## Search found 722 matches

Wed Jun 27, 2012 10:26 pm UTC
Forum: Mathematics
Topic: Identifying subgroups of (Z/nZ)*
Replies: 4
Views: 3054

### Re: Identifying subgroups of (Z/nZ)*

There might be some clever math (chinese remainder theorem + extended euclidean algorithm, etc) you can do to solve this problem, but here's a straightforward brute-force way to do it that I think should perform fine if you only need n in the hundreds of thousands. Use a disjoint union-find data st...
Wed Jun 27, 2012 12:08 am UTC
Forum: Mathematics
Topic: Entropy of a Rubik's cube
Replies: 13
Views: 6380

### Re: Entropy of a Rubik's cube

The moves of a Rubik's cube form a group (in fact the picture on the wiki page for group is a Rubik's cube). For all groups G, if G has n elements in it (which for the Rubik's group is very very large), then for any element g of G, g n = e. If that's too much reading, in summary groups are sets wit...
Tue Jun 26, 2012 11:29 pm UTC
Forum: Mathematics
Topic: Entropy of a Rubik's cube
Replies: 13
Views: 6380

### Re: Entropy of a Rubik's cube

Here is something with a similar flavour: If you start with the solved state and you apply any sequence of moves, which I will call g (for example you could have g = F, turning the front face once, or g = M2E2S2, etc.), repeatedly applying g will eventually lead you back to the solved state. If g = ...
Tue Jun 26, 2012 11:06 pm UTC
Forum: Mathematics
Topic: Identifying subgroups of (Z/nZ)*
Replies: 4
Views: 3054

### Identifying subgroups of (Z/nZ)*

For my work right now I have a process which produces a subgroup of (Z/nZ)*. It is produced as a list of integers between 0 and n-1. I want it as a set of generators, but I can't think of an efficient way to find them. I post this partially because an efficient solution would be useful to me, but al...
Fri May 25, 2012 10:20 pm UTC
Forum: Mathematics
Topic: Quickly finding a subfield of a given cyclotomic field
Replies: 0
Views: 1399

### Quickly finding a subfield of a given cyclotomic field

I need a way to quickly determine subfields of small degree (3, 5, or 7) of a given cyclotomic field of large degree, something like Q(e^{2\pi i/d}) for d \approx 100 000 . Eight now even for d \approx 1000 I'm having trouble, using Sage's built in subfield finding function. Typically d is e...
Wed Apr 25, 2012 12:34 am UTC
Forum: Coding
Topic: Finding one of many solutions to a big system of equations
Replies: 4
Views: 1546

### Re: Finding one of many solutions to a big system of equatio

This is related to coursework so I'd rather not talk too much about it now. After everything is submitted I'll get back to you.
Tue Apr 24, 2012 7:40 am UTC
Forum: Coding
Topic: Finding one of many solutions to a big system of equations
Replies: 4
Views: 1546

### Re: Finding one of many solutions to a big system of equatio

Thanks eta! This is great.
Tue Apr 24, 2012 12:20 am UTC
Forum: Coding
Topic: Finding one of many solutions to a big system of equations
Replies: 4
Views: 1546

### Finding one of many solutions to a big system of equations

I have a system of 21 equations in 35 variables and I'd like Maple to find a solution. Just one will do. However I rarely use Maple so I'm not sure how to get it to do this. fsolve complains when it has more variables than equations, and I can't seem to find any alternatives in the documentation. Mo...
Sun Apr 01, 2012 5:58 pm UTC
Forum: Mathematics
Topic: Anyone got a handy proof that a^((p-1)(q-1)) = 1 mod pq ?
Replies: 4
Views: 2472

### Re: Anyone got a handy proof that a^((p-1)(q-1)) = 1 mod pq

I one common way to prove this is with the chinese remainder theorem . You can also prove it with Euler's theorem , which is a generalization of FLT. Beware, these are both very mainstream theorems, so this question might be a way to build up to them, and using them to answer your question might be ...
Thu Mar 22, 2012 1:27 am UTC
Forum: Mathematics
Topic: Gamma Function vs. Factorial Question
Replies: 4
Views: 2469

