That's not a rule, that's a tautology.

Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?

- Wed Feb 08, 2017 9:14 am UTC
- Forum: Mathematics
- Topic: "There is an Exception to every rule"
- Replies:
**32** - Views:
**2961**

That's not a rule, that's a tautology.

Fine, for every prime p, and every integer a prime to p, a raised to power p minus a is divisible by p. Not a tautology, definitely no exceptions. Happy now?

- Mon Dec 26, 2016 12:29 pm UTC
- Forum: Mathematics
- Topic: Probability distribution on the power set of natural numbers?
- Replies:
**12** - Views:
**1005**

You should be able to apply Kolmogorov's zero-one law:

https://en.wikipedia.org/wiki/Kolmogorov's_zero%E2%80%93one_law

The Hewitt--Savage zero-one law is even better for this problem I would think.

https://en.wikipedia.org/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law

https://en.wikipedia.org/wiki/Kolmogorov's_zero%E2%80%93one_law

The Hewitt--Savage zero-one law is even better for this problem I would think.

https://en.wikipedia.org/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law

- Mon Dec 26, 2016 12:21 pm UTC
- Forum: Mathematics
- Topic: Can't keep up with all the math behind machine learning
- Replies:
**3** - Views:
**742**

I want to know how something works on a deeper level when I use it, otherwise I feel dissatisifed and "stupid". This page on Sparse Coding makes absolutely no sense to me in terms of the math: http://ufldl.stanford.edu/wiki/index.php/Sparse_Coding . It's not that it doesn't make sense, it...

- Sat Feb 07, 2015 2:33 pm UTC
- Forum: Individual XKCD Comic Threads
- Topic: 1483: "Quotative Like"
- Replies:
**83** - Views:
**8988**

Most people are, frankly, quite abysmal at quoting accurately from memory. Hence it is usually more correct to say “She was like” than “She said”, because most of the time she didn’t actually say precisely what is claimed, but only something broadly similar. So it's basically a shorter version of &...

- Tue Nov 11, 2014 10:44 pm UTC
- Forum: Mathematics
- Topic: Congressional apportionment
- Replies:
**14** - Views:
**3498**

That's interesting (though having spent quite a bit of time thinking on election methods, not overly surprising. Impossibility theorems seem to be the name of the game). I'd definitely like to look closer at the claim that every divisor method violates quota, and I don't particularly trust rangevot...

- Thu Nov 06, 2014 11:36 pm UTC
- Forum: Mathematics
- Topic: how to calculate these intersections without having to write
- Replies:
**3** - Views:
**1694**

The first and third of these sets have equal size, and there's a nice bijection between them, if you can spot it.

- Thu Nov 06, 2014 11:05 pm UTC
- Forum: Mathematics
- Topic: Multiply 2 terminating numbers and get a repeating decimal?
- Replies:
**6** - Views:
**2172**

If you have two numbers a and b that have a terminating n-ary expansion, then so does their product. The quick way to see this is to multiply a and b by a suitably large power of the base so that they become integers, then product them to get an integer, obviously. Dividing by the (square of the) su...

- Wed Oct 01, 2014 8:40 pm UTC
- Forum: Mathematics
- Topic: The Ironic Paradox of Normalcy [Is my argument sound?]
- Replies:
**42** - Views:
**7371**

OK, speaking as a professional mathematician, if a potential student put that under my nose I'd yell "CRANK!" and run. The reason? A wall of text posing as something mathematical and talking about non-mathematical concepts, naming Cantor, this definitely spikes my crank-ometer. Don't do it...

- Fri Aug 08, 2014 11:40 am UTC
- Forum: Mathematics
- Topic: is mathematics a religion?
- Replies:
**82** - Views:
**11976**

And actually, for a person with zero formal training, that's pretty good. I think we could all profit from a mutual open exchange of ideas here. I think he could profit from learning a bit about the marvels that the pros have discovered in the past 2500 years. And I thing we can profit from a fresh...

