## Search found 758 matches

Thu May 17, 2012 2:50 pm UTC
Forum: Mathematics
Topic: Infinite power tower of i?
Replies: 16
Views: 5784

### Re: Infinite power tower of i?

I actually fiddled with the convergence a bit, thinking geometrically, and couldn't get anywhere. Anyone have an easy way to see it?
Wed May 09, 2012 3:41 pm UTC
Forum: Mathematics
Topic: confused with integration
Replies: 7
Views: 1842

### Re: confused with integration

(1) The integral of a f'(x) isn't quite f(x), but it's equal to a difference. The theorem is this: \int_a^b f'(x) dx = f(b) - f(a) So the integral of the derivative from a to b is the difference of the function. Remember that f'(x)dx intuitively represents very tiny chang...
Tue May 08, 2012 11:31 pm UTC
Forum: Mathematics
Topic: f(g(x)) = g(f(x)) (commuting mappings)
Replies: 16
Views: 4104

### Re: commutating maps

I would guess that a "nontrivial nice description" doesn't exist, else you could provide "nontrivial nice descriptions" of centralizers for arbitrary groups.
Mon May 07, 2012 6:43 am UTC
Forum: General
Topic: What is your favorite number?
Replies: 39
Views: 7348

### Re: What is your favorite number?

The Euler-Mascheroni constant is a neat little guy.
Mon Apr 23, 2012 7:39 pm UTC
Forum: Mathematics
Topic: Need to confirm my thinking on a probability problem
Replies: 5
Views: 1509

### Re: Need to confirm my thinking on a probability problem

Right, so I believe it suffices to assume that A is of the form [imath]\xi^2 \in B[/imath] for a measurable subset B of the real line. Now you want to use the symmetry of the normal distribution.
Mon Apr 23, 2012 1:50 pm UTC
Forum: Mathematics
Topic: Need to confirm my thinking on a probability problem
Replies: 5
Views: 1509

### Re: Need to confirm my thinking on a probability problem

How was E(X|Y) defined for you?
Sat Apr 14, 2012 4:28 pm UTC
Forum: Mathematics
Topic: Limit of a series
Replies: 26
Views: 6107

### Re: Limit of a series

Are we sure Dopefish's original reasoning isn't valid? Decomposing a sequence into subsequences isn't the same as rearranging it. Right, but you can't decompose the series into two sums and conclude the whole thing diverges because one of the subseries does. For instance, you could decompose the se...
Sat Apr 14, 2012 7:37 am UTC
Forum: Mathematics
Topic: Limit of a series
Replies: 26
Views: 6107

### Re: Limit of a series

Proginoskes wrote:(z4lis slipped his in as I was writing this up!)

Hey, we even have the same solution. That's no fun.
Sat Apr 14, 2012 7:31 am UTC
Forum: Mathematics
Topic: Limit of a series
Replies: 26
Views: 6107

### Re: Limit of a series

It diverges. Group the terms together like \frac{1}{3n-2} + \frac{1}{3n-1} - \frac{1}{3n} = \frac{9n^2 + 2}{27n^3 - 27n + 6n} The intuition is that this goes like 1/3n and so diverges, but you can rigorously finish the proof with the limit comparison test. For amusement, you can use the same techni...
Tue Apr 10, 2012 2:25 am UTC
Forum: Mathematics
Topic: Quotient map locally homeomorphic to R^2?
Replies: 8
Views: 2869

### Re: Quotient map locally homeomorphic to R^2?

Pick a point in the plane that's not the origin and consider a small box around it. What does the equivalence class of that small box look like?
Tue Apr 03, 2012 2:12 am UTC
Forum: Mathematics
Topic: Mean of infinite sets
Replies: 10
Views: 2692

### Re: Mean of infinite sets

Right, well your only tool to estimate the average is to do computations, using the law of large numbers. However, there's not any guarantee about how fast the sample means will converge. If you can estimate the variance, then you can get some idea of how fast your sample means are converging. Also,...
Mon Apr 02, 2012 10:44 pm UTC
Forum: Mathematics
Topic: Mean of infinite sets
Replies: 10
Views: 2692