### Re: Gamma Function vs. Factorial Question

Here is a MO post discussing this question.
Mon Mar 19, 2012 12:55 am UTC
Forum: Mathematics
Topic: Is there a name for this?
Replies: 12
Views: 3288

### Re: Is there a name for this?

That said, I’m pretty happy in my ability to calculate cube roots of up to 9-digit numbers without a calculator. As in, have someone start with a 3-digit number, cube it, and show me the result. I can get the cube root back in a few seconds (well, I’m out of practice so maybe more like half a minut...
Sat Mar 17, 2012 11:26 pm UTC
Forum: Mathematics
Topic: Is there a name for this?
Replies: 12
Views: 3288

### Re: Is there a name for this?

One way to phrase your observation is "The quadratic residues modulo 10 are 0, 1, 5, and 6". Quadratic residues are an important topic in number theory. The most important piece of machinery in place to figure out what is a quadratic residue and what is not is the law of quadratic reciproc...
Mon Feb 27, 2012 2:33 am UTC
Forum: Science
Topic: Uniform charge density in all of space
Replies: 5
Views: 2279

### Uniform charge density in all of space

What E-field is produced by a charge distribution which is uniform in all of R 3 ? I don't know, but here are some observations: Because the distribution is isotropic I think it's tempting to say that the E-field will be 0 (the only isotropic vector field). However this E-field doesn'...
Fri Jan 20, 2012 12:59 am UTC
Forum: Mathematics
Topic: Nonstandard proofs for simple theorems
Replies: 34
Views: 7324

### Re: Nonstandard proofs for simple theorems

This is a really neat trick. Thanks very much.
Tue Jan 17, 2012 3:29 am UTC
Forum: Mathematics
Topic: Nonstandard proofs for simple theorems
Replies: 34
Views: 7324

### Nonstandard proofs for simple theorems

In an upcoming assignment I have to prove that the sum of consecutive squares is n(n+1)(2n+1)/6. The problem is definitely below the difficulty level of the course so I'd like to have some fun and present an uncommon proof of it. The two common ones I'm familiar with are by induction and summing (n+...
Thu Jan 05, 2012 12:36 pm UTC
Forum: Mathematics
Topic: What's the point of rationalizing?
Replies: 61
Views: 9314

### Re: What's the point of rationalizing?

I simplify a lot of my answers because oftentimes putting answers into certain forms will make it easier to use them later, and sometimes certain forms will be familiar to me and help me figure things out more quickly about my answer. Consequently I think a lot of conventions regarding simplificatio...
Wed Nov 23, 2011 5:51 pm UTC
Forum: Mathematics
Topic: Requesting Inequalities and Tips to Solving Them
Replies: 9
Views: 2507

### Re: Requesting Inequalities and Tips to Solving Them

Art of Problem Solving tends to have some useful resources for competition math. As a place to start, try their wiki page about inequalities.
Fri Nov 18, 2011 9:52 pm UTC
Forum: Mathematics
Topic: The notation "a/bc"
Replies: 24
Views: 3294

### Re: The notation "a/bc"

If a/bc would be a/(bc), what about a/b*c+d? a/bc(d+e)+f/gh^2? With proper fractions or handwriting, it is no problem to write fractions without ambiguity. In ascii, I prefer to look at "/" for multiplications like "-" is for additions. a-b+c is (a-b)+c and not a-(b+c), so a/b*c...
Thu Nov 17, 2011 1:58 am UTC
Forum: Mathematics
Topic: Quickie -- bounding factorials
Replies: 4
Views: 3971

### Re: Quickie -- bounding factorials

Spoiler:
Take the log of n! to get [imath]\sum_{i=1}^n log(i)[/imath]. This is an upper Riemann sum for [imath]\int_0^n log(x) dx = nlog(n) - n[/imath], which is precisely the log of (n/e)^n
Fri Oct 07, 2011 11:05 pm UTC
Forum: Mathematics
Topic: Three questions
Replies: 12
Views: 2834

### Re: Three questions

Regarding the etymology of normal: yes I meant in the context of normal subgroups.
Thu Oct 06, 2011 1:38 am UTC
Forum: Mathematics
Topic: Three questions
Replies: 12
Views: 2834

