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[Jul. 19th, 2008|09:14 am] |
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I read the other day that the expected distance between two randomly picked points on the infinite-dimensional hypersphere is sqrt(2) (which has to be simultaneously one of the MUCH cooler results I've heard in a while!). Do any of you know how to derive it? |
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| Out Of Print Blogspot |
[Jul. 18th, 2008|12:16 am] |
http://outofprintmath.blogspot.com/.
Basically a web site for an out-of-print mathematics texts wish list.
I'm not certain what this is supposed to accomplish. Presumably there might be some effort to bring the most wished-for book back in print, but they don't actually say.
Still, in case you want to make a suggestion, there's the link. |
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[Jul. 16th, 2008|03:50 am] |
| [ | mood |
| |  excited | ] |
Hey!!
I'm the president of the Math/Stat club at my university and I'm starting to plan this year's events. We try to evenly divide our time between math related things and stat related activities.
We're planning on bringing in 4 speakers this year, 2 about mathematics and 2 about stats. Are there any speakers that you would suggest? If you don't have a speaker in mind, what are some awesome topics of discussion? Keep in mind the vast majority of our members are not grad school bound, so nothing too intense. :)
Also, what are some other activities that would be fun? We usually have a contest the week leading up to Pi Day with the professors. Those who want to participate each have a cup with their name on it and the professor with the most money from each department gets a Pi(e) in the face...corny I know, but it makes for a good time.
Any other suggestions?? Let me know!!!
Thanks! :) |
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| Transportability of concrete categories |
[Jul. 15th, 2008|05:50 pm] |
I don't know if this belongs here or in ![[info]](http://p-stat.livejournal.com/img/community.gif) math_help; it's a request for help, but not for help with a class.
I'm attempting to read through Abstract and Concrete Categories: The Joy of Cats for my own personal enlightenment, and am just now trying to grok the definition of a (uniquely) transportable concrete category.
"A concrete category (A,U) over X is (uniquely) transportable provided that for every A-object A and every X-isomorphism k : UA --> X there is a (unique) A-object B with UB = X such that k : A --> B is an A-isomorphism."
By thinking of this in terms of typical constructs I think can make it make sense: a construct is transportable if, given an object A, any bijection from its underlying set to some other set X can be turned into an isomorphism from A to some damn fool object with underlying set X. Is this about right?
(I had more questions but I am being cut short by the fact that my workplace is closing.) |
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| Discrete Math |
[Jul. 15th, 2008|02:09 am] |
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I'm a middle school math teacher taking grad courses to be certified in secondary math education. My graduate program requires its students to take a series of discrete math courses. I'm currently in the introductory course, but my professor does not teach it as such and really skipped over the basics of proof writing, set theory, etc. We're now starting to work with integers and modular arithmetic. I understand some things, but I'm still having issues with subsets, unions, compliments, and writing proofs.
My professor is one of those people who is "too brilliant to teach" in that he obviously understands it, but has trouble explaining on a more basic level to those of us that aren't getting it.
We use the text Discrete Math with Graph Theory and my professor basically re-does problems from the text without really going over or explaining them any more in depth than the book does.
My question is: are there any good books/resoures out there that covers basic set theory and proof writing? Please keep in mind that before this class, I hadn't taken any math classes in a while and the highest level of math I'd taken was calc I. I pick up math relatively easily... it might just be that I'm out of practice with higher-level thinking? Middle school math doesn't require much more than...well...a working knowledge of basic algebra. Thanks for any help, in advance! |
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[Jul. 14th, 2008|11:00 pm] |
Does anyone else here think that Fréchet space sounds like a special place in your mind you go when you order deli-style offerings at a fast food joint?
"Get in your Fréchet space, at Wendy's!" |
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| Getting ready to do Algebra. |
[Jul. 14th, 2008|09:29 pm] |
I'm getting ready to take grad level Algebra I. My adviser gave me a book to read over the summer. He said "read the first 50 pages." I audited undergrad Algebra I, but I'm wondering what concepts should I focus on for my self-study preparation?
1. What are the hardest things to understand about Algebra I. I'm talking about concepts like "compactness" in Analysis (that took a few days for me to really grasp.)
2. What are the most important theorems? You know the big ones you really need to know.
3. What is "cool" or "fun" about this subject? What can I look forward to?
Are there any internet resources or books you would recommend?
My plan right now is to read the first 50 pages. Memorize the definitions. Do the problems I can find answers for on the web and maybe learn two of the important proofs by heart. (not memorize them, but so I can pretty much do them on my own.) I hate feeling overwhelmed, and since my undergrad degree was in Drama (yes Drama) I often find I need to do a lot of back-tracking to fill in my gaps. (but this is getting better after my first year.) So, I want to be extra prepared and lessen the misery. If possible. |
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| There should be an easier proof |
[Jul. 14th, 2008|11:01 am] |
This is from Brocker and tom Dieck's Representations of Compact Lie Groups.
One of the first problems asks you to show that GL(n,C) naturally injects into GL+(2n,R), where they suggest the map should from from thinking of Cn as R2n.
Now, taking their advice, write a complex matrix as A + Bi, where A and B are real n x n matrices.
