The Art of Problem Solving

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The Art of Problem Solving

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[Nov. 19th, 2009|08:15 pm]

You have a coin, a pyramid, and a die. The coin has the numbers 1 and 2 on it, the pyramid has the numbers 1, 2, 3, 4 written on it, and the die has the usual 1 through 6 on it. An item is chosen at random and flipped/rolled. The coin has probability .5 of being chosen, the pyramid has probability .3, leaving the die .2 probability of being chosen. Let Y be the number showing after flipping/rolling.

Find E(Y) and Var(Y).

My thought: let X be a random variable so that X is 1 if the coin is chosen, 2 if the pyramid is chosen, and 3 if the die is chosen. Then E(Y|x) = 1.5 if x=1, 2.5 if x=2, and 3.5 if x=3. So then I should find E[E(Y|X)] ??

This question has me all tangled up. I'm not sure I really understand conditional expectation. Any help is appreciated, thanks!

10 comments|post comment

You know you're a math geek when...

[Nov. 17th, 2009|09:07 am]

... you accidentally misread "northern lights" as "noetherian lights" and start thinking about DCC.
... you squirt mayonnaise on your sandwich in the shape of an aleph.
... you post such exploits on an online math community.
... you add to this list.

16 comments|post comment

CS Grad School Advice -- What to choose as a field of interest?

[Nov. 15th, 2009|11:19 pm]

I couldn't find an active Computer Science LJ and figured there were some CS students here =), so here goes:

I'm applying to computer science graduate schools, but I'm not sure what subfield (e.g., AI, Theory, Computer Architecture) to talk about in my research interests. Does anyone know how much it matters?

It's not that I don't know what I want to study -- I do. (Theoretical Computer Science, in particular, complexity theory.)

The problem is, I focused a lot more on machine learning and NLP as an undergrad (most of my undergrad research was in it, and 2 of my recs are from my supervisors), so I think my application looks a lot stronger to a Machine Learning person than to a theory person. (I do have a decent theory background -- I majored in math, took some advanced theory classes, and did research with a combinatorics professor [though it didn't go anywhere] -- but I don't think I'm currently as strong].

So should I say that I want to study Machine Learning (I'm stronger in it, but not really interested anymore) or Complexity Theory (less experienced, but it's what I want to study). How much does it really matter -- is it easy to switch once you're actually admitted? (Not sure how choosing an advisor really works...)

Or should I say I want to study both? (It's a good possibility, but I'm wondering whether it will dilute my research statement and make it sound rather rambly and less focused.)

Thanks!

3 comments|post comment

graph theory book recommendation

[Nov. 12th, 2009|11:23 pm]

I'm looking for a good book on graph theory whose definition of a path allows repeated vertices. Anyone have any suggestions?

6 comments|post comment

a stochastic process

[Nov. 12th, 2009|06:41 pm]

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John opened a checking account in a bank on day d_0 and deposited there x_0 dollars.
Then on d_1 he deposited x_1.
Then on d_2 he deposited x_2.
...

Is there a theory of this kind of stochastic processes?

E.g. if differences d_1 - d_0, d_2 - d_1, ... distributed exponentially (Poisson process) and x_0, x_1, x_2...
lognormally (or Gauss, or Bernoulli, or ...), what would be distribution of $ total on day d ?

2 comments|post comment

[Nov. 12th, 2009|11:41 am]

A couple more questions

1. If T: H-->K has finite rank, then T*:K-->H has finite rank and dim(Im(T)) = dim(Im(T*)).

I'm not really sure how to approach this.. =/

2. Show that an idempotent operator is compact if and only if it has finite rank.

The "only if" is quite easy, because any finite rank operator is compact, but the other direction has me stuck

Thanks for any help!

6 comments|post comment

for those who haven't already emigrated

[Nov. 11th, 2009|10:41 pm]

Do you know about the new website for mathematicians? It's called mathoverflow, and was released less than a month ago. It's accessible mostly to people who have at least started math grad school (indeed having recently acquired that status, I only understand about a quarter of the questions in my field), but the idea is to post questions that should be interesting to other mathematicians, or to answer them. Read the FAQ and join. Or have you already?

1 comment|post comment

Seeking Reference Material

[Nov. 8th, 2009|08:17 pm]

Hi everyone,

I am seeking any reference material that eventually explains Hopf algebras. Really, I'd take anything that does a good job at explaining related topics starting with the definition of a module. I need something designed for a non-specialist.