- Sat May 31, 2014 12:46 pm UTC
- Forum: Mathematics
- Topic: hypothesis on primes
- Replies:
**9** - Views:
**3732**

First,a (late) thank to >-) for his answer Then,I 'd like to put the following thoughts under discussion by anyone who might care (all numbers are natural): 1.Let p b a prime. Between two succesive multiples of p 2 (that is between kp 2 and (k+1)p 2 ), you can always find exactly p-1 numbers d such...

- Thu Apr 24, 2014 11:02 pm UTC
- Forum: Mathematics
- Topic: how many ways-married couples
- Replies:
**7** - Views:
**2314**

Hello!!! Could you help me at the following exercise? With how many ways can we choose a man and a woman that are not married to each other from n married couples? I thought that it is (n-1) n ,but I am not sure.Can you tell me if it is right? So firstly, how many ways are there of choosing a man a...

- Fri Mar 14, 2014 5:05 am UTC
- Forum: Mathematics
- Topic: algebraic degree of sin(1 degree)
- Replies:
**4** - Views:
**2114**

Since totient(180) = 48 (totient being the number of numbers coprime to 180 less than one 180 - or the order of the multiplicative group mod 180), this makes sense; the minimum polynomial is presumably the 180th cyclotomic polynomial (though I don't know how one would prove that) If it's true that ...

- Thu Feb 27, 2014 11:03 am UTC
- Forum: Mathematics
- Topic: Axiomatic mathematics has no foundation
- Replies:
**158** - Views:
**27505**

Forest Goose wrote:Presburger Arithmetic is consistent and complete.

So is the theory of algebraically closed fields. Just thought I'd throw in some actual mathematics to try to stem the tide of pseudo-philosophical manure in this thread.

- Thu Feb 27, 2014 10:58 am UTC
- Forum: Mathematics
- Topic: A probability(?) problem
- Replies:
**13** - Views:
**2934**

I don't see how that follows at all. It seems like you're implicitly relying on some other distribution information to make that statement about the size of X? That was pretty much my thought too. Unless you are implicitly assuming that X takes a form similar to {1, 2, ..., n} then I don't see how ...

- Sun Feb 23, 2014 3:03 pm UTC
- Forum: Mathematics
- Topic: A probability(?) problem
- Replies:
**13** - Views:
**2934**

Here's another reason why you might say 2: Change the question slightly to: let X be a finite set of natural numbers. Choose an element randomly from X with uniform probability (can be done since X is finite). This element is 2. What does this tell you about X? Well, one obvious thing that it tells ...

- Fri Feb 14, 2014 9:50 pm UTC
- Forum: Mathematics
- Topic: quarkcosh1's math coincidences thread
- Replies:
**32** - Views:
**6486**

I'd suggest that before delving into details like this, try to determine whether quarkcosh knows anything whatsoever about the Monster group apart from the frequency with which certain numbers show up in that Wikipedia list. Just because someone is stringing together meaningless concepts without an...

- Fri Feb 14, 2014 10:29 am UTC
- Forum: Mathematics
- Topic: quarkcosh1's math coincidences thread
- Replies:
**32** - Views:
**6486**

The = means approximately equal. I was looking at the subgroups of the monster group and 11 appears 6 times which seems kind of high since numbers slightly lower than it don't appear as many times. The best explanation I can think of as to why that is is because pi^3 - e^3 = 11 but I can't explain ...

- Fri Jan 24, 2014 9:40 am UTC
- Forum: Mathematics
- Topic: y = sum of subset of divisors of x
- Replies:
**2** - Views:
**1760**

In essence, you just want to know which subsets of X can add together to make a number n. I don't know the most efficient algorithm to do this, but I needed this to solve a problem about possible decompositions of modules for algebraic groups, and thought up the following easy recursive algorithm. f...

