@Pedro since I can fill the cycle structure in 5*4*3*2 ways, then I divided by 2*2*2 because it doesn't matter which element is first in each of the cycles, or which of the two cycles are first
$x \in V(IJ) \Leftrightarrow (f_i \cdot g_j)(x)=0$, where $f_i \in I$ and $g_j \in J$
$\Leftrightarrow f_i(x) \cdot g_j(x)=0 \Leftrightarrow f_i(x)=0 \text{ OR } g_j(x)=0 \Leftrightarrow x \in V(I) \text{ OR } x \in V(J)$ $\Leftrightarrow x \in V(I)\cup V(J)$
@Pedro what confuses me, is that it seems like you have to count elements (--)() and ()(--) even if they are really the same. For example, that you have to count (1 2)(3 4 5) and (3 4 5)(1 2) as two separate element :S
@Pedro that's very elegant. I'll have to mock around with it. I'm still not 100% clear on why the strategy in my book comes with a caveat. Hold on, let me show you a question I posed here earlier
I have a bit of a problem. I'm currently reading about permutations, and I have a little exercise that asked me to find all cycle structures in $S_6$. I came up with the following
$
( -)\\
(- -)\\
(- - -)\\
(- - - -)\\
(- - - - -)\\
(- - - - - -)\\
(- -)(--)\\
(- -)(---)\\
(- -)(----)\\
(- -)(--...
Its this bit "Note that we do not divide by an extra factor of 2 since 2-cycles and 3-cycles are different" I don't really understand. As in, that sounds to me like we are counting (ab)(cde) and (cde)(ab) is distinct elements :S
That's a start, @usukidoll: note, however, that $(1+2i)(1-2i) = 5$, so $5$ can't well be a Gaussian prime, can it? Try following Pedro's hint: this is the last I'll give.
@usukidoll: I think you've been given many hints and need to do some focused thinking. Try writing some things down. But first, I don't think your definition of a Gaussian integer is quite right, if I understand you. So here's a clearer definition: a Gaussian integer is a complex number $a+bi$ such that $a$ and $b$ are integers.
$a$ and $b$ are not random, and you are not "choosing them", per se. I think you need to carefully read the task at hand. Here is one final hint: try to see when $(a+bi)(c+di)$ is an integer. That may help you determine why you're interested in numbers representable as a sum of squares in the first place.
but also the question is which rational primes less than 50 are also Gaussian primes? so we need to know which one on the list is a Gaussian Prime which is a Gaussian Integer divisible by two distinct... Gaussian Integers
ac+adi+bci+bd(i^2)
$ac+adi+bci-bd$
becomes an integer when the i's get lost or something
What was wrong with the example of $5$, along with the other suggestions given? Did you notice anything special about the factorization $(2+i)(2-i)$?
Math is difficult. The answer isn't always going to be evident after a few minutes. You have to be patient, sit down, and write down a few things. Don't take "the obvious" for granted. Working out examples can help you figure out why things work in one situation, but don't in another.
What about the norm? Why is it relevant? I'm not saying it's not, but you can't just jumble together concepts that are loosely related in your head. Try to fashion these into a coherent understanding of the problem, starting with understanding what you are trying to do in the first place.
I got two questions. What's the term for the polynomial that has roots the powers of the generator of a group, and what's the fastest way to find a group knowing its generators
skull I am going to throw PDEs if you don't knock it off. Obviously, PDEs are way more lengthy than number theory
I don't see anything special with the 5... argh
let's try this again... gonna gather what I have let's see we have integers a and b... Gaussian integer is a complex number $a+bi$ all primes less than 50 - 1 3 5 7 11 13 17 19 23 29 31 37 39 41 43 47 need to find out which rational primes on this list is a Gaussian prime. so we need to find out when $(a+bi)(c+di) $ is an integer that I need and determine if it's a Gaussian prime which is a Gaussian integer p such that if p=ab, where a and b are Gaussian integers, then a or b is a unit. So I need to apply all those numbers and see if it's a Gaussian prime or not.
hmm... if we have 2 and 5 and represented as sum of two squares then we have $2^2+5^2 = 4+25=29$.. well that's a prime in the set of integers
so would that mean that I would have $(2+5i)$ as a Gaussian prime?
