also, you're at a pretty elite place (@Daminark). I've been to a lot of places now, and there is always a lot of worry at the elite places compared to other places.
it's kind of like when my social life declines I sometimes feel a little more relaxed because I'm not surrounded by constant worrying lol.
but then the flip side is your social life declines rip.
Possