Search found 248 matches

by DavCrav
Fri Sep 06, 2013 9:25 pm UTC
Forum: Mathematics
Topic: basic doubt about groups
Replies: 19
Views: 4960

Re: basic doubt about groups

The integers under addition are finitely generated and are not a product of cyclic groups. I'm genuinely curious what definition of cyclic you use that excludes Z, because it can't be "a group other than Z which is generated by one element". E.g., a group G is cyclic if there exists x in ...
by DavCrav
Sun Aug 25, 2013 12:36 pm UTC
Forum: Mathematics
Topic: When will $1 be the smallest denomination?
Replies: 51
Views: 10347

Re: When will $1 be the smallest denomination?

gmalivuk wrote:It is a fake boxing stance.


Aha. I still prefer the mid fist pump explanation...
by DavCrav
Sun Aug 25, 2013 12:45 am UTC
Forum: Mathematics
Topic: When will $1 be the smallest denomination?
Replies: 51
Views: 10347

Re: When will $1 be the smallest denomination?

This is just a general question gmalivuk, but your photo looks like one of the following three things are happening: 1) You are dancing. 2) You are mid fist pump. 3) You are attempting (badly) a boxing stance. I was just interested as to which. I have nothing of value (ha!) to add to this thread, by...
by DavCrav
Sun Aug 25, 2013 12:42 am UTC
Forum: Mathematics
Topic: S subset N, exists an r in N such that if r in S then S=N
Replies: 14
Views: 3943

Re: S subset N, exists an r in N such that if r in S then S=

Anyway I do think that we often fail to give due respect to the awesome power of the universal quantifier. It lets us quantify over uncountable sets, most of whose members we could never name. That's a lot of power ... use it wisely! :-) Luckily I rarely have to invoke uncountable anything in my re...
by DavCrav
Sat Aug 24, 2013 4:24 pm UTC
Forum: Mathematics
Topic: S subset N, exists an r in N such that if r in S then S=N
Replies: 14
Views: 3943

Re: S subset N, exists an r in N such that if r in S then S=

This sort of thing is OK in a nice set like the natural numbers, but suppose we replace N by R. Now this sentence becomes a bit more interesting. Let S be the subset of all definable numbers. This can mean definable in *any way*: by computer programs, by algorithms, using words, anything reasonable...
by DavCrav
Fri Aug 23, 2013 10:09 pm UTC
Forum: Mathematics
Topic: S subset N, exists an r in N such that if r in S then S=N
Replies: 14
Views: 3943

Re: S subset N, exists an r in N such that if r in S then S=

This sort of thing is OK in a nice set like the natural numbers, but suppose we replace N by R. Now this sentence becomes a bit more interesting. Let S be the subset of all definable numbers. This can mean definable in *any way*: by computer programs, by algorithms, using words, anything reasonable....
by DavCrav
Fri Aug 23, 2013 8:49 pm UTC
Forum: Mathematics
Topic: S subset N, exists an r in N such that if r in S then S=N
Replies: 14
Views: 3943

Re: S subset N, exists an r in N such that if r in S then S=

The first line says: let S be a subset of N, so you have a fixed subset before you start. It's the difference between the two sentences: 1) For any number x, there is a number y such that y is larger than x. 2) There is a number y such that, for any number x, y is larger than x. Since S is chosen to...
by DavCrav
Thu Aug 22, 2013 6:41 pm UTC
Forum: Mathematics
Topic: Group theory II
Replies: 46
Views: 13633

Re: Group theory II

Cool. Can you check whether the group <a,b|a 2 , b 3 , (ab) 7 , [a,b] 10 , ([a,b] 4 b) 7 > is infinite? I just found out that when I ask for the order, and magma says 0, it does not mean infinite, it means it does not know. Therefore the online magma calculator is of no use to this problem. Also, i...
by DavCrav
Thu Aug 22, 2013 3:58 pm UTC
Forum: Mathematics
Topic: Group theory II
Replies: 46
Views: 13633

Re: Group theory II

Wait, so do you have magma (the real version not the online calculator)? Yes. I am a professional mathematician, so I have this sort of access, along with my own server to run this stuff on. Also, for 1) Thanks for proving that the Janko group J 1 is in fact a quotient of the group <a,b|a 2 , b 3 ,...
by DavCrav
Wed Aug 21, 2013 11:43 am UTC
Forum: Mathematics
Topic: Group theory II
Replies: 46
Views: 13633