### Re: Mean of infinite sets

So F should be uniform, you say? How are you choosing your function f?
Mon Apr 02, 2012 9:00 pm UTC
Forum: Mathematics
Topic: Mean of infinite sets
Replies: 10
Views: 2692

### Re: Mean of infinite sets

There's definitely an average across an infinite set - that's exactly what the expectation of a random variable is. Now, there's not a uniform probability distribution on those sets, so you'll need some other definition of "average" to compute what you're talking about with the integers. I...
Mon Apr 02, 2012 4:32 pm UTC
Forum: Mathematics
Topic: Anyone got a handy proof that a^((p-1)(q-1)) = 1 mod pq ?
Replies: 4
Views: 2518

### Re: Anyone got a handy proof that a^((p-1)(q-1)) = 1 mod pq

I'm studying number theory as well at about the same level you are, and I've found the best way for me to memorize and use the Chinese remainder theorem is to use the abstract algebra formulation. The ring Z/pqZ can be factored into the rings Z/pZ and Z/qZ. So, we can take an arbitrary element of Z/...
Sat Mar 31, 2012 3:02 pm UTC
Forum: Mathematics
Topic: Motivation for self-study
Replies: 3
Views: 1414

### Re: Motivation for self-study

I think it's definitely unreasonable for the typical person to be able to remember all the theorems and proofs they read, especially after not seeing them for eight months. I've always believed that precisely remembering results is not as important as being able to recognize when something can be us...
Sat Mar 31, 2012 2:50 pm UTC
Forum: Mathematics
Topic: Simplify
Replies: 15
Views: 4562

### Re: Simplify

Homework or not?

Either way, I would try to get rid of the fractions as much as possible. For instance,

$\frac{\frac{1}{x^2} + \frac{1}{y^2}}{\frac{1}{x^2}-\frac{1}{y^2}} = \frac{x^2 + y^2}{y^2 - x^2}$ by multiplying top and bottom by x2y2. Try that for the other terms and see if it helps.
Thu Mar 22, 2012 4:42 pm UTC
Forum: Mathematics
Topic: Being Fast At Calculation
Replies: 14
Views: 3285

### Re: Being Fast At Calculation

If you're slow at calculation, you can do what anyone has to do to become faster: practice. Identify what's taking you a long time, make up some problems similar to those, and practice solving them quickly and without errors.
Thu Mar 01, 2012 2:52 am UTC
Forum: Mathematics
Topic: Usage of matrix multiplication
Replies: 10
Views: 4333

### Re: Usage of matrix multiplication

Matrix multiplication is used in multivariable calculus, where the derivative matrix of a composition g(f(x)) is the matrix product of g and f by the chain rule.
Wed Feb 29, 2012 5:54 pm UTC
Forum: Mathematics
Topic: Funny mathematical terms and statements
Replies: 21
Views: 8525

### Re: Funny mathematical terms and statements

Normal Person: Would you like the window open or closed? Mathematician: Yes. This is only potentially ambiguous in writing. When speaking, the difference between exclusive and inclusive "or" is apparent from intonation. --- Examples that come to mind are "compact", "real&qu...
Tue Feb 28, 2012 3:31 pm UTC
Forum: Mathematics
Topic: Funny mathematical terms and statements
Replies: 21
Views: 8525

### Re: Funny mathematical terms and statements

Normal Person: Would you like the window open or closed?

