Search found 251 matches

by DavCrav
Sun May 17, 2009 11:45 pm UTC
Forum: Mathematics
Topic: ITT: Why Math Is Awesome.
Replies: 51
Views: 6327

Re: ITT: Why Math Is Awesome.

I like how abstract and precise it is. It's consistency is kind of reassuring, while all the same it's fun trying to find holes in that consistency. *Shushes Gödel, who sits in the corner rocking backwards and forwards...* Edit: Why do I like mathematics? I don't know any more. I'm not sure I do, a...
by DavCrav
Fri May 15, 2009 5:36 pm UTC
Forum: Mathematics
Topic: Courses, Abridged
Replies: 19
Views: 2427

Re: Courses, Abridged

I'll try to list the single most important thing from each subject (imho) as requested. Just be mindful that presented in this way they probably won't make much sense nor will this list replace a REAL course on the subject. Linear algebra: For any n-dimensional vector space, V, one can always find ...
by DavCrav
Fri May 15, 2009 5:25 pm UTC
Forum: Mathematics
Topic: Submitting a paper
Replies: 14
Views: 1683

Re: Submitting a paper

quintopia wrote:
DavCrav wrote:I've had professors ask me where I think a certain paper should be submitted.


Which is what I said I would do in the professor's place. What's the problem here?


Well, I'm not a student (was until recently though). I was talking about them asking me for advice on where to put their papers...
by DavCrav
Thu May 14, 2009 9:40 am UTC
Forum: Mathematics
Topic: Submitting a paper
Replies: 14
Views: 1683

Re: Submitting a paper

I don't understand some people on here. If this person is an undergraduate, there is no way they could pick their way through the ridiculous waltz of journals. That's not easy for grad students and even postdocs, and advice is sought from other people whatever the level of the academic. I've had pro...
by DavCrav
Sun May 10, 2009 10:53 pm UTC
Forum: Mathematics
Topic: Semisimple Lie algebras
Replies: 5
Views: 852

Re: Semisimple Lie algebras

Since L is a subalgebra of sl, it inherits a natural action on the n-dimensional ambient space. This is apparently a simple application of weight spaces, but since I don't have the lecture notes for the course, I cannot work out how it applies. (The guy who set the exam question tells me that it is ...
by DavCrav
Sun May 10, 2009 11:05 am UTC
Forum: Mathematics
Topic: Semisimple Lie algebras
Replies: 5
Views: 852

Re: Semisimple Lie algebras

mike-l wrote:Isn't the subalgebra of diagonal matrices abelian and hence solvable?


Ah yes. This is what happens when I miss a word out of the question. Let's try again.

Let [imath]L[/imath] be a subalgebra of [imath]\mathrm{sl}_n(\mathbb{C})[/imath], where [imath]n\geq 2[/imath], and [imath]L[/imath] acts irreducibly. Show that [imath]L[/imath] is semisimple.
by DavCrav
Fri May 08, 2009 12:48 am UTC
Forum: Mathematics
Topic: Semisimple Lie algebras
Replies: 5
Views: 852

Semisimple Lie algebras

I'm quite embarrassed to have to ask this question, but this is a backup in case the person who is meant to give me the answer fails to do so. Let L be a subalgebra of \mathrm{sl}_n(\mathbb{C}) , where n\geq 2 . Show that L is semisimple. This should be done with basic Lie algebra theory (wh...
by DavCrav
Fri Apr 24, 2009 4:51 pm UTC
Forum: Forum Games
Topic: Name a smaller (closer to zero) number.
Replies: 52
Views: 4845

Re: Name a smaller (closer to zero) number.

headprogrammingczar wrote:[imath]\frac{1}{\aleph_0}[/imath]
It technically isn't infinity...


Erm?
by DavCrav
Fri Apr 24, 2009 9:30 am UTC
Forum: Mathematics
Topic: Dimension of magic square vector spaces
Replies: 26
Views: 4971

Re: Dimension of magic square vector spaces

If one does not require that the diagonals add up to the magic constant, one actually has that the product of two magic squares is a magic square, and so if the coefficients come from a ring R, then all magic squares form an R-algebra. In http://people.maths.ox.ac.uk/~craven/oldweb/magicalgebras.pdf...
by DavCrav
Thu Apr 23, 2009 9:46 pm UTC
Forum: Mathematics
Topic: Famous professors thread
Replies: 10
Views: 2066