### Re: Three questions

Thanks guys. Any insight on the first two questions? The second is just a terminology question, but I can't seem to find any sources discussing it.
Tue Sep 27, 2011 8:27 pm UTC
Forum: Mathematics
Topic: Three questions
Replies: 12
Views: 2834

### Re: Three questions

It's not homework. I just think they're interesting questions (though they may not be so interesting if they can be mistaken for homework). Sorry for the confusion.
Mon Sep 26, 2011 11:41 pm UTC
Forum: Mathematics
Topic: Three questions
Replies: 12
Views: 2834

### Three questions

1. Why is the "smallest" solution (x,y) to x^2 - 61y^2 = 1 (Pell's equation for d = 61 ) so large compared to the smallest solutions for nearby numbers? (Smallest in this case means the first given by the continued fraction expansion of \sqrt{61} .) 2. Why are the symmetric and nor...
Sun Sep 25, 2011 11:03 pm UTC
Forum: Mathematics
Topic: I need another set of eyes on this integral...
Replies: 3
Views: 1148

### Re: I need another set of eyes on this integral...

At a glance your situation doesn't look to bad. For one, the integrals in the first question are definitely given by I(0), I'(0), and I''(0) because they're exactly what you get if you differentiate under the integral sign. The expression you got at the end of the...
Sun Sep 25, 2011 10:41 pm UTC
Forum: Mathematics
Topic: easy way to integrate (x^2)*(cos x)^3
Replies: 7
Views: 5633

### Re: easy way to integrate (x^2)*(cos x)^3

You don't need integration by parts if you antidifferentiate under the integral sign. I'd rather do it this way instead of messing with trig identities and IBP, but it's a matter of taste. I'll do one of the four terms of the integral here, not because the other three are any harder, but because it ...
Sat Sep 24, 2011 6:10 pm UTC
Forum: Mathematics
Topic: easy way to integrate (x^2)*(cos x)^3
Replies: 7
Views: 5633

### Re: easy way to integrate (x^2)*(cos x)^3

Try writing [imath]\cos x[/imath] as [imath]\frac{e^{ix} + e^{-ix}}{2}[/imath].
Mon Sep 19, 2011 10:41 pm UTC
Forum: Gaming
Topic: Deus Ex: Human Revolution
Replies: 357
Views: 45792

### Re: Deus Ex: Human Revolution

I think I may livestream (or just record) and add some commentary to my next run. The official rules will probably be this: Iron man rules: the game may only be reloaded from a save at the start of each major mission or hub segment. The goal is to minimize the total number of reloads. The game may ...
Sat Aug 27, 2011 5:44 pm UTC
Forum: Logic Puzzles
Topic: Logic Poll
Replies: 12
Views: 4051

### Re: Logic Poll

I don't think "How many times was your answer chosen? (i.e. 42 people chose the number 42)" should be included because it depends heavily on how many times the quiz has been taken. All the other ideas are very cool.
Tue Aug 23, 2011 1:28 pm UTC
Forum: Logic Puzzles
Topic: Do I have the best solution?
Replies: 6
Views: 2535

### Re: Do I have the best solution?

I agree that if you decide to call everyone yourself inviting everyone will take xy and not y^x (the units are wrong. Consider the case where y = 1 \mbox{ minute} and x = 10 (and don't lose track of units!). Now try y = 60 \mbox{ seconds} and x = 10 ). However you can do better by taking advantage o...
Mon Aug 22, 2011 3:58 pm UTC
Forum: Logic Puzzles
Topic: Logic Poll
Replies: 12
Views: 4051

### Re: Logic Poll

Looks neat. Question: one of the questions asks "What % of people...". Do you want an answer in the form of "x%" or just "x" (e.g. if I think the correct answer is 101%, should I write "101" or "101%" in the box)?
Sat Aug 20, 2011 1:37 pm UTC
Forum: Mathematics
Topic: The nth root of a number modulo a prime
Replies: 12
Views: 9860