Then the map is (A+Bi) -> the block matrix
[ A -B
B A]
Now, it's easy to check this is an injective group homomorphism (and thus, that the image is in GL(2n,R), as opposed to M(2n,R)). What's confusing me is that fact that this lands in the positive determinant matrices. Note that 1) it's possible that neither A nor B are invertible and also 2), we can't conclude the determinant of this block matrix is det(A)^2 + det(B)^2.
I have been able to show it lands in positive determinant matrices by way of a topological argument (I proved GL(n,C) is connected, so it lands in the identity component of GL(2n,R), which clearly has positive determinant. In fact, the identity component of GL(2n,R) is precisely those matrices of positive determinant.) But I feel there should be a more straightforward argument - This is one of the first problems in the book and it occurs before a single topological fact about any Lie Group is proved.
Note also that intuitively, since a choice of a basis vector in Cn determines 2 vectors in R2n, any change of basis of Cn means changing basis of R2n twice, so that the net effect is an orientation preserving change. I've just been unable to work this out into something mathematically rigorous.
Ideas? |
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| PDE question |
[Jul. 14th, 2008|10:32 am] |
Reading a paper I came across something that the author said but didn't seem to deem worthy of indicating how it's prove.
Let $D(z, 1)$ denote the disk of radius $1$ with center $z \in \mathbb{C}$. Let $G(w, \eta)$ be the Green's function for $D(z, 1)$. Then $\int_{D(z, 1)} \left|G(w, \eta) - G(\sigma, \eta)\right| dA(\eta) \leq K$ for some $K > 0$ that doesn't depend on $w, \sigma \in D(z, 1)$ or $z$. Here of course dA is just ordinary area measure.
I have NO idea how this is proven. |
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| geometry and pizza slices |
[Jul. 12th, 2008|02:55 pm] |
I'm far from a mathematician, though I did always enjoy doing word problems in school (no, seriously) and, as my fabulous college physics professor demonstrated to me, I can do calculus if I'm tricked into it.
What I am mostly is a writer. I'm working on a story in which one character is a savant with the ability to calculate precisely equal sections of polygons. He's lucky enough to have found a job working in a pizzeria, and the restaurant has become known for serving its pizza cut either into the same number of slices as people at the table, or any number the customers ordering it choose (so long as it's not more than, say, 4 slices per person, or some absolute limits depending on the size of the pizza). I don't need to "show my work" since the character skips all the usual calculation steps most people would need to do to get, for example, 7 equal slices out of a 12" round pizza.
I do, however, need to know whether a slicing pattern other than radial cuts on a round pie can be made to yield slices of equal area for all values of the number of slices for a given size and shape of pizza. Specifically:
- Can slices of equal area be cut out of a (roughly; I figure if he can look at a pizza and mentally divide its area into any number of slices, he can adjust for irregularities in the shape of the pizza's circumference or perimeter, too) rectangular pie regardless of how many slices are requested? Prime numbers seem like they'd be tricky.
- Does it make a difference if slices from a rectangular pizza must be rectangular, versus only needing to be quadrilateral?
- What about triangular slices from a rectangular pizza?
- What about quadrilateral slices from a *round* pizza?
I know basic geometry formulae (okay, I know some and can google the rest) but I suspect there's some fancier math than what I'm adept at that might be able to help with doing the hard way what my savant character does. I could pick some variable values and see if I could make a given set (or three or five or twelve) work, but that still wouldn't necessarily prove that it could be done for all the values in my set. (Sorry if I'm abusing mathematical terminology, and please correct me if I seem to be using the wrong term anywhere.) Mainly, I need to know how customers could be allowed to order their pizza sliced, and what if anything I can't have my savant character be able to do.
As I said, I don't *need* to show the calculations the character skips over... but in addition to needing answers to my logistics questions, I'd really enjoy seeing *how* the answers could be worked out. Feel free to geek out, in other words. (:
If this is the wrong community, I'll be happy to re-post if someone directs me to a better forum. (Since it isn't a homework assignment, I didn't think math_help was the best place, but it didn't seem high-enough level math for math_research, either.) |
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[Jul. 11th, 2008|11:26 pm] |
Ok, this is just perverted:
THEOREM: The set P = {all prime numbers} is infinite.
Nonstandard proof: We must show that *P must contain nonstandard members. To this effect, consider a hypernatural K in *N that is divisible by every natural number. One such K could be (1, 2, 3, ..., n, ...) if the ultrafilter F contains {all even numbers, all multiples of 3, all multiples of 4, all multiples of 5, ...}. Next, consider the hyperprime number p in *P that divides K + 1. Such p exists because every hypernatural number greater than 1 has at least one hyperprime factor, by the transfer principle (next section). Then p must be nonstandard, for if it weren't, it would divide K, by assumption on K, and since p would then divide K and K + 1, it would also divide their difference, 1, which is not true for standard primes. This proves the theorem.
(Taken from here.) |
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[Jul. 10th, 2008|04:08 pm] |
I've got a question about conditioning on Gaussian random processes/fields. The finite-dimensional situation is elementary: if (X,Y) is an (n+m)-variate Gaussian, then X conditioned on Y is also Gaussian. See the Wikipedia article for all the formulae.