Re: Group theory II

In an infinite group? Not that I know of. It seems to me that even in a finitely presented group, there could be infinitely many normal subgroups of bounded order... (Don't quote me on this.) This is false. In any finitely generated group, there are only finitely many subgroups of a given index. Ed...
by DavCrav
Thu Mar 14, 2013 9:25 am UTC
Forum: Mathematics
Topic: Groups & Their Presentations
Replies: 10
Views: 2529

Re: Groups & Their Presentations

While what letterX and stewbasic are saying is technically true, in reality your problem is much more tractable, since you don't want to change the generators. In this case it is a case of applying Tietze transformations to see whether one may reduce the word, and Magma has done that for you in the ...
by DavCrav
Wed Mar 13, 2013 6:46 pm UTC
Forum: Mathematics
Topic: Groups & Their Presentations
Replies: 10
Views: 2529

Re: Groups & Their Presentations

I typed the group into Magma and applied ReduceGenerators to it, and it came up with two relations for your group, namely b^-2aba^-1b^-1a^-1b^2a and b^-1ab^-1a^-1baba^-1b^-1aba^-1, so that G=<a,b|b^-2aba^-1b^-1a^-1b^2a,b^-1ab^-1a^-1baba^-1b^-1aba^-1>, with the same generators as before. Magma refuse...
by DavCrav
Sat Feb 02, 2013 2:58 pm UTC
Forum: Mathematics
Topic: Next great mathematical invention
Replies: 17
Views: 4321

Re: Next great mathematical invention

Yeah, the Laplace example is great. Nothing that was discovered in relativity and quantum theory could be said to have been "hiding in" the work he was familiar with. It was all quite novel. I cannot find such a quote by Laplace. On the contrary: What we know is not much. What we do not k...
by DavCrav
Sat Feb 02, 2013 2:31 pm UTC
Forum: Mathematics
Topic: Next great mathematical invention
Replies: 17
Views: 4321

Re: Next great mathematical invention

Carrying on the thought of fishfry, since calculus we had analysis, set theory (ZFC, GB, etc.), then category theory, and from what I can gather, although I'm not an expert on this, intensional dependent type theory. And that's just in foundations of mathematics.Entirely new fields of mathematics ha...
by DavCrav
Mon Oct 29, 2012 10:31 am UTC
Forum: Mathematics
Topic: group theory
Replies: 64
Views: 24908

Re: group theory

There's an online Magma calculator here: http://magma.maths.usyd.edu.au/calc/ I doubt if the SmallGroup database goes that far though. The paper that constructs all such groups is by A.E. Western, called Groups of Order p3q, which appeared in Proc. LMS back in 1898. The description is pretty complic...
by DavCrav
Thu Oct 25, 2012 9:50 am UTC
Forum: Mathematics
Topic: group theory
Replies: 64
Views: 24908

Re: group theory

The problem of finding all the non-isomorphic groups of order n is a fundamental question in group theory, so I think that what I am doing is group theory, even though it is not using very advanced techniques. With all due respect, no it isn't a fundamental question. The number of groups of order p...
by DavCrav
Mon Oct 22, 2012 10:21 am UTC
Forum: Mathematics
Topic: group theory
Replies: 64
Views: 24908

Re: group theory

A couple of things from earlier in the thread that I cannot find answers/responses to, so I'll post them here. Sorry if this is re-supplying information. 1) The automorphism group of (C_p)^n is GL_n(p), and in general if Z is a cyclic group, Aut(Z^n)=GL_n(Z). The easiest way to see this is that the ...
by DavCrav
Sun Oct 21, 2012 10:12 am UTC
Forum: Mathematics
Topic: The projective Weil representation of SL_2(Z/nZ)
Replies: 1
Views: 617

Re: The projective Weil representation of SL_2(Z/nZ)

The problem lies because you are misinterpreting Schur's Lemma. It says that *endomorphisms* of an irreducible representation are scalars, but not maps between isomorphic representations. For example, take the representations of a cyclic group of order 2 given by diag(1,-1) and diag(-1,1). These are...
by DavCrav
Wed Aug 01, 2012 6:34 pm UTC
Forum: Mathematics
Topic: Prime Ideals in C[x,y]
Replies: 3
Views: 3898

Re: Prime Ideals in C[x,y]

I'm concerned: you seem to imply that you think that every prime ideal of C[x,y] is maximal. However, there are plenty of ideals whose quotient is C[x] (e.g., (y)), which is an integral domain, so (y-a) is a prime ideal, for example.
by DavCrav
Wed Aug 01, 2012 6:27 pm UTC
Forum: Mathematics
Topic: group theory
Replies: 64
Views: 24908