Re: Famous professors thread

the anecdotes are interesting and all, but i see no one has decided to follow my convention of linking to an important result and instead opted for the generic bio and home pages. I probably have an unfair advantage, because I'm coming from Oxford... In no particular order, I've been lectured to by...
by DavCrav
Thu Apr 23, 2009 6:36 pm UTC
Forum: Mathematics
Topic: Famous professors thread
Replies: 10
Views: 2066

Re: Famous professors thread

I probably have an unfair advantage, because I'm coming from Oxford... In no particular order, I've been lectured to by John Thompson, Jon Alperin, jon Carlson, Dave Benson, Efim Zelmanov, Michael Aschbacher, Gus Lehrer, Boris Zil'ber, Dominic Joyce, Raphael Rouquier, Peter Neumann, Dan Segal, Marcu...
by DavCrav
Wed Apr 22, 2009 7:36 pm UTC
Forum: Mathematics
Topic: A Toughie
Replies: 24
Views: 2050

Re: A Toughie

The problem with statements about determining whether a number is rational or not is that the numbers you start off with, the computable numbers (those for which there is some algorithm to work out their decimal expansion, essentially) form a countable subset of the reals, and so the question doesn'...
by DavCrav
Wed Apr 22, 2009 7:30 pm UTC
Forum: Mathematics
Topic: question that I have no Idea how to go about
Replies: 10
Views: 1199

Re: question that I have no Idea how to go about

Well, this hitting thing is confusing, as in it's not adequately explained, but it must seriously affect the probabilities, because otherwise the probability is 1/1296 = 0.77%...
by DavCrav
Sun Mar 29, 2009 10:19 pm UTC
Forum: Mathematics
Topic: Singularics - Claims to prove RH by dividing by 0
Replies: 21
Views: 3497

Re: Singularics - Claims to prove RH by dividing by 0

Any set of random (or seemingly random) periodic values ρx can be infinitely defined (meaning that there are an infinite number of solutions for the order of the values) in reverse with a simple function Ν (m), the Neutronic Function. I'm not even sure what that means. Obviously a PhD in mathematic...
by DavCrav
Tue Mar 17, 2009 6:38 pm UTC
Forum: Mathematics
Topic: How can (0,1) exist?
Replies: 11
Views: 1810

Re: How can (0,1) exist?

Is it incorrect to say that there exists no n such that \sum_1^n{\frac{9}{10^n}} is not in the interval (0,1)? To be nitpicky, yes. What you meant to say is that there exists no natural number n such that \sum_{k=1}^n{\frac{9}{10^k}} is not in the interval (0,1). Actually, assuming that n is at lea...
by DavCrav
Mon Mar 09, 2009 10:38 am UTC
Forum: Mathematics
Topic: Find if a number has a root
Replies: 8
Views: 1245

Re: Find if a number has a root

FWIW, a year or so ago, I was looking for perfect powers that are close neighbours. Such pairs (& triplets) are reasonably common in the small numbers (eg, 121, 125, 128) but they start to thin out fairly quickly. Such groupings tend to occur with perfect squares or powers of 2 (which I suppose...
by DavCrav
Sun Mar 08, 2009 2:58 pm UTC
Forum: Mathematics
Topic: dexa- rotation of mathematical proportion/equations
Replies: 7
Views: 1281

Re: dexa- rotation of mathematical proportion/equations

I seriously hope that at some point I don't jump off the deep end like this guy...
by DavCrav
Sun Mar 08, 2009 10:52 am UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2112

Re: Representation theory help

For instance, in abelian groups all subgroups are kernels of homomorphisms. In general groups, if we want to 'quotient out' a non-normal subgroup, the best we can do (AFAIK) is map to the set of right cosets. But in what sense could this map be considered as if it's a homomorphism, and the resultin...
by DavCrav
Sun Mar 08, 2009 1:22 am UTC
Forum: Mathematics
Topic: Real life arguments solved with math
Replies: 12
Views: 1459

Re: Real life arguments solved with math

Right: please contrast What it means at its most basic level with Huh?? The geodesic flow on a compact surface of constant negative curvature is plenty continuous, and the distance between any two unit vectors is bounded above (certainly, it is less than the diameter of a fundamental region of the a...
by DavCrav
Sun Mar 08, 2009 1:11 am UTC
Forum: Mathematics
Topic: group theory question
Replies: 14
Views: 2108

