Search found 248 matches

by DavCrav
Sat Apr 10, 2010 5:57 pm UTC
Forum: Mathematics
Topic: New value of pi?
Replies: 45
Views: 6727

Re: New value of pi?

My favorite bit is but this does not account for the infinitesmal area under the curve that just does not disappear, forcing us to conclude that the true value of Pi must be a fraction more that what we thought it was. This is why NASA are using a secret value of Pi that is a fraction more than the...
by DavCrav
Thu Apr 08, 2010 11:00 am UTC
Forum: Mathematics
Topic: Commuting permutations
Replies: 5
Views: 4109

Re: Commuting permutations

You are looking for the centralizer of a given permutation. If you don't know about wreath products this will be difficult to describe, but if you want to know: Suppose that the cycle type is 1^{a_1}2^{a_2}\ldots n^{a_n} . The centralizer is the direct product of the centralizers of the individual c...
by DavCrav
Wed Apr 07, 2010 5:33 pm UTC
Forum: Mathematics
Topic: Four color theorem
Replies: 82
Views: 20823

Re: Four color theorem

What about more or less dimensions? I suppose in 1D you only need two colours (think alternately coloured "dashed" line), but in 3D...do you need an infinite number of colours? I think you must. Yes, you can embed K_n in 3 space because, among other things, a curve doesn't split the space...
by DavCrav
Thu Mar 25, 2010 2:31 pm UTC
Forum: Mathematics
Topic: Four color theorem
Replies: 82
Views: 20823

Re: Four color theorem

Classification of finite sets, its 1500+ pages, but each part of the proof has been checked and verified Assuming that you meant Classification of FInite Simple Groups, as you appear to have done so, CFSG took many more than 1500 pages. Try adding an extra 0 on the end. The Aschbacher--Smith quasit...
by DavCrav
Wed Mar 24, 2010 7:29 am UTC
Forum: Mathematics
Topic: Four color theorem
Replies: 82
Views: 20823

Re: Four color theorem

Two proofs that come to mind are ... and the categorisation of finite simple groups. ... the second is incredibly gigantic and mindbendingly complicated. It wasn't Pornography Storage Apparatus aided, and it would be hard to make a machine check it. I should point out that the original classificati...
by DavCrav
Thu Mar 18, 2010 10:30 am UTC
Forum: Mathematics
Topic: Ideal Currency Denominations
Replies: 47
Views: 5697

Re: Ideal Currency Denominations

The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.) British currency FTW! Yeah, when the Brits finally decided to decimalize their currency, they saw how well the 1-2-5 system was already working for dozens of countries, and so copied th...
by DavCrav
Wed Mar 17, 2010 9:42 pm UTC
Forum: Mathematics
Topic: Ideal Currency Denominations
Replies: 47
Views: 5697

Re: Ideal Currency Denominations

gmalivuk wrote:The 1-2-5 pattern seems to work well for lots of countries. (Repeated, of course: 1, 2, 5, 10, 20, 50, 100, etc.)


British currency FTW!
by DavCrav
Tue Mar 16, 2010 11:29 am UTC
Forum: Mathematics
Topic: Studying logic
Replies: 9
Views: 1594

Re: Studying logic

If you're looking to learn the more basic propositional logic then I can provide a link to the lecture notes I'm using at the moment in my prop logic module. They're all hand written (I guess the lecturer doesn't like computers too much) but they have an upside of example sheets (with solutions) an...
by DavCrav
Mon Mar 08, 2010 9:44 am UTC
Forum: Mathematics
Topic: how to approach a math competition
Replies: 9
Views: 1555

Re: how to approach a math competition

...My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1....tldr; can an...
by DavCrav
Sun Mar 07, 2010 11:34 am UTC
Forum: Mathematics
Topic: how to approach a math competition
Replies: 9
Views: 1555

Re: how to approach a math competition

...My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1....tldr; can an...
by DavCrav
Wed Mar 03, 2010 9:21 am UTC
Forum: Mathematics
Topic: Wrong on the Internet: John Gabriel
Replies: 185
Views: 65922