Mathematician: Yes.
Wed Feb 22, 2012 7:02 pm UTC
Forum: Mathematics
Topic: Nonstandard proofs for simple theorems
Replies: 34
Views: 7419

### Re: Nonstandard proofs for simple theorems

Another proof that there are infinitely many primes:

Spoiler:
$\sum_{n=1}^\infty \frac{1}{n^2} = \prod_{\mbox{all primes p}} \bigg(1 - \frac{1}{p^2}\bigg)^{-1}$ by Euler. If there were only finitely many primes, the RHS would be rational, but the LHS is [imath]\pi^2/6[/imath], which is irrational.
Thu Feb 16, 2012 5:11 pm UTC
Forum: Mathematics
Topic: A Most Amazing Number
Replies: 33
Views: 17419

### Re: A Most Amazing Number

That number is as mysterious as dihydrogen monoxide is dangerous!
Mon Feb 13, 2012 7:18 pm UTC
Forum: Mathematics
Topic: Number Theory Help
Replies: 4
Views: 964

### Re: Number Theory Help

I'm familiar with the things you've all said, but every technique I attempt ends up with me needing n >= ab to be true, so I'm stumped. I realize it's probably something simple that I just haven't noticed, but at this point I wouldn't mind just having a solution.
Mon Feb 13, 2012 4:32 am UTC
Forum: Mathematics
Topic: Number Theory Help
Replies: 4
Views: 964

### Number Theory Help

So I'm working through a book on number theory and will probably have a lot of questions about the exercises I just can't see the solution to. If any of you could point me in the right direction, I'd appreciate it. The first problem: Suppose a and b are positive integers with gcd(a,b) = 1. Suppose n...
Tue Feb 07, 2012 7:01 pm UTC
Forum: Mathematics
Topic: A Probability Conundrum
Replies: 7
Views: 1314

### Re: A Probability Conundrum

I might be missing an easy symmetry argument or looking something over, but the probabilities of the second draw are not the probabilities of the first draw, as you've removed a card from the deck. Most of what I see above assumes replacement. Normally, a hand of cards is taken without replacement.
Tue Feb 07, 2012 6:58 pm UTC
Forum: Mathematics
Topic: Hausdorff distance up to isometry? [solved]
Replies: 11
Views: 3758

### Re: Hausdorff distance up to isometry?

I don't think you have to assume the shapes are only moved about in their ambient space. If you have two metric spaces X, Y (that might be subspaces of some larger space, that they're still spaces in their own right), for every pair of isometries f:X -> Z and g:Y -> Z, you can define d(f(X),g(Y)) in...
Fri Feb 03, 2012 5:50 pm UTC
Forum: Mathematics
Topic: Help on some estimates...
Replies: 2
Views: 540

### Re: Help on some estimates...

++\$_ wrote:The taylor series for sine is x - x3/6 + x5/120 + ....

I think you forgot the 6.

I knew it was something stupid like that, thanks. Everything works out fine now.
Fri Feb 03, 2012 4:57 pm UTC
Forum: Mathematics
Topic: Help on some estimates...
Replies: 2
Views: 540

### Help on some estimates...

So this is for probability homework, but I doubt anyone will give a response before it's due. What I need to do is estimate the product \prod_{j=1}^n \frac{\sin(\frac{ju}{n^{3/2}})}{\frac{ju}{n^{3/2}}} The limit should be e^{-u^2 \frac{1}{18}} , but I get e^{-u^2 \frac{1}{3}} . My method was...
Thu Feb 02, 2012 4:48 pm UTC
Forum: Mathematics
Topic: Temporal distortion, geodesics(?), and My Little Pony
Replies: 24
Views: 3342

### Re: Temporal distortion, geodesics(?), and My Little Pony

I never had much experience with calculus of variations, and it's been a long time since I've done anything with it, but I believe you want to do something like the following: 1) Define F[p] to be that integral you have over a path p. Now, we want a path that minimizes this function, so if we could ...
Sat Jan 28, 2012 4:59 pm UTC
Forum: Mathematics
Topic: Linear algebra (homework)
Replies: 5
Views: 1253

### Re: Linear algebra (homework)

Just wanted to mention that the bilinear form won't be exactly bilinear, in the complex case one of the slots is linear and the other spits out the complex conjugates of constants.
Mon Jan 23, 2012 7:22 pm UTC
Forum: Mathematics
Topic: The natural structure of mathematics
Replies: 7
Views: 1803