Re: group theory question

I think I need to know what `wreath product problems' are first, before directing you anywhere... The wreath product is a nice construction, and interacts well with representations.
by DavCrav
Sun Mar 08, 2009 1:10 am UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2112

Re: Representation theory help

There's a book in our college library called 'Cohomologie non-abelienne'. It was too difficult for me to get much out of out, but from what I have read elsewhere, non-abelian cohomology gives rather little reward for a huge amount of work, compared to all the wonderful structure that exists for abe...
by DavCrav
Sat Mar 07, 2009 8:59 pm UTC
Forum: Mathematics
Topic: Real life arguments solved with math
Replies: 12
Views: 1459

Re: Real life arguments solved with math

Once I was arguing with a friend about controlled chaos. My friend (who debated in high school) stated that there is no such thing, and was convinced that there was no logical argument that could prove him wrong. I am reminded of a quotation of Charles Babbage: "I am not able rightly to appreh...
by DavCrav
Sat Mar 07, 2009 8:53 pm UTC
Forum: Mathematics
Topic: Proof that the sqrt of 2 is irrational?
Replies: 28
Views: 3989

Re: Proof that the sqrt of 2 is irrational?

@Everyone arguing about this stuff. Seriously, you're doing it wrong. It is easy to define an even number to be one of the form 2n, and an odd number to be one of the form 2n+1. Then using well-ordering, you can prove that every number is either even or odd, and cannot be both. Then algebraic manipu...
by DavCrav
Sat Mar 07, 2009 4:24 pm UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2112

Re: Representation theory help

Time to split responses, I suppose To DavCrav: Thanks for the suggestion - I know the Diaconis notes, but was unaware of the book... though on checking it is in a part of the library I pass through frequently. If you know of any applications of modular representation theory, you would be a hero. Er...
by DavCrav
Sat Mar 07, 2009 2:32 pm UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2112

Re: Representation theory help

So the overall concept was explained to me on a particularly long plane flight, but there's still a lot I don't understand. What confuses me the most is the process of finding and using a representation. Given a specific group, how does one go about deciding upon a representation. And then, what is...
by DavCrav
Sat Mar 07, 2009 2:26 pm UTC
Forum: Mathematics
Topic: dexa- rotation of mathematical proportion/equations
Replies: 7
Views: 1281

Re: dexa- rotation of mathematical proportion/equations

cameronv wrote:I would like help from mathematicians in ths forum that we know each other...


I would like to help, but... no I cannot think of anything suitable. I'm wondering if English is perhaps not your first language. And it appears that mathematics is not one of your languages at all...
by DavCrav
Sat Mar 07, 2009 2:17 pm UTC
Forum: Mathematics
Topic: Representation theory help
Replies: 23
Views: 2112

Re: Representation theory help

I'm not familiar with Fourier analysis, but I'd have thought that the special thing about the integers, the torus and the reals is that they're all abelian, and in this case you have Pontryagin duality and whatnot. I don't know if the methods of classical Fourier analysis carry over all that well t...
by DavCrav
Wed Mar 04, 2009 12:32 am UTC
Forum: Mathematics
Topic: Favourite Erroneous "Proofs"
Replies: 194
Views: 43432

Re: Favourite Erroneous "Proofs"

Yes, that sort of thing. I don't know whether the diagonal fallacy is called that everywhere, but that is what I know of it as. It comes from the assumption that every subobject of the product of two objects is a product itself; the name comes from this fallacy applied to 2-dimensional vector spaces...
by DavCrav
Tue Mar 03, 2009 11:44 pm UTC
Forum: Mathematics
Topic: Favourite Erroneous "Proofs"
Replies: 194
Views: 43432

Re: Favourite Erroneous "Proofs"

I can't believe I've read through four-and-a-half pages of erroneous proofs with all to show for it some divide-by-zero problems. How about the diagonal fallacy? Broadly speaking, this comes from the assumption (for example) that in the direct product of two groups, G and H , every subgroup is of th...
by DavCrav
Tue Mar 03, 2009 11:55 am UTC
Forum: Mathematics
Topic: Working on Gilbreath's Conjecture
Replies: 9
Views: 1921

Re: Working on Gilbreath's Conjecture

I'm afraid I can't see how your method can be easily rescued. Your statement on the first half of the second page (If f(j) is changed but a_{i,j} still does not exceed 2^{j-i} , level i = 1 starts with a 2 and then is all odd numbers, then a_{j,j} still must be equal to 1, when j > 1 ) appea...
by DavCrav
Wed Feb 25, 2009 2:16 pm UTC
Forum: Mathematics
Topic: Strong Induction and Recursive Functions
Replies: 4
Views: 1093