Now, consider a continuous Gaussian process/field f(x) on R2 (that is, for any x1, ..., xn ∈ R2, (f(x1, ..., f(xn)) is an n-variate Gaussian). If we condition the field on finitely many values f(y1) = b1, ..., f(ym) = bm, then by the above discussion, the field is still Gaussian. But what if we condition on a far more complicated event or σ-algebra?
( Read more... ) |
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| Geodesics on Projective Spaces |
[Jul. 10th, 2008|03:50 pm] |
Dear Friends,
Does anybody know any good reference where geodesic equations on CP^n are investigated? In particular, I'm interested in CP^3. |
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| Segal Bargmann Space Question |
[Jul. 9th, 2008|05:22 pm] |
Hi, first sorry for crossposting this on MathResearch. It would just save me a lot of time if I got an answer asap.
Anyway, time for a fun obscure analysis question :)
The setting I'm in is the Segal Bargmann space $H^p(\mathbb{C}^n, d\mu)$ for $p > 1$, which is the Banach space of entire functions $f$ on $\mathbb{C}^n$ where $\int_{\mathbb{C}^n} |f(w)|^p d\mu (w) < \infty$ and $d\mu(w) = e^{-\frac{|w|^2}{2}} dV(w)$ and $dV$ is ordinary volume measure.
The question is: Is the dual space of $H^p(\mathbb{C}^n, d\mu)$ just $H^{p'}(\mathbb{C}^n, d\mu)$ under the standard pairing where $p'$ is the conjugate exponent? I think so, but I find absolutely no reference for this.
This is not as trivial as finding the dual of say, the bergman space of the unit ball. For $p = 2$ we get a reproducing kernel Hilbert space and the kernel is $K(z, w) = e^\frac{\langle z, w\rangle}{2}$ and the orthogonal projection $P$ onto $L^2(\mathbb{C}^n, d\mu)$ is the integral operator with this kernel.
In our case, $P$ is a bounded operator from $H^p(\mathbb{C}^n, d\mu)$ to $H^p(\mathbb{C}^n, d\mu)$ iff $p = 2$, so unlike the Bergmann space case, we can't use a simple Hahn Banach arguement to get the standard duality.
I'd be very surprised if this is not known, it's just that I can't seem to find a reference.
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| more linear algebra questions |
[Jul. 9th, 2008|01:19 pm] |
Maybe I'm not reading it correctly, but Sylvester's theorem looks like a tautology. It reads
"Let V be a finite dimensional vector space over R, with a scalar product. There exists an integer r > 0 such that if {v1,...,vn} is an orthogonal basis of V, then there are precisely r integers i such that [vi,vi] > 0."
Well, obviously. Either there are some vi with that property or not. If there aren't, r = 0, and if there are, then r is just that number. I guess I'm just not seeing the significance of r, though Lang mentions it is the index of positivity of the scalar product.
Or is this just a theorem telling us that the number of i doesn't depend which orthogonal basis is used? |
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| Mathy Programmer Looking for a Topic |
[Jul. 9th, 2008|11:19 am] |
| [ | mood |
| |  curious | ] |
I've got a decent-beginning-grad-student-ish handle on algebra
and topology, solid programming skills, a new-found love of Lisp,
and a drive to learn/do more.
I am looking for some area of algebra or topology that could
benefit from some better software. For a time earlier this year,
I thought it might be calculating sizes of subalgebras of universal
algebras. But, the UACalc maintainers are really responsive and
really on top of that game.
Does anyone have any suggestions? Things that one wishes there
were better tools to tinker around or better tools to compute
something or better tools to validate something?
I'd prefer things algebra, topology, or maybe number theory, but
I am open to other suggestions. I'm open to starting from a problem
to develop algorithms or starting from algorithms where there isn't
really an implementation readily available. Edit: I should also say
I am willing to work with folks who need custom code to progress
on their dissertations and such.
Thanks. |
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| Complex Variables in C^2 |
[Jul. 8th, 2008|04:21 pm] |
I'm doing a little research project. I'm looking for any results in Several Complex Variables that are true in C2 but not in Cn (n>2). Have you seen any?
Such a thing is a little difficult to search for. I have found a few things, though. E. M. Chirka proved such a result in 2001. Following Chirka, G. Bharali recently did some work along the same lines. Also, Rudin has such a result in Function theory in the unit ball of Cn. |
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| Grad school questions |
[Jul. 8th, 2008|09:20 am] |
Okay, hi. I'm looking at putting together an application for graduate school for the fall (yes, I know, it's really late, the department said they'd still consider an application). I have many questions which I hope you all may be able to give me some perspective on.
( Cut to save your friends page )
EDIT: Indecision is lame. :S |
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| World series |
[Jul. 8th, 2008|10:31 am] |
And to think I always thought this was convergent. Damn.
UPDATE: Better do this to avoid division by zero...
And define the sequence to be zero for y=1904 and 1994. |
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| pythagorean theorem, illustrated |
[Jul. 7th, 2008|02:10 am] |
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