### Re: The natural structure of mathematics

I'm not certain that subdividing mathematics into a hierarchy of more and more advanced material, each thing depending on the last thing, branching out into little independent trees, is the best way to view the subject. While we've already labelled and divided knowledge into various fields (algebra,...
Mon Jan 09, 2012 3:29 am UTC
Forum: Mathematics
Topic: Lie Groups
Replies: 1
Views: 1197

### Lie Groups

Just want an opinion about a "proof". It's not for homework. I assume that I know that if f:G --> G' is group homomorphism between the Lie groups that's differentiable at 1, then Tf:g --> g' preserves the Lie algebra. Now, I want to show that [X,Y] = -[Y,X]. Now, if G has product p(x,y), I...
Fri Dec 30, 2011 6:22 pm UTC
Forum: School
Topic: Calling my professor?
Replies: 2
Views: 1994

### Calling my professor?

So I'm sitting three days away from the due date of graduate school applications, and one of my LOR writers has not turned in anything. I sent him a reminder email yesterday, but still have no response. I'm desperately trying to get someone willing and able for backup, but no luck with them, either....
Mon Dec 19, 2011 3:20 pm UTC
Forum: General
Replies: 26
Views: 4566

### Re: Give me your opinion

Sorry if this is a shock, but you'll find that in the real world nobody cares. The title is confusing, the "Nearly White" bit made me think of dirty drapes or fabric, could be seen as a racist term. The "Et Cetera Etc." doesn't make any sense and isn't particularly funny in this ...
Sat Dec 10, 2011 1:50 pm UTC
Forum: Mathematics
Topic: Mathematics and Computer Science
Replies: 2
Views: 776

### Re: Mathematics and Computer Science

How many books are you willing to read about mathematics? If the number is very low, someone probably has some good general overviews that capture lots of the amazing features of mathematics without becoming needlessly technical or going too in depth. If the number is higher, then you could read mor...
Fri Dec 09, 2011 4:24 pm UTC
Forum: General
Topic: The Sun
Replies: 19
Views: 3553

### Re: The Sun

You didn't stare directly at it until the star itself turned purple and began wobbling around? You chicken. I did this all the time when I was a child and only need corrective lenses for one eye.
Tue Dec 06, 2011 1:54 am UTC
Forum: Mathematics
Topic: Putnam (aka halp!)
Replies: 17
Views: 2559

### Re: Putnam (aka halp!)

I feel stupid for totally missing that the only prime p such that p-2 and p+2 are prime is 5. I basically proved that other than 2, any prime p that appears in more than seven does so iff 4 other numbers are also prime, two of which were p+2 and p-2. Maybe I'll get some points.
Tue Dec 06, 2011 1:32 am UTC
Forum: Mathematics
Topic: Probability Theory - generated Sigma Algebra
Replies: 3
Views: 4881

### Re: Probability Theory - generated Sigma Algebra

The sigma algebra generated by a collection C of subsets is just the smallest sigma algebra that contains C. Since you just have a single subset to worry about, it's a pretty small set.
Sun Dec 04, 2011 4:40 pm UTC
Forum: Mathematics
Topic: Putnam (aka halp!)
Replies: 17
Views: 2559

### Re: Putnam (aka halp!)

Proginoskes wrote:Is discussion of the Putnam Exam itself forum on these forums? I expected a thread with about 100 replies by now.

I don't think we're supposed to discuss it until later today. And I've noticed that there usually isn't that much Putnam discussion on here...
Thu Dec 01, 2011 4:12 am UTC
Forum: Mathematics
Topic: Multiple Choice Problem
Replies: 9
Views: 2070

### Re: Multiple Choice Problem

As a general rule, don't delete your post after your question was answered. Others using the search function might find it useful, in the future. It's something like this comic.