### Re: The nth root of a number modulo a prime

Another idea: Taking discrete logarithms is hard (the Diffie-Hellman key exchange is based on it), but supposing you can, you can solve your problem too. Suppose you have a primitive root r (i.e. an element r of Z/pZ for which r^x = g \,\,(mod \, p) has a solution for all g ). Write x = r^m,...
Fri Aug 19, 2011 10:08 pm UTC
Forum: Mathematics
Topic: The nth root of a number modulo a prime
Replies: 12
Views: 9860

### Re: The nth root of a number modulo a prime

In practice polynomials over Z/pZ can be factored fairly efficiently using probabilistic methods. Unfortunately I have no reference for this, but here is a sketch (with holes) of how it works: The idea is that you pick a j in Z/pZ and define two polynomials f_j = (x - j)^{\frac{p-1}{2}} + 1,...
Fri Aug 19, 2011 7:33 am UTC
Forum: Science
Topic: At what level does quantum uncertanty vanish?
Replies: 34
Views: 5376

### Re: At what level does quantum uncertanty vanish?

See here for a problem about how long one can balance a pencil on its tip while taking into account quantum effects. A solution can be found here.
Sat Aug 13, 2011 10:30 pm UTC
Forum: Coding
Topic: O(you make baby Jesus cry)
Replies: 19
Views: 4787

### Re: O(you make baby Jesus cry)

As an aside, I fixed my code that generates "the derived group" of G (I read http://en.wikipedia.org/wiki/Group_(mathematics) and http://en.wikipedia.org/wiki/Commutator finally)! I basically get stuck on: why does a_ia_ja_i^{-1}a_j^{-1} ALWAYS reduce to e , unless the group is communitiv...
Fri Aug 05, 2011 7:18 am UTC
Forum: Coding
Topic: O(you make baby Jesus cry)
Replies: 19
Views: 4787

### Re: O(you make baby Jesus cry)

Yakk: I see no errors in the part of my OP that you quote. g is a group element of the free group G . Despite the misunderstanding it looks like you understand the problem. I agree that the solution you give works. It would be very nice to show that the shortest solutions are always dependent on []....
Fri Jul 29, 2011 4:41 pm UTC
Forum: Mathematics
Topic: Functions that some math students think are equal but aren't
Replies: 32
Views: 6460

### Re: Functions that some math students think are equal but ar

I think the most common examples are assuming that a function is linear when it's not. The Freshman's Dream is a direct example of this, and your a/b + c/d = (a+c)/(b+d) is as well if you think of a fraction as a function of two arguments (the numerator and the denominator): F(a,b) + F(c,d) = F(a+b,...
Sun Jul 24, 2011 8:05 am UTC
Forum: Mathematics
Topic: A question on infinite series
Replies: 7
Views: 2109

### Re: A question on infinite series

No problem. Kudos on coming up with the complete solution on the first* try. Here's how I would look at it (spoilered because you might decide you want to keep thinking about it, i.e. figure out why your solution works if you haven't already): The sums the video is talking about are geometric series...
Sun Jul 24, 2011 6:32 am UTC
Forum: Mathematics
Topic: A question on infinite series
Replies: 7
Views: 2109

### Re: A question on infinite series

Your formula says that if you start with the fraction \frac{1}{a+1} , a sum that solves your problem (the only one, actually, since in the video the sums are all a certain kind) is what you posted in the OP. What I mean to say is that your equation still holds even if a is not an integer. I'm not su...
Sun Jul 24, 2011 5:24 am UTC
Forum: Mathematics
Topic: A question on infinite series
Replies: 7
Views: 2109

### Re: A question on infinite series

Why does a have to be a natural number for your equation to hold? (Rhetorical, it doesn't, but those kinds of questions are good to ask). You can always write \frac{p}{q} as \frac{1}{q/p} , and that might be enough to find what you're looking for, though I'm not sure since I don't know how you're tr...
Sun Jul 24, 2011 4:00 am UTC
Forum: Coding
Topic: O(you make baby Jesus cry)
Replies: 19
Views: 4787

### Re: O(you make baby Jesus cry)

My (optimistic) end goal is to be able to describe all elements with the property for arbitrary n. The program was written to give me some example solutions so that I might get inspired. For the program to be useful I think it should produce a lot of solutions so that finding patterns is easier, but...