1-q^{2n-1}} The fractions...

And that was given, so it's just very simple. Yes.

Homework Statement Royden Chapter 4, Problem 10a Show that if f is integrable over E then so is |f| and \left|\int_E f \right| \leq \int |f|. Does the integrability of |f| => the integrability of f? Homework Equations f^+ = max\{f, 0\} f^- = max\{-f, 0\} |f| = f^+ + f^- A...

I was thinking about this same question today. Though, to be very honest I'm yet to find a construction of a non-measurable set that I "get" -- I can follow the steps of the proof but I don't see the big picture. You know? It is very interesting to me that something sensible like the axiom of...

No, for nXn there are \sum_{i=0}^{n} (n-i)^2. Think of a 2x2 grid. there are 4 little squares and one big one, on a 3x3 grid you have 9 little squares, 4, 2x2 squares and 1 big one... see the pattern? (Though, I still don't know what the thing about rectangles is getting at.)

"how many rectangles can you contain." What do you mean by this?

Thanks! It looks pretty cool.

Yes. This is just the question.

I'm asking if for every finite group is there at least one polynomial in Q that has that group as its Galois group. That is we did a lot of "give a polynomial find the Galois group." can you do always the reverse? maybe this is obvious or something...

What course is this from? 2.6 is not an exact answer so maybe they want you to use Newton's method? There are a few ways of solving this-- but the method will depend on what course you are taking.

We want [(a^x)(b^y)(c^z)]^p = 1 (a^x)^p = 1 only when x=1 (b^y)^p = 1 only when y=1 (c^z)^p = 1 only when z=p so: a*1*1 a*b*1 a*b*c^p 1*b*1 1*b*c^p 1*1*c^p We have 6 possible generators.

We are looking for groups of order p, so yes, they will be cyclic. Your notation is confusing me. "Hence we need to find all the (a,b,c) in CpxCpxCp^2 such as lcm(o(a),o(b),o(c)) = p" Why are we looking at 3 elements at once? Aren't we just looking for the elements with order P? The elements...

Not that I know of. Finding sub groups can be hard. On to B. "How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime?" UM. We have the three cyclic subgroups. Cp, Cp and C(p^2) C(p^2) will always have Cp as a subgroup. So that's three so far. Can there be others? I...