# Search results for query: *

1. ### Functional Analysis HMWK

What if I re-index so that n\in \mathbb{N} so that d_0=e_0, d_1=e_1, d_2=e_{-1},\dots. Then by Bessel's inequality we have \sum_{n=0}^{\infty}|(f,d_n)|^2\le \parallel f\parallel ^{2}<\infty. Hence \sum_{n=0}^{\infty}|(f,d_n)|^2 converges absolutely and whence (f,d_n)\to 0 and thus (f,e_n)\to\ 0...
2. ### Functional Analysis HMWK

What should I be careful with? Does L^2-convergence not imply absolute convergence? Sorry, just a little confused by your statement.
3. ### Functional Analysis HMWK

Yep we have. Could I say that: Since \{e_n\} is complete it follows that f=\lim_{n\to\infty} \sum_{m=-n}^{n}(f,e_m)e_m. Thus, the latter sum converges and hence \lim_{n\to\infty}\sum_{m=-n}^{n}|(f,e_m)|^{2}<\infty. Thus |(f,e_m)|^2 \to 0 so |(f,e_m)|\to 0 and whence (f,e_m)\to 0, as required...
4. ### Functional Analysis HMWK

Wait, if {e_n} is complete, then \lim_{n\to\infty}\sum_{m=-n}^{n}(f,e_m)e_m converges to f absolutely, so the coefficients necessarily converge to zero. Would this work?
5. ### Functional Analysis HMWK

Homework Statement Let e_{n}(t)= \frac{1}{ \sqrt{2\pi}}\cdot e^{int} for n\in\mathbb{Z} and -\pi\le t\le\pi. Show that for any f\in L^{2}[-\pi,\pi] we have that (f,e_{n})=\int_{-\pi}^{\pi}f(t)\cdot e^{-int}dt\to0 as |n|\to \infty. The Attempt at a Solution I want to use dominant convergence...
6. ### What is the best way to think about an R-algebra?

I know that thinking a R-module is simply a ring R acting on a set (following the usual axioms), would it be safe to think of an R-algebra as the ring R acting on another ring? This may seem convoluted, but I'm just having a little trouble getting through all the different definitions of an...
7. ### Simple Tensor Product

Thanks for the help! I got it.
8. ### Simple Tensor Product

We have 0= 1\otimes 0 = 1\otimes 12= 12(1\otimes 1)=12\otimes 1=(2\otimes 1)+ (10\otimes 1)=(2\otimes 1)+0=2\otimes 1. Hence, m\otimes 1 = k(2\otimes 1)=0.
9. ### Simple Tensor Product

Actually, I guess that doesn't really prove anything since it isn't assumed that \phi is injective. Hrmm...
10. ### Simple Tensor Product

Well it is trivial if m=0, so suppose m\neq 0 even. Then it follows that m=2k hence m \otimes 1 = 2k\otimes 1 = 2(k\otimes 1) Hence, for any morphism of \mathbb{Z}-modules \phi : (\mathbb{Z}_{10}\otimes_{\mathbb{Z}}\mathbb{Z}_{12})\to \mathbb{Z}_{2}, it follows that...
11. ### Simple Tensor Product

yepp 1 \otimes 10 = 10(1\otimes 1)=10\otimes 1 = 0\otimes 1=0.
12. ### Simple Tensor Product

Homework Statement Show that \mathbb{Z}_{10}\otimes_{\mathbb{Z}}\mathbb{Z}_{12} \cong \mathbb{Z}_{2} The Attempt at a Solution Clearly, for any 0\neq m\in\mathbb{Z}_{10} and 0\neq n \in \mathbb{Z}_{12} we have that m\otimes n = mn(1\otimes 1), and if either m=0 or n=0 we have that m\otimes n...
13. ### Hilbert Spaces

omg how do i call myself a math major. thank you.
14. ### Hilbert Spaces

Homework Statement Let H be a Hilbert space. Prove \Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} The Attempt at a Solution First suppose x = 0. Then we have \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y)...
15. ### Post Your Summer/Fall 2012 Class Schedules