Example2; $\langle \{(1,2,3,4),(1,2),(1,2,3,4,5,6,7)\}\rangle$ are the generators. What's the fastest way to see if they'll form a cyclic group and find $n$ in $\Bbb Z_n$?(numbers are random)
so I'm finding all primers less than 50 that I can represent as a sum of two squares and must satisfy p=1 mod 4 where p = prime before I can represent it as a (a+bi)(c-di) combination?
hmmmm......is 37 prime in the Gaussian integer (I'm seeing that it's not :/)? does it mean that I need p =3mod4 for it to be considered a gaussian integer
You JUST wrote $37=(6+i)(6-i)$. That's a factorization of $37$ into a pair of Gaussian integers, neither of which are units. Now reread what Mike told you.
@MikeMiller, might be a stupid question, but someone was in here earlier asking why (1,2,3)(4,5) and (4,5)(1,2,3) are considered two different elements in $S_5$, and I realized I didn't know myself. I didn't have the time to stick around to see the explanation though. Why is that the case?
I mean, it depends what you mean. If you let $g = (123)$ and $h = (45)$, then you're writing $gh$ and $hg$ which are not a priori the same. But in this case, since the cycles are disjoint, they commute, so $gh = hg$.
Hello guys. A moderator on a math forum pointed out errors in some stuff of mine, but I can't actually see what he refers to. It turns out we was wrong in saying a lemma was incorrect, that is, "you need more conditions if you want to keep the same conclusion." Now I'm looking at the other two remarks.
> "It seems that the first major mistake is the first inequality, where you absorb the factor of 1/2 without justification. In fact I can't find any number for which the assertion holds -- I searched to 10 million." Here's what I did:
Absolutely @MikeMiller. I'm actually a big fan of autodidact-ing. Personally, I just don't find foundations interesting enough to self-study in any serious depth, lol.
Yes you can, @VincenzoOliva. Look at "Latex in chat" on the right.
@beginner The difference between ZF and naive set theory is that in naive set theory, given a statement $\phi$, there's always a set $\{x : \phi(x) \text{ is true}\}$; for instance, $\phi$ could be $\phi(x) = \text{every injective map from } x \text{ to itself is a bijection}$
If $\phi(x)$ is "$x$ is a subset of $\Bbb Z$" then the set you get above is $\mathcal P(\Bbb Z)$, the power set of $\Bbb Z$ - this set is totally alright, it exists, everything's fine.
This is known as the principle of unrestricted comprehension. But Russels paradox shows that there isn't a set $\{x : \phi(x) \text{ is true}\}$, where $\phi$ is "$x$ does not contain itself".
So our assumptions are contradictory! That's bad. So we recast set theory with a new set of axioms - the ones we liked most ended up being called ZFC - that didn't contain the principle of unrestricted comprehension.
so i have all the subsets of integers, so infinite sets of increasing number of elements in each, all in one infinitely big set. why is this the principle of unrestricted comprehension?
@DavidZ No question there, but I don't think the OP was thinking about distributions.
That's why I had the adjective 'calculus' :)
@DavidZ It also just doesn't look like a generally good question... shouldn't migrations be for questions that would be well-received on the target network?
I try to encourage good questions with upvotes instead of downvotes personally. I almost always upvote questions with a reasonable amount exposition or apparent effort on behalf of the OP.
Is this somewhere I can work on my problems and get answer feedback?
Is there somewhere I can type up work as I go, like a journal, and other people will give me feedback, and I can give feedback on their journals?
Thank you very much.
Could I write up my work for a set of exercises and have i...
For whatever reason, I've noticed a huge influx of questions that not only demonstrate no effort on behalf of the OP, but are also so full of grammar and formatting related errors that I cannot even parse them.
Not very theory oriented, but I taught myself most of the integration techniques that I know from that book. I'd say it's good enough to teach yourself calculus out of assuming you have a good grasp of trig and algebra.
@MikeMiller Two most recent ones are weird. Downvoted. Other than that, I don't see a cause for concern. Physics enthusiasts can be like that, no offense to David Z here.
Just remember that you aren't going to get all the theory that you should be exposed to, so you'll want to go back and supplement it with other books later on.
Is this somewhere I can work on my problems and get answer feedback?
Is there somewhere I can type up work as I go, like a journal, and other people will give me feedback, and I can give feedback on their journals?
Thank you very much.
Could I write up my work for a set of exercises and have i...