Re: Wrong on the Internet: John Gabriel

It always seemed obvious to me that the reals are countable. The principle of well ordering guarantees that any set of real numbers has a least element. Start with the set of all reals, print the least element, then remove it from the set. Rinse and repeat an infinite number of times. Maybe there's...
by DavCrav
Mon Mar 01, 2010 9:44 am UTC
Forum: Mathematics
Topic: WolframAlpha keeps calculating after it prints an answer
Replies: 5
Views: 1355

Re: WolframAlpha keeps calculating after it prints an answer

I've had this happen. It has a go at integration, gives some numerical answer, and then tries to give a nice expression for it afterwards. It's kind of cool.
by DavCrav
Fri Feb 26, 2010 9:30 am UTC
Forum: Mathematics
Topic: Voting Systems
Replies: 12
Views: 1150

Re: Voting Systems

There is the decision problem -- "does this set describe a set of winning coalitions". It looks like it is in NP (it might not be, if the required vote strengths goes super-exponential, but for some reason I doubt that...) I had an idea about this: is the largest voting strength that a pe...
by DavCrav
Thu Feb 25, 2010 11:47 am UTC
Forum: Mathematics
Topic: Voting Systems
Replies: 12
Views: 1150

Re: Voting Systems

Can we take a set of "winning coalitions", and produce a voting scheme from it? There are some rules that the sets have to have. The one that is easiest to notice is that if A \subseteq B and A is a winning coalition, then so is B. Problem 2 mentions a further pretty natural constraint. I...
by DavCrav
Wed Feb 24, 2010 10:54 am UTC
Forum: Mathematics
Topic: Voting Systems
Replies: 12
Views: 1150

Re: Voting Systems

Well, for reasonably large n, the answer to 1) is"many". My first estimate would be all possible sets of subsets of {1,..,n}. This has a lot of overcount, of course, because for example we can have {1} but not {1,2}. So you need to include all oversets of any given set. Hence you can think...
by DavCrav
Sat Feb 20, 2010 6:08 pm UTC
Forum: Mathematics
Topic: The Life of a Modern Mathematician
Replies: 3
Views: 1207

Re: The Life of a Modern Mathematician

MartianInvader wrote:You'll spend time going to conferences (which are typically awesome).


Oh yeah! Conferences rock.
by DavCrav
Sat Feb 20, 2010 6:06 pm UTC
Forum: Mathematics
Topic: Can't Appreciate the Unknown
Replies: 18
Views: 2233

Re: Can't Appreciate the Unknown

Nah the best bit is when you're stuck on a problem for days and all of a sudden you realize A. How to do it with ease and B. That you're an idiot for not seeing it before. :) Well, I was stuck on a problem for a few days, but now I know how to do it. I wouldn't say it was easy, but the proof is onl...
by DavCrav
Thu Feb 18, 2010 1:55 pm UTC
Forum: Mathematics
Topic: Can't Appreciate the Unknown
Replies: 18
Views: 2233

Re: Can't Appreciate the Unknown

I hear it's been referred to as an O-level by some teachers, but I choose to ignore those comments because I don't have any idea whatsoever what an O-level is. Back in a mists of time the British system had Ordinary and Advanced qualification levels, O-levels and A-levels. Then the O-levels were re...
by DavCrav
Tue Feb 09, 2010 10:07 am UTC
Forum: Mathematics
Topic: What is the error in this 2=1 "proof"?
Replies: 22
Views: 3466

Re: What is the error in this 2=1 "proof"?

Sure. If you are doing certain subjects, you might not like this. But then, since it isn't smooth things are going to get horrible. The same is probably true about something like \sin(n\pi x)/n between 0 and 1, right? (I'm an algebraist so not really used to all this stuff.) This latter exam...
by DavCrav
Tue Feb 09, 2010 9:29 am UTC
Forum: Mathematics
Topic: What is the error in this 2=1 "proof"?
Replies: 22
Views: 3466

Re: What is the error in this 2=1 "proof"?

Why exactly is the limit of the step-like function not equivalent to a straight line? I mean, it obviously looks more and more like a straight line the more steps we add. If you zoom in though, it still looks step-like, no matter how many times you subdivide the steps. Compare that with what happen...
by DavCrav
Mon Feb 08, 2010 10:00 am UTC
Forum: Mathematics
Topic: What is the error in this 2=1 "proof"?
Replies: 22
Views: 3466

Re: What is the error in this 2=1 "proof"?

