Oh, well, I'd prefer to do this chapter properly, so that I don't have any trouble with the second groups chapter (which is after modules). Then linear algebra. Then fields (and Abel-Ruffini mmm).
once you develop the language of group actions, everything becomes an almost-tautology.
@iwriteonbananas thought i'd share : the nice thing about mayer-vietoris sequence is that it gives you an algorithm to compute homology of arbitrary 3-manifolds
i think there's an exercise in hatcher which asks to compute homology of the 3-manifold obtained from pasting boundaries of two torii using the identity map $x \mapsto x$
it's a fact (as far as i know) that every 3-manifold can be decomposed into handlebodies (solid g-torii) glued along their boundaries by a diffeomorphism. this is called heegard decomposition. you can use this to compute homology of any 3-manifold via mayer-vietoris on the handlebodies, once you know the gluing diffeomorphism
@iwriteonbananas close enough, close enough. not quite, though, as $S^3$ is two solid torii pasted along the boundary via the diffeo which identifies the longitudal circle of one with the latitudal one of the other.
I am not sure what the best is using just the fact the group is order $n$. A sort of similiar idea is you can show that if $H$ is a subgroup of $G$ then $G \setminus H$ generates the group $G$. @SohamChowdhury
I think it should actually be pretty easy to show that, but it shows that every subgroup of $S_n$ is generated by $n/2$ elements, and that is a sharp bound
I guess the "what are the subgroups of $S_n$" is not really completely answered now that I think about it :P (since you don't know anything about the groups on more than $n$ elements)
Admittedly, not sure how the p-adics got the rights on the notation $\mathbb Z_p$, they should be forced to use inverse limit notation, or some short hand close to that.
I think that is the new name for the padic integers, pints. And why should algebraist write \mathbb{Z} / n \mathbb{Z} or even the \Bbb variant (of course one can just make a newcommand for that)
then $k_2 \cdot a \cdot m \pmod{n} = k_2 \cdot k_1 \cdot t \cdot m \pmod{n} = (k_2 \cdot t) \cdot k_1 \cdot m \pmod{n} = n \cdot k_1 \cdot m \pmod{n} = 0 \pmod n$
@TheArtist That would be one approach to a case study, but I think in general it is just an analysis of what you are looking at, so you can look at things that lead to the failure if you want.
I don't see why not. You could maybe look at a project and see why was it done on time and under budget, how is this working better than other solutions, how come this car is doing so well compared to other similar cars, etc.. It is probably less common, because I am guessing that most things that are successful were designed to work that way, and that was not unexpected success which I am guessing is when case studies would be interesting @TheArtist
I am also guessing that there is a lot of "mini case studies" for prototypes to see how they work in practice
see what was done well, see if there are unexpected problems or successes
Sure. Projectively, all nonsingular conics (curves) are equivalent over $\Bbb R$. But there are two different surfaces, again over $\Bbb R$. Over $\Bbb C$, they're all the same. [Emphasis on nonsingular.]
One of my favorite classic questions, @Balarka. You can do it using what we're talking about in a reasonably elementary way or you can do it using topology of the Grassmannian ("enumerative geometry"). How many lines meet 4 lines in $\Bbb P^3$ in general position?
@TedShifrin $\Bbb P^1 \times \Bbb P^1$? I am not sure, but look : two skew lines span two copies of $\Bbb P^1$ inside $\Bbb P^3$, making a nonzero angle ('cause they're skew).
And now we are looking at lines which intersects both of these copies.
I haven't written down anything yet, but let me do that.
@BalarkaSen You mean using a handlebody decomposition? It's hard to parse what he means there. Certainly Heegaard diagrams are useful, in the sense that there's a thing that's useful for contact topology that has them in its definition. But I've never seen someone do much contact topology with a Heegaard decomposition. Open book decompositions are a more common tool.
@MikeMiller No, I mean there's a standard $\Delta$-complex structure on $S^3$ which is obtained from identifying opposite sides of a 3-simplex. This can be proved to be $S^3$ by cutting up the tetrahedra along a small square from the middle. I was fiddling a lot with this when trying to visualize the Hopf fibration. The guy sits right in front of me in prof's office, and he said that this is a good exercise in the sense that a lot of notions of contact topology emerges (roughly) from looking
He's a student of Etnyre : that's how I came to know that Etnyre was a famous symplectic topologist, to which Ted expressed his surprise during a discussion here.
Let A be a convex polygon in the 2D plane. Let O be the center of mass of the polygon. If one draws a line that goes through O, does it split A in two polygons that have the same area ?
Je pense que la réponse est oui, mais je vois pas trop comment le prouver
@N3buchadnezzar If you're not using relative references, I would suggest using $E$3:$E$6. I would also discourage you from using numeric names for variables.
So there's a two-parameter family of lines meeting each of the two skew lines. Can you see that they cover all of $\Bbb P^3$, @Balarka? ... Now, what about the locus of all lines meeting three pairwise skew lines?
@TedShifrin what about the converse: is there a polygon such that any line that goes through the center of mass splits the polygon in two sides with equal areas? If that is not true, are there closed curves other than the circle that have this property ?
There are wonderful curves of constant breadth, @leGrandDodo, like the Wankel engine shape, but I doubt it's true for them (other than a circle). I'll think about it.
@Clarinetist :p I think I will just go back to latex. I was trying to transpose a row of values, but when I did not specify the endpoint it threw me an error. The values were just zeros.
@TedShifrin I am not sure if I see that. I'll think about it after I finish eating. Excuse me if I am being silly, but I rarely think about these stuff. The word "projective space" naturally reminds me of the cell structure instead of Gr(1, k).
@N3buchadnezzar Yeah, basic idea is that INDIRECT([STRING]) turns [STRING] into a cell reference, and then you go ahead and SUM over the cell reference
Great to know! I have very limited experience with excel and word, now we use it a fair bit on group projects and organization. Hence I must know the very basics of it :p It finaly worked, so thanks a bunch =)
No problem. @N3buchadnezzar Feel free to ask questions whenever you feel like it. I just left an actuarial job where I was using Excel and Access every day, so I'm quite familiar with them