Thanks in advance.

16 comments|post comment

Manifolds question-zilla!

[Nov. 7th, 2009|09:39 pm]

All right, so normally I wouldn't be posting like three questions at the same time, but I had swine flu this week (make sure you get the vaccine if you haven't already, because I promise you don't want the swine flu, it sucks) and couldn't go to class or meet with people. So with no further ado:

1) We're supposed to show that a simple Lie group is linear. In the hint sheet the professor gave us, he says, let C_g be the map that is conjugation by g, and let Ad(g) = T_1 C_g, the differential of this map. (I think this is called the adjoint representation or something.) I have no idea where this hint is going, and I don't see anything extra special about simple Lie groups that helps us buy this result, except perhaps that they are connected (because the identity component of a Lie group is a normal subgroup).

2) Next thing I'd like to show is that the differential of the inversion map (call it inv) is negative the identity. The hint sheet says to use the fact that exp is a homomorphism from lines through the origin in the Lie algebra to the Lie group. Certainly it's the case that exp(-v) = inv(exp(v)), and I think this is probably what we want, but I don't know how to get T_1 inv in there.

3) Last one for now I promise. Why would the differential of the determinant be trace?

Thanks for your help. I'm going to go hang my head in manifolds-induced shame now.

17 comments|post comment

[Nov. 6th, 2009|02:51 pm]

I'm re-taking the Math GRE tomorrow morning (ugh), and I want to take a couple peppermints. Does anyone know if small things such as this are permitted? I can't seem to find an answer to this specific question...

Edit: Well, I definitely did better this time, that much is for sure. I still don't really understand why graduate schools care if I can do 3 hours worth of hero-level integration, though....

8 comments|post comment

Places to look for postdoc jobs

[Nov. 3rd, 2009|12:17 am]

Ok, so I'm pretty much finished with my PhD and have started to look for postdoc positions. So far I've looked at mathjobs.org and EIMS and will contact people in my field (harmonic analysis and operator theory) and ask them to inform me if any kind of postdoc jobs open up (this is for the departments that aren't advertising an open postdoc position, though I'll directly email people of interest who are in departments that are advertising, just to inform them that I'm applying.)

So the question is, where else should I look for postdoc jobs (or really just academic jobs in general) to apply to?

7 comments|post comment

[Nov. 2nd, 2009|12:33 pm]

I've seen the result all over the place, but no proof: The L^p-norm of a function f approaches the L-infty norm of f as p approaches infinity. Any illuminating clues?

EDIT: We assume f is in L^p and L^infty, so that it's in L^q for all q > p.

5 comments|post comment

more analysis....

[Nov. 1st, 2009|10:44 pm]

Two question:

1. Let B(H,K) denote the space of bounded linear operators H --> K. Show B(H,K) is complete under the operator norm ||T|| = sup{ ||Th|| : h in H and ||h||≤1 }.

I seem to really struggle with this operator norm guy. Obviously I want to start with a Cauchy sequence of operators and show that it converges to some operator in the space.... But I'm just not clear on how to do all that

2. Let g be in L^1(R). The operator L: L^2(R) --> L^2(R) defined by Lf(x) = ∫ g(x-y)f(y)dy is a bounded linear operator.

My initial thought is to try to realize g(x-y) as the kernel of some integral operator, but I'm not sure how to go about doing that exactly.

Any thoughts?

Thanks

4 comments|post comment

Safety Schools?

[Oct. 31st, 2009|12:54 pm]

Hi folks. Long time lurker here, and today I have a question. What sort of graduate schools would fit as "safety schools" for me when applying to graduate school?

I'm currently an undergraduate in a school usually ranked in the top 25 so far as math departments go. I have an overall GPA over the last couple of years of 3.88 and a math GPA of 3.96. I haven't gotten my subject GRE scores back, but if I remember right I answered 55 questions on it and only remember feeling shaky about one of them, so my score will probably be somewhere around 800.

I've taken undergraduate courses in Advanced Calculus, Real Analysis, and Topology/Differential Geometry, I've taken a graduate course in Modern Algebra, and I'm currently taking graduate courses in Complex Analysis, Real Analysis, and Manifolds. I've made the Annual Deans List before and been given a "top junior in math" award. During the summer I participated in an REU program. Right now, I'm working as a TA. And I have my three recommendations lined up from good faculty.