There is absolutely no reason to assume without proof that if the length of each curve in a sequence converges that it converges to the length of the limit curve. Indeed, it's easy to see with a staircase that goes up 10m and across 10m. No matter how small the steps, the total length is 20m, but t...
by DavCrav
Fri Feb 05, 2010 7:07 pm UTC
Forum: Mathematics
Topic: Double Majoring Physics/Math
Replies: 23
Views: 6739

Re: Double Majoring Physics/Math

I'm pretty ambivalent about the well-roundedness thing -- I do have passing curiosity directed toward other subjects outside of math and physics, and I can definitely imagine that it is easier to pursue those curiosities during undergrad, but I tend to lean toward DavCrav's stance, ultimately. I wo...
by DavCrav
Thu Feb 04, 2010 2:09 pm UTC
Forum: Mathematics
Topic: Infinity
Replies: 31
Views: 4048

Re: Infinity

andrewxc wrote:Also as was stated above, if ln(x) has a limit, then so should its inverse function, e^x.


I might be wrong, but if [imath]\ln(x)[/imath] has a limit, wouldn't that mean that [imath]\mathrm{e}^x[/imath] attained infinity at a finite value, so wouldn't be defined after a certain point? A bit like [imath]\tan(x)[/imath].
by DavCrav
Thu Feb 04, 2010 10:47 am UTC
Forum: Mathematics
Topic: Double Majoring Physics/Math
Replies: 23
Views: 6739

Re: Double Majoring Physics/Math

If I were you I would try to find a second major or minor in something completely different from mathematics, such as: history, creative writing, pol sci, etc. It will help you think about the world and mathematics in very different, and nontrivial ways. Also it is harder to take interesting classe...
by DavCrav
Thu Feb 04, 2010 10:29 am UTC
Forum: Mathematics
Topic: Wreath Products (Run away!)
Replies: 8
Views: 2017

Re: Wreath Products (Run away!)

About semidirect products: Let G be the dihedral group D_8 , and let A=\langle a\rangle be a subgroup of order 4 and b an involution (element of order 2) outside of A . Clearly A is a normal subgroup and B=\langle b\rangle is a subgroup, we have A \cap B=1 , and AB=G . These are almost the criteria ...
by DavCrav
Sun Jan 31, 2010 11:33 pm UTC
Forum: Mathematics
Topic: Quick matrix question.
Replies: 7
Views: 1036

Re: Quick matrix question.

achan1058 wrote:Row 3 and 4 forms an invertible submatrix, so there's actually no need to work anything out.


Any two rows form an invertible submatrix...
by DavCrav
Sun Jan 31, 2010 2:02 pm UTC
Forum: Mathematics
Topic: [solved]Commuting fucntions problem
Replies: 4
Views: 705

Re: Commuting fucntions problem

For part (c), first apply the fact that p and q commute, and then specialize x to be a fixed point under q. Do a bit of cancelling and see what happens.
by DavCrav
Fri Jan 22, 2010 10:31 am UTC
Forum: Mathematics
Topic: Cardinality of the gaussian integers
Replies: 16
Views: 2360

Re: Cardinality of the gaussian integers

One way to order them is a + bi < c + di if a < b or (a=b and c < d). It is easy to see this is a strict well-ordering and isomorphic to \left \langle \omega,< \right \rangle . Okay, first of all you mean a<c or a=c and b<d, not what you wrote. But this is still false, since you have essentially de...
by DavCrav
Fri Jan 22, 2010 2:22 am UTC
Forum: Mathematics
Topic: Why and How
Replies: 12
Views: 1211

Re: Why and How

The gas problem is subtle, but the more people who are going, the less that effect is going to be. If you think about it, if one person goes versus two people, the gas money is halved, whereas if twenty-two people go versus twenty-one, the difference isn't very much at all. Basically, for everything...
by DavCrav
Fri Jan 22, 2010 2:10 am UTC
Forum: Mathematics
Topic: Cardinality of the gaussian integers
Replies: 16
Views: 2360

Re: Cardinality of the gaussian integers

Alternatively, embed the Gaussian integers into the rationals by sending a+bi to p/q, where p is the a-th power of 2 and b is the bth power of 3. By swapping 2 for 5 and 3 for 7, you can deal with minus signs for either a or b, and maybe (a+1)-th power and (b+1)-th power to deal with a+0i and 0+bi. ...
by DavCrav
Fri Jan 22, 2010 2:07 am UTC
Forum: Mathematics
Topic: Give these Word Problems a Try
Replies: 19
Views: 1895

