Homework Statement
Give an example of a ring R with infinitely many non-isomorphic simple modules.
The Attempt at a Solution
I was thinking of setting
R=\mathbb{Z}_{p_1}\times \mathbb{Z}_{p_2}\times \mathbb{Z}_{p_3}\times \cdots
where p_1,p_2,p_3,\ldots is an infinite increasing list...
Homework Statement
Perhaps I should say first that this question stems from an attempt to show that in the group
\langle x,y|x^7=y^3=1,yxy^{-1}=x^2 \rangle,\ \langle x \rangle is a normal subgroup.
Let G be a group with a subgroup H . Let G be generated by A\subseteq G . Suppose...
Ah. Brain malfunction on my part. For some reason I was thinking of I\times I as the boundary of the square and not the whole square. That's sorted then. Thanks!
Homework Statement
In Lee's "Topological Manifolds", there is a result on page 193 called "The Square Lemma" which states that if I denotes the unit interval in \mathbb{R}, X is a topological space, F\colon I\times I\to X is continuous, and f,g,h,k are paths defined by...
Homework Statement
This question is not the assignment problem but I think that if the result I mention here is true, then my assignment problem will be solved.
Let (X,\Sigma,\mu) be a measure space.
Suppose that (h_n)_{n=1}^\infty is a sequence of non-negative-real-valued integrable...
No I think it is. I quote Corollary 2.3 from "Fourier Analysis" by Stein and Shakarchi -
"Suppose that f is a continuous function on the circle and that the Fourier series of f is absolutely convergent, \sum_{n=-\infty}^\infty |\hat{f}(n)|<\infty. Then, the Fourier series converges...
I was just looking to see if it converged. So yes, you're absolutely right. I was going well off-beam with my attempt.
And about converging to x^2; I guess it does so uniformly since the function is continuous on the circle and the Fourier series converges absolutely.
Thank you!
I'm trying to show that the Fourier series of f(x)=x^2 converges and I can't. Does anybody know if it actually does converge? (I'm assuming that f(x)=x^2 for x\in [-\pi,\pi]).
The Fourier Series itself is \displaystyle\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos nx
I tried...
Homework Statement
This is a question from a sample exam, rather than a homework problem.
\text{Let }f \text{ be the periodic function with period } 1 \text{ defined for }-1/2\leq t<1/2 \text{ by }f(t)=t^2\\ \text{ and let }g \text{ be the periodic function with period } 1 \text{ defined...
Hi, I'm just wondering this:
If A and B are subsets of the real numbers, is
inf A + inf B = inf(A+B)?
I've proved that
inf A + inf B <or= inf(A+B)?
I've tried the rest of the proof but it won't work. Is it even true?
I have the following statement:
Let A\subseteq \mathbb{C} be open and let f\colon A \to \mathbb{C} be holomorphicic (in A). Suppose that D(z_0,R)\subseteq A. Then
f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k\ \forall z\in D(z_0,R), where
a_k=\displaystyle\frac{1}{2\pi...