Re: group theory

Just to let you know, inside GL_2(q) there are elements of order q+1, q-1 and q. This follows from the cyclotomic Sylow theory of groups of Lie type (but in the GL_2 case, much more easily) which says that, whenever \Phi_d(q)^a divides the order of GL_n(q) as a polynomial in q, there exists...
by DavCrav
Wed Aug 01, 2012 6:20 pm UTC
Forum: Mathematics
Topic: Investing in different countries
Replies: 9
Views: 3440

Re: Investing in different countries

What you are talking about is the carry trade.

http://en.wikipedia.org/wiki/Carry_%28investment%29

The reason that the Yen was so strong throughout the 90s and much of the 00s is because people borrowed in Yen and converted to other currencies.
by DavCrav
Wed Aug 01, 2012 6:17 pm UTC
Forum: Mathematics
Topic: Proof by Exhaustion - Is this sufficient?
Replies: 4
Views: 2385

Re: Proof by Exhaustion - Is this sufficient?

Alternatively, notice that the operation of taking remainders when dividing by 4 "commutes" with multiplication, in the sense that if I take the remainder of xy, it is equal to the remainder of x times the remainder of y (possibly after you take remainders again, as 3*3=9, which should be ...
by DavCrav
Sun Jul 22, 2012 9:37 pm UTC
Forum: Mathematics
Topic: group theory
Replies: 64
Views: 24908

Re: group theory

Just to let you know, inside GL_2(q) there are elements of order q+1, q-1 and q. This follows from the cyclotomic Sylow theory of groups of Lie type (but in the GL_2 case, much more easily) which says that, whenever \Phi_d(q)^a divides the order of GL_n(q) as a polynomial in q, there exists ...
by DavCrav
Wed Jul 06, 2011 10:36 am UTC
Forum: Mathematics
Topic: Research adviser question.
Replies: 17
Views: 2549

Re: Research adviser question.

As this is a maths forum, I'm thinking maths journals here: Ann of Math: 2-4 years is standard from submission to print. Invent Math: has an 18-month backlog, from appearing online to print. Math Z: 15 month backlog (my paper appeared in print last month, online in March 2010). Advances in Math: not...
by DavCrav
Tue May 10, 2011 11:39 pm UTC
Forum: Individual XKCD Comic Threads
Topic: 0896: "ZOmbeis"
Replies: 125
Views: 32299

Re: 0896: "Marie Curie"

(being a bit of a feminist, but in a good way) Just so you all know, this is the most abhorrent statement I have seen today. Don't get caught in the idea that being a feminist is bad, lest your brain get devoured by utahraptors. Or, just continue being a complete fuckhead, and have a nice fucking d...
by DavCrav
Tue May 10, 2011 9:19 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0896: "ZOmbeis"
Replies: 125
Views: 32299

Re: 0896: "Marie Curie"

On the subject of Emmy Noether, her appointment at Göttingen was only blocked by a bunch of historians, not the scientists. So don't blame science, blame the historians... However, this is all rubbish. Emmy Noether is as well remembered in mathematical history now as she deserves. If anybody here co...
by DavCrav
Mon Apr 25, 2011 9:53 am UTC
Forum: Mathematics
Topic: Jordan Normal Form
Replies: 7
Views: 1914

Re: Jordan Normal Form

jestingrabbit wrote:Also, I wouldn't call that matrix idempotent.


I guess he meant nilpotent...
by DavCrav
Mon Apr 18, 2011 5:43 pm UTC
Forum: Mathematics
Topic: Getting my undergrad in math. How about you?
Replies: 51
Views: 8191

Re: Getting my undergrad in math. How about you?

I hope they are sparse! Nope. That's why I gave it to a big computer. But the entries are from F_3, which helps. Strassen? Big-iron, or GPU based? I only meant a big computer, not a supercomputer. It's only multiplying some matrices together: they aren't even over 100,000 dimensions! I ...
by DavCrav
Mon Apr 18, 2011 12:20 pm UTC
Forum: Mathematics
Topic: If you learned statistics would you never buy insurance?
Replies: 92
Views: 10640

Re: If you learned statistics would you never buy insurance?

Well through the normal curve I honestly can't imagine a kind of insurance where claims are normally distributed, though. (Well, the totals are for the insurers, because of the previously mentioned central limit theorem. But individual claims are never normally distributed.) Honestly, this looks mo...
by DavCrav
Mon Apr 18, 2011 8:12 am UTC
Forum: Mathematics
Topic: If you learned statistics would you never buy insurance?
Replies: 92
Views: 10640

Re: If you learned statistics would you never buy insurance?