Algebraic Topology Algebraic Geometry I Linear Algebraic Groups Functional Analysis II Rings and Modules Field Theory Commutative Algebra Lie Algebras Differential Equations Algebraic Geometry II I bet you wouldn't guess that I'm applying to an algebra group for graduate studies...
16. ### Sets of Measure Zero

Homework Statement Let \sigma (E)=\{(x,y):x-y\in E\} for any E\subseteq\mathbb{R}. If E has measure zero, then \sigma (E) has measure zero. The Attempt at a Solution I'm trying to show that if \sigma (E) is not of measure zero, then there exists a point in E such that \sigma (\{e\}) that...
17. ### Measure Theory Question

Nevermind, I'll just use Monotone Convergence on a sequence of simple functions.
18. ### Measure Theory Question

So my question is if I'm allowed to say this: \int f\, d\nu = \sup \{\int \phi\, d\nu :0\le\phi\le f,\, \phi \,simple\} =\sup \{\int \phi g\,d\mu :0\le\phi\le f,\,\phi\,simple\} = \int fg\,d\mu?
19. ### Measure Theory Question

I'm using Royden, where the integral of a non-negative function is defined as \int f\, d\,\mu =\sup \{\int \phi \, d\mu :0\le\phi\le f,\,\phi \, simple\}
20. ### Measure Theory Question

Hey, thanks for the reply! I've already done this for simple functions, I'm just stuck on how to show that any non-negative measurable function satisfies the equality. Should i take approximations from above and below? or should what I gave in my first reply be sufficient?
21. ### Measure Theory Question

Should I do some inequalities for simple functions above and below? I feel like that last equality should be an inequality. Hrmf.
22. ### Measure Theory Question

I've shown for non-negative simple functions that \int \phi\,d\nu=\int \phi g \,d\mu. Now I wish to show it in general for non-negative measurable functions. So I say let f be a non-negative measurable function on X. Fix \phi as a simple function such that 0\le\phi \le f. Hence we have \int...
23. ### Number Theory Help

Again, thanks all! This helped so much!
24. ### Number Theory Help

I'm still confused as to where I'm supposed to go with all of this. Let p be an odd prime. For each d|p-1, let f(d) be the amount of elements of \mathbb{Z}_{p}^{\times} such that those elements have order d. Clearly \sum_{d|p-1}f(d)=p-1. Fix a \in\mathbb{Z}_{p}^{\times} such that ord(a)=d, for...
25. ### Number Theory Help

Then a^{p-1}\equiv 1 \mod (p^{2}) Why is this a problem?
26. ### Measure Theory Question

Homework Statement Let (X,\mathcal{B},\mu) be a measure space and g be a nonnegative measurable function on X. Set \nu (E)=\int_{E}g\,d\mu. Prove that \nu is a measure and \int f\, d \nu =\int fg\,d\mu for all nonnegative measurable functions f on X. The Attempt at a Solution I'm basically...
27. ### Number Theory Help

Yep, we had to use the fact that a+p\in\mathbb{Z}_{p^{2}}^{\times} is a generator, hence showing that it is cyclic. My original question was the starting point to this proof. I'm still confused on how exactly we showed that \gcd(k,p)=1. Also, thank you so much for your help otherwise!
28. ### Number Theory Help

Yep that's the question exactly. What is it exactly that you're trying to show that makes the claim untrue? I'll take it up with my prof if it indeed isn't a true claim.
29. ### Number Theory Help

OOOOOH Since p odd prime it follows that it cannot be that both p|a^{m}-1 and p|a^{m}+1. I understand why we want to do this, but what if p^2|a^{m}+1?
30. ### Number Theory Help

I'm still stuck on your last hint :(
31. ### Number Theory Help

No, for any odd prime p it is not the case for both p|n and p|n+2, for any n\in\mathbb{Z}.
32. ### Number Theory Help