- Mon Nov 04, 2013 10:53 pm UTC
- Forum: Mathematics
- Topic: Permutations and Combinations
- Replies:
**7** - Views:
**1714**

Hmm. So independent and mutually exclusive look like the same thing in this case. In which case, it isn't robustened, or whatever somebody earlier said. If they are different things, what is the difference in the context of combinations?

- Mon Nov 04, 2013 10:42 am UTC
- Forum: Mathematics
- Topic: Permutations and Combinations
- Replies:
**7** - Views:
**1714**

I don't think that's clear. You do need independence of some sort, not just not mutually exclusive. Here is an example, in the spirit of the OP: we'll take clothing and sex. There are (generically) two options for sex, but depending on the choice of sex, only certain clothing options might be availa...

- Sun Sep 08, 2013 9:59 am UTC
- Forum: Mathematics
- Topic: Group theory II
- Replies:
**46** - Views:
**12164**

I put an extra zero on the end of MaxRelations but it seemed to take a long time and then my connection died and my SSH session closed. The trouble with things like this is, there's no reason to think that it will take less storage than there are atoms in the Universe...

- Fri Sep 06, 2013 9:57 pm UTC
- Forum: Mathematics
- Topic: Group theory II
- Replies:
**46** - Views:
**12164**

I ran the code, and R fails to be confluent.

- Fri Sep 06, 2013 9:25 pm UTC
- Forum: Mathematics
- Topic: basic doubt about groups
- Replies:
**19** - Views:
**4152**

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 ...

- Sun Aug 25, 2013 12:36 pm UTC
- Forum: Mathematics
- Topic: When will $1 be the smallest denomination?
- Replies:
**51** - Views:
**7577**

gmalivuk wrote:It is a fake boxing stance.

Aha. I still prefer the mid fist pump explanation...

- Sun Aug 25, 2013 12:45 am UTC
- Forum: Mathematics
- Topic: When will $1 be the smallest denomination?
- Replies:
**51** - Views:
**7577**

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...

- 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:
**3074**

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...

- 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:
**3074**

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...

- 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:
**3074**

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....

- 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:
**3074**

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...

- Thu Aug 22, 2013 6:41 pm UTC
- Forum: Mathematics
- Topic: Group theory II
- Replies:
**46** - Views:
**12164**

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...

- Thu Aug 22, 2013 3:58 pm UTC
- Forum: Mathematics
- Topic: Group theory II
- Replies:
**46** - Views:
**12164**

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 ,...

- Wed Aug 21, 2013 11:43 am UTC
- Forum: Mathematics
- Topic: Group theory II
- Replies:
**46** - Views:
**12164**

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...

- Thu Mar 14, 2013 9:25 am UTC
- Forum: Mathematics
- Topic: Groups & Their Presentations
- Replies:
**10** - Views:
**1755**

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 ...

- Wed Mar 13, 2013 6:46 pm UTC
- Forum: Mathematics
- Topic: Groups & Their Presentations
- Replies:
**10** - Views:
**1755**

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...

- Sat Feb 02, 2013 2:58 pm UTC
- Forum: Mathematics
- Topic: Next great mathematical invention
- Replies:
**17** - Views:
**3226**

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...

- Sat Feb 02, 2013 2:31 pm UTC
- Forum: Mathematics
- Topic: Next great mathematical invention
- Replies:
**17** - Views:
**3226**

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...

- Mon Oct 29, 2012 10:31 am UTC
- Forum: Mathematics
- Topic: group theory
- Replies:
**64** - Views:
**22401**

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...

- Thu Oct 25, 2012 9:50 am UTC
- Forum: Mathematics
- Topic: group theory
- Replies:
**64** - Views:
**22401**

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...

- Mon Oct 22, 2012 10:21 am UTC
- Forum: Mathematics
- Topic: group theory
- Replies:
**64** - Views:
**22401**

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 ...

- Sun Oct 21, 2012 10:12 am UTC
- Forum: Mathematics
- Topic: The projective Weil representation of SL_2(Z/nZ)
- Replies:
**1** - Views:
**469**

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...