I think the point is that you (and indeed I) have insurance only for things that I am required to (car) and things for which the cost of it going wrong is too high (house). I don't insure my phone, as I can replace it, and I don't have extended warranties, since these are essentially insurance for t...
by DavCrav
Mon Apr 18, 2011 12:10 am UTC
Forum: Mathematics
Topic: Getting my undergrad in math. How about you?
Replies: 51
Views: 8191

Re: Getting my undergrad in math. How about you?

It's a representation of a finite group, in particular the Suzuki sporadic simple group. I want to understand how certain group elements act under this representation, so I have to multiply the matrices together to get the action of that element. That's the easy bit really. The hard bit is finding t...
by DavCrav
Sun Apr 17, 2011 7:26 pm UTC
Forum: Mathematics
Topic: Basic sin(x)cos(x)dx integral question
Replies: 4
Views: 2800

Re: Basic sin(x)cos(x)dx integral question

Why don't you start with [imath]2\cos x\sin x=\sin 2x[/imath]?
by DavCrav
Sun Apr 17, 2011 5:28 pm UTC
Forum: Mathematics
Topic: Getting my undergrad in math. How about you?
Replies: 51
Views: 8191

Re: Getting my undergrad in math. How about you?

Yakk wrote:I hope they are sparse!


Nope. That's why I gave it to a big computer. But the entries are from F_3, which helps.
by DavCrav
Sun Apr 17, 2011 3:33 pm UTC
Forum: Mathematics
Topic: Getting my undergrad in math. How about you?
Replies: 51
Views: 8191

Re: Getting my undergrad in math. How about you?

Age: 29
Education level: Ph.D. in group representation theory.
What I'm doing: postdoc at Oxford.
What I'm doing right now: waiting for a computer program to finish multiplying together 33000-dimensional matrices...
by DavCrav
Sun Apr 17, 2011 3:30 pm UTC
Forum: Mathematics
Topic: Free Will & Reimann Hypothesis?
Replies: 11
Views: 1735

Re: Free Will & Reimann Hypothesis?

...Also, humans are by nature finite creatures, so our set of beliefs at any given moment (or over the course of our lives) is finite. Just thought I'd have a wonder about this: I believe "n+1 is greater than n" with any integer in place of n. This means that I have an unbounded set of be...
by DavCrav
Wed Apr 13, 2011 9:27 am UTC
Forum: Mathematics
Topic: Writing a Calculus Textbook
Replies: 6
Views: 2188

Re: Writing a Calculus Textbook

Do any of you think this is feasible? Quite honestly, no. I have written a book, and you should expect, for a day's work, to have at most 3-5 pages of decent work, which will then have to be rewritten again. In addition, by this time I had had years of practice writing mathematics, so in three week...
by DavCrav
Wed Apr 13, 2011 9:17 am UTC
Forum: Mathematics
Topic: Random Integer
Replies: 18
Views: 2734

Re: Random Integer

... I think the second could will have labeled the planet with life on it with some infinite number and somewhere in the universe there is a first and last planet... even if there are an infinite number of them. It's similar to, say, how there are an infinite number of reals between 0 and 1 and yet...
by DavCrav
Wed Mar 30, 2011 9:38 am UTC
Forum: Mathematics
Topic: Finite group extensions
Replies: 37
Views: 3859

Re: infinite group extensions

I obviously accept your word on this. I recall being surprised by the language, because it feels more natural to me that the object being extended is a subobject of the extended object. I don't know why in homological algebra texts a different usage has been adopted, and I was rather looking for an...
by DavCrav
Sun Mar 27, 2011 6:38 pm UTC
Forum: Mathematics
Topic: Finite group extensions
Replies: 37
Views: 3859

Re: Finite group extensions

mike-l wrote:And on the off chance the person you're talking to isn't a group theorist, if it isn't clear from context, it's probably best to ask.


I also said that this term doesn't exist in group theory (I mean a finite group extension), so it's good to ask in any case!
by DavCrav
Sat Mar 26, 2011 12:03 pm UTC
Forum: Mathematics
Topic: Finite group extensions
Replies: 37
Views: 3859

Re: Finite group extensions

Explanation: Wikipedia is wrong. E.g., Robinson's A Course in the Theory of Groups, p310, Segal's Polycyclic Groups, p38. I trust Derek and Dan over Wikipedia. Oh, and other source: I'm a professional group theorist. PlanetMath has it your way, Wolfram has it my way. Jyrki quoted a text that has it...

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