Re: Strong Induction and Recursive Functions

Another method is to notice that f(0) and f(1) are correct, and to prove that for the f given, it satisfies the equation given. Then appeal to the fact that the function so defined in the question is unique.
by DavCrav
Sun Feb 01, 2009 1:59 pm UTC
Forum: Mathematics
Topic: Two somewhat difficult questions (not-homework related)
Replies: 21
Views: 2104

Re: Two somewhat difficult questions (not-homework related)

In general, this problem of finding the black islands in the situation is the idea of so-called percolation theory. To give a demonstration that this area is hard, if you had chosen 60% rather than 50% chance, you would have exactly one, infinite component (as n\to \infty ) with probability 1. (That...
by DavCrav
Mon Jan 19, 2009 11:39 am UTC
Forum: Mathematics
Topic: Bourbaki
Replies: 2
Views: 546

Re: Bourbaki

They are very useful as reference material, but they aren't textbooks by any real measure. If you want to learn a topic, read another book. If you already know a topic, and want to refresh your memory or glance at things, they are very useful. I have three of them myself, and whenever Springer do on...
by DavCrav
Mon Jan 19, 2009 11:34 am UTC
Forum: Mathematics
Topic: Where to publish
Replies: 10
Views: 1458

Re: Where to publish

In my spare time I wrote a short mathematical paper deriving an expression for what seems to be a mathematical coincidence, thus showing that it isn't one. First I submitted it to the Canadian Mathematical Bulletin (I'm Canadian), but they rejected it in three minutes and told me to try the America...
by DavCrav
Mon Jan 19, 2009 11:24 am UTC
Forum: Mathematics
Topic: MATH PUZZLE!
Replies: 25
Views: 2468

Re: MATH PUZZLE!

After a math lecture in Harvard the students went to the cafeteria and each one of them took a cup of tea and cheese sandwich. They all paid $18.94. How many students where they? THANKS!! As many as you want. You simply said that each of them paid $18.94, which tells you nothing. However, I'd menti...
by DavCrav
Sun Dec 28, 2008 9:59 am UTC
Forum: Mathematics
Topic: group theory question
Replies: 14
Views: 2108

Re: group theory question

I asked a Scandinavian person, and she said that it's pronounced something like 'Zee-loff'. And while were at it, Lie (as in Lie groups and Lie algebras) is pronounced 'Lee-uh'. Anyway...
by DavCrav
Mon Dec 22, 2008 12:27 am UTC
Forum: Mathematics
Topic: question about permutation cycles
Replies: 3
Views: 1379

Re: question about permutation cycles

In general, conjugation of one permutation by another is very easy to do. (This is why computation in symmetric groups is much easier than in other groups.) If you are conjugating x by g, then write out x as cycles. Now, apply g to each of the numbers in the cycle, but don't alter the cycle . This i...
by DavCrav
Mon Dec 22, 2008 12:13 am UTC
Forum: Mathematics
Topic: Looking for a Group
Replies: 11
Views: 2335

Re: Looking for a Group

OK, this is getting quite complicated, so let's try to distill what's happened here: The OP was looking for an infinite, finitely generated, amenable group, whose abelianization is finite, and is residually finite. More generally, he was looking for one of those groups such that all subgroups of fin...
by DavCrav
Thu Dec 11, 2008 9:33 pm UTC
Forum: Mathematics
Topic: group theory question
Replies: 14
Views: 2108

Re: group theory question

OK, let's go step by step: Let x be the generator for the infinite cyclic subgroup of index 2. Let y be an element outside x. By considering G/<x>, one easily sees that y 2 is in G, since (<x>y) 2 =<x>, (as G/<x> is cyclic of order 2). The map g-> g y (i.e., conjugation by y) is a homomorphism (in f...
by DavCrav
Thu Dec 11, 2008 12:16 pm UTC
Forum: Mathematics
Topic: group theory question
Replies: 14
Views: 2108

Re: group theory question

Let the infinite cyclic group be generated by x, and let y be an element of G not in <x>. Consider the element x^y: where can that be sent, given that y^2 is in <x>, and so must act trivially on x. If y acts trivially on x, then G is abelian, and you should be able to do it. So y acts non-trivially ...

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