Re: Give these Word Problems a Try

<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous> That's why you should take pure mathematics linear algebra, rather than applied! :) Goddammit, what sort of a wuss are you? I was doing gaussian elimination when I was in highschoo...
by DavCrav
Thu Jan 21, 2010 12:23 pm UTC
Forum: Mathematics
Topic: Give these Word Problems a Try
Replies: 19
Views: 1895

Re: Give these Word Problems a Try

jestingrabbit wrote:<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>


That's why you should take pure mathematics linear algebra, rather than applied! :)
by DavCrav
Thu Jan 21, 2010 12:37 am UTC
Forum: Mathematics
Topic: I'm bad at maths. Any ideas why?
Replies: 36
Views: 4493

Re: I'm bad at maths. Any ideas why?

IMO, many of such "flashes" are wrong, and a few of them are right. We simply don't count the wrong ones because of our bias. There is some truth to this viewpoint, I have to concede. But I remember an afternoon discussing Deligne--Lusztig theory and Broué's conjecture where every suggest...
by DavCrav
Wed Jan 20, 2010 8:23 pm UTC
Forum: Mathematics
Topic: Epsilon-Delta proof
Replies: 10
Views: 4506

Re: Epsilon-Delta proof

OK, in order to understand the epsilon-delta definition of continuity, you need to analyze what a discontinuity in a function is. Take a graph of f, and suppose that it is discontinuous at x=1, say f(x)=2x for x\leq 1 , and f(x)=2x+1 for x>1 . Drawing this on a piece of paper might b...
by DavCrav
Wed Jan 20, 2010 7:50 pm UTC
Forum: Mathematics
Topic: Very Big Numbers
Replies: 14
Views: 1708

Re: Very Big Numbers

Syrin wrote:The first *infinite* ordinal is pretty crazy, if you ask me.

editted because of pedants


Edited? :twisted:
by DavCrav
Wed Jan 20, 2010 7:49 pm UTC
Forum: Mathematics
Topic: Give these Word Problems a Try
Replies: 19
Views: 1895

Re: Give these Word Problems a Try

I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go. That's not really solving them though - it's asking a computer to solve them for you. Gaussian elimination is the same, whether Maple does it or you do it. Computers are a wonderful inv...
by DavCrav
Wed Jan 20, 2010 7:47 pm UTC
Forum: Mathematics
Topic: Coolest Topic in Math?
Replies: 6
Views: 1231

Re: Coolest Topic in Math?

I agree with t0rajir0u. Even third-year undergraduate research projects (such as in my institution) are generally on uninteresting and unimportant areas, with very few exceptions (and generally the person involved was far beyond the average third-year level). There are some really interesting areas ...
by DavCrav
Wed Jan 20, 2010 7:30 pm UTC
Forum: Mathematics
Topic: I'm bad at maths. Any ideas why?
Replies: 36
Views: 4493

Re: I'm bad at maths. Any ideas why?

You can still practice to be creative, you know, so I disagree with most of what you said. Of course, if your practice is limited to a small type of problems, it's your fault for practicing wrong, not the fault of practicing. Also, someone who is not familiar with the tools often cannot be creative...
by DavCrav
Wed Jan 20, 2010 2:02 pm UTC
Forum: Mathematics
Topic: I'm bad at maths. Any ideas why?
Replies: 36
Views: 4493

Re: I'm bad at maths. Any ideas why?

I have it on good authority (i.e., Chinese parents) that one surefire way for any reasonably intelligent student to be good at math is to practice. Unfortunately this is the viewpoint of many people who start mathematics degrees. If you are reasonably intelligent and practise, then you can get very...
by DavCrav
Tue Jan 19, 2010 10:48 am UTC
Forum: Mathematics
Topic: I'm bad at maths. Any ideas why?
Replies: 36
Views: 4493

Re: I'm bad at maths. Any ideas why?

Not liking maths because you find matrices boring is like not liking carpentry because you think hammers are boring. Hammers are fairly boring, it's what you can do with hammers that is interesting. And even then, to build interesting things, you need a lot more complicated tools than hammers. That...

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