Hmm. I am actually not familiar with any kind of interpretation of $q$-eta in group theory. In fact, there doesn't seem to be any relevant material out there, @alexander
you look at $\operatorname{Irr}(G)$ a lot in representation theory of finite groups. my coauthor Mark Lewis is really into looking at prime graphs of character degrees.
the point of the paper is there is a type of groups called $M$-groups, which is defined character theoretically. the paper obtains a group theoretic definition. the objective, basically, is to make it so you can talk about $M$-groups without characters, e.g. you could put them in a D&F-level book.
the only piece missing is that he can't find a way to prove this certain relation is an equivalence relation without characters. that's the hole i've been trying to fill in, trying to prove that equivalence relation group theoretically. :)
I don't know how really @RobJohn converted this equation $\frac {1}{2ix}log (\frac {1-e^{-ix}}{1-e^{ix}})$ to $\frac {1}{2ix}log (\frac -e^{-ix})$ and then to $\frac {\pi - x}{2x} for x\in (0,2\pi)$
@Alexander I have a bit of an issue with a certain question. I managed to solve it for even n, but I can't find the correct pattern for an odd n. May I bother you with it or not a good time?
Given the complete graph $K_n$, {$v_1, v_2,.., v_n$}, we define the following direction for each edge: for j>i, $v_i v_j$ is from $v_i$ to $v_j$ if |i-j| is odd, and the opposite if |i-j| is even. We define the capacity function c($v_i v_j$)=|i-j| and set $v_1$ as the source, $v_n$ as the sink. I need to find maximal flow and minimal cut. Now, my path..
@AlexanderGruber I think it is not amenable to such a search. the groups need to be insoluble (to avoid hall subgroups) -- they cannot be p-soluble for any p dividing their order. they need to have very few supersolvable subgroups.
(so the groups need to be big, so they have tons of subgroups, so they don't work)
@AlexanderGruber I was always tending to be extremely proof-based in math, these days I feel like I'm just killing myself, it seems you've got to just learn the application of some theories and move on do research on the things which are not fully being investigated yet.
@Alexander What I did for odd n, I defined a flow where every even vertex X goes to n with full capacity, 1 goes to every even vertex X with full capacity, and an edge between even x and even (n-x+1). I showed that this flow is feasible (the conservation constraints work), and it obviously has the same value as the capacity of cutting all the nodes except n (t) itself.
I am trying to find a similliar pattern for an even n. I saw it's always possible to have a flow where there's full capacity from the odd X's to n there too, but the paths that have to be added so that such flow is feasible get extremely complicated when n>6 (3, 4, 5 paths with no pattern to follow). Any ideas on how to work on the even n will really help me @Alexander
i'm not sure if i can help you actually, @Studentmath. I'm not too good with flow problems. I'll think about it and let you know if I come up with something.
@AlexanderGruber but the small index maximal subgroups are also a danger, since they may be more soluble. basically i'm looking for H, K so that there are no subgroups X ≤ H and Y ≤ K with X x Y of order n.
@JackSchmidt @AlexanderGruber Using the definition that $\left[\begin{matrix}n\\k\end{matrix}\right]_q$ is the number of $k$ dimensional subspaces of $\Bbb F_q^n$, I was trying to show that $$\left[\begin{matrix}n\\k\end{matrix}\right]_q= q^k\left[\begin{matrix}n-1\\k\end{matrix}\right]_q +\left[\begin{matrix}n-1\\k-1\end{matrix}\right]_q$$
@user2179021 with 7 upvotes and no enlightening comments, it should be a hard question. I looked at it, and I don't know. It would be useful to get some background about it
@user2179021 if Op was tasked to do this in a combinatorial course or a linear algebra course, whether it requires only freshman knowledge, this kind of background
I'm not actually in this class, and I still haven't really formally been introduced to differential forms. I've been crawlin' through the web looking for examples...
cause sure, you can calculate the area of a surface by double integrating, if you paramaterize the surface in two dimensions and double integrate over those dimensions
If I can show a cut is minimum in any way, it shows there exists a maximum flow of the same value immediately without the need to show of any flow of the same value, right?
@Anthony i think the best thing would be to learn how to do it brute force, freshman calculus style, and then figure out how the whole theoretical, differential form framework ties into it
@user2179021 thinking back at it, it seems useful to consider vectors in the vector space ${(\mathbb Z/3\mathbb Z)}^n$ and consider also the orthogonal complement of $v$ (these are finite dimenion vector spaces)
Although I don't usually write problems explicitly from earlier stuff on later exams, certainly knowledge of that prior material is needed to do the later stuff.
Given that the two-forms we were given are $dx_i \wedge dx_j$ ... How do I denote the projection of a two form onto another two-form? @TedShifrin I'm really confused...
no, @Anthony. You have $dx\wedge dy$, $dy\wedge dz$, and $dx\wedge dz$ (or $dz\wedge dx$). Those compute, in turn, the signed area of the projection of the parallelogram spanned by $v$ and $w$ in the $xy$-, $yz$- and $xz$- (or $zx$-) planes.
here is how i would adapt it. we have 13 weeks in our course, which corresponds more or less to 13 topics. each topic gets a grade, and the final grade is determined by whatever your highest score in each topic was over the course of the semester.
In some ways that's really good, @685-252, as most math students find that they learned the material on exam X sometime during the material covered on exam Y. So to go back and show that you've actually learned what you didn't quite get earlier isn't a bad idea.
Be careful, @Alex. That might reward short-term memorization.
I prefer to tell students that I want to know what they know at the end ... and knowledge of topic X on the final that was flubbed on exam X mitigates to some extent a poor performance. But generally you'll find that math grades go down on finals because students have "learned" by short-term memorization and bomb the final.
Doesn't this belong in the new education site which I have only been to once? :D
I took it last year, and I was kind of introduced to differential forms, my housemate is in it this year and I was trying to get reacquainted. @TedShifrin
@TedShifrin my other idea was to use something like what my calc IV teacher did during our summer course... homework is worth something small like 10%, everything else is three exams, which are all cumulative. If you get a higher score on a later exam, it wipes out scores from previous exams. so if you get an A on the final, for example, you get an A in the class. if you get a higher grade on the second midterm than the first, the grade on exam 1 is forgotten, and the average of the 2nd and
that way rewards long term memorization and always allows for a "comeback," but i don't like it as much because it doesn't emphasize mastery on each individual topic
Be careful, @Alex. The trouble with that vague wipe-out policy is that a student may fail to know how to do related rates or max/min or whatever repeatedly, but the final exam has a smaller amount of it (which he still botches) and that score replaces his bum score. I only want to replace when you show your knowledge has been substantiated.
They don't really do much with differential forms, though, @Anthony, do they? You need a serious treatment of it for weeks and weeks to really learn it. (See my lectures. :) )
@TedShifrin what do you think about combining the ideas in this way? like in the first, each topic has a grade, but instead of taking the highest, i do a wipe-out policy.