The biggest deficiencies in my record are that I started out originally in community college, so my time at my current school will come to three years, and I have no time right now so I don't see a senior thesis on the horizon.

Maybe I'm being dense, but I don't feel comfortable judging which Ph.D. programs would be close to a sure thing when applying. University of North Carolina at Chapel Hill? Or Clemson University? What do you think?

Thanks in advance!

7 comments|post comment

image of a compact operator...

[Oct. 30th, 2009|07:45 pm]

Two posts in one day, sorry.

Is the image of a compact linear operator (say, from H to K) necessarily open or closed?

My instinct is that neither of these is necessarily true, but I'm having a tough time thinking of any counterexamples...

Any hints?

Thanks

5 comments|post comment

looking for an example

[Oct. 30th, 2009|10:40 am]

I was working through Fary's proof of the Fary-Milnor theorem, and in one of the theorems he proofs on the way, he includes the hypothesis that k(C_n) ≤ k(s).

To explain:
Given a curve closed C, the closed planar curve C_n is the projection of C onto a plane with normal vector n. The quantity k(s) is the curvature of C at a point s. So the statement above says that the total curvature of C_n is less than the curvature of C at each point.

In a footnote Fary says that this is necessary because it is possible to find a curve C and a projection C' of C such that k(C) < ∞ and k(C') = ∞; that is, the total curvature of C is finite, while the total curvature of C' is not. However, he gives no example nor any indication as to how we can find such a curve.

I've been trying to come up with an example of such a curve, but without any luck. If anyone here by chance knows an example, or just has any general thoughts on the matter, it'd be a tremendous help

Thanks!

4 comments|post comment

[Oct. 29th, 2009|11:46 pm]

Hi! I'd love help with a statistics problem, if anybody has time. Thanks in advance!

The problem reads:

We have a certain population. The true but unknown proportion of units with some specific characteristic is pi = 0.70. We want a random sample of size "n" units to estimate pi. What is the minimum value of "n" for a sampling distribution of "p" reasonable approximated by a normal distribution?

As far as I understand, with a normal distribution, you get a standard deviation of 1. And the equation I have for standard deviation involving pi is: (pi*(1-pi))/n, where n = sample size.

But when I tried that:

(pi*(1-pi))/n = 1
(pi*(1-pi)) = n
.7*(1-.7) = n = .21?

Clearly you can't have a sample size of .21. So I must be doing something wrong.

If anybody could help me out, that'd be awesome! Thank you.

2 comments|post comment

Opinions, advice?

[Oct. 29th, 2009|06:39 pm]

Can someone "lose" their ability to think mathematically? Has there ever been any study done where anti-depressants or certain medications can interfere with one's ability to think clearly? The reason I ask is because I seem to have lost all math capability the past year.

Now, I don't know if I can attribute this loss to medication, depression, or lack of interest, but I used to LOVE mathematics. There was a time I could do mathematics ALL day. Now I feel less sure, and for some reason I don't WANT to study math anymore.

Can you believe this?

I know there is an anti-anxiety med (Buspar) that one can take to improve memory and cognitive functioning. The only drawback to it, though, is that you have to take it TWICE a day at exactly the same times.

What a crappy world we live in...

15 comments|post comment

[Oct. 18th, 2009|05:13 pm]

1. Show that a multiplication operator satisfies M_φ² = M_φ if and only if φ is a characteristic function.

The "only if" direction is straightforward, but I'm unclear how to go the other way

2. Define V: L^2([0,1]) --> L^2([0,1]) by (Vf)(x) = ∫₀^x f(y)dy. Compute the adjoint of this operator.

I've played around with this without much success. I feel like there's some simple manipulation that I'm just not seeing

Thanks!

7 comments|post comment

Math education programs

[Oct. 16th, 2009|12:26 pm]

At the end of this school year, I'll be graduating with a master's degree in pure math. I want to go into teaching, though, particularly at an early college (algebra, trig, calculus, etc.) level. I am thus looking for good, reputable math education programs to apply to for a Ph.D. So far I've come up with UCSB's Center for Mathematical Inquiry, Michigan's IBL, Wisconsin, UCSD/SDSU's MSED, and Oregon State. (Texas would be really cool, but they want you to have like five years of teaching experience, which I don't.) Any other places you folks think I should look at? Thanks for your help!

12 comments|post comment

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