Either p|(a^{m}+1) or p|(a^{m}-1). If it is the latter, then p|(a^{m}-1) implies that a^{m}\equiv 1\mod p, but that would mean that the order of a is m, which contradicts the fact that a is a generator of \mathbb{Z}_{p}^{\times}. Hence, it must be that a^{m}\equiv -1\mod p.
33. ### Number Theory Help

Well we can say kp=(a^{m})^2-1=(a^{m}-1)(a^{m}+1), or is there something that I'm missing?
34. ### Number Theory Help

Well if 1\le k<p, then clearly \gcd(k,n)=1. But if p<k, then its not so clear.
35. ### Number Theory Help

We can take d=p-1 and hence there exists \phi(p-1) many elements of \mathbb{Z}_{p}^{\times} whose order is p-1. But since the order of \mathbb{Z}_{p}^{\times} is p-1, it follows that these elements are exactly the generators of \mathbb{Z}_{p}^{\times}. Hence, we've show that there exists at...
36. ### Number Theory Help

Yep since f(d) is either \phi(d) or 0; both of which are less than or equal to \phi(d). And since \sum_{d|p-1}f(d)=\sum_{d|p-1}\phi(d), it follows from the previous sentence that f(d)=\phi(d). So now we know that \phi(d) is exactly the number of elements in \mathbb{Z}_{p}^{\times} that have...
37. ### Number Theory Help

Yep :) \sum_{d|p-1}\phi(d)=p-1
38. ### Number Theory Help

For every d|p-1, it follows that every f(d) is either \phi(d) or 0, because we haven't verified the fact that there exists an element of order d for each d|p-1.
39. ### Number Theory Help

If we take d=p-1, then there are only \phi (p-1) many generators of \mathbb{Z}_{p}^{\times}. Is this right? Since d|p-1, it follows that \phi(d)|\phi(p-1). Hence there exists k\in\mathbb{Z} such that \phi(p-1)=k\phi(d).
40. ### Number Theory Help

\phi (d) many.
41. ### Number Theory Help

The size of \mathbb{Z}_{p}^{\times} is \phi (p)=p-1
42. ### Number Theory Help

Any element a^{r} such that \gcd(r,d)=1 is a generator.
43. ### Number Theory Help

Well I know its obvious that a^{p-1}\equiv 1 (\mod p) since [a]\in\mathbb{Z}_{p}^{\times}. Hence there exists k\in\mathbb{Z} such that a^{p-1}=1+kp. How do I show that \gcd (k,p)=1?
44. ### Number Theory Help

Homework Statement Let p be an odd prime. Show that there exists a\in\mathbb{Z} such that [a]\in\mathbb{Z}^{\times}_{p} is a generator and a^{p-1}=1+cp for some c coprime to p. Homework Equations The Attempt at a Solution I honestly have no idea where to even start with this. Any help will...
45. ### Proving the Sequence of Real Numbers is Not Cauchy

Well if you're trying to show that it is not Cauchy, state what it means for a sequence to not be Cauchy. That is where I would start :)
46. ### What equation or pattern applies to this kind of combinatorial possiblity?

If order didn't matter, I'd look up Bell numbers, they arise as a way to count the amount of partitions there are to any given set. Since order does matter, you know that your number you will want to count should be less than this. I hope that helps.
47. ### Quantum Physics - My adventure and obsession.

Is undergraduate QM a little hand-wavy? Is there a good text i could supplement with that treats it a little more axiomatically maybe?
48. ### Quantum Physics - My adventure and obsession.

Just a side note, I'm 3rd year pursuing a math degree. I plan on taking QM courses either in supplement to my Master's (I'm hoping to get into quantum computing through non-commutative geometry). Would undergraduate QM courses be relatively more accessible with a background in linear algebra...
49. ### Post your spring schedule

Group Theory General Topology Measure Theory Cryptography Combinatorics YAAAAAY.
50. ### Fit in some Mathematical Finance in before end of degree?

I'll be in my 4th year of a Pure Math degree next year, and I'm think of what courses to fill up my timetable with. I'll be taking an overload, so I'll be taking 6.0 credits worth of courses (3.0 per semester). I'm only required to take 3.0 more courses in Pure Math to get my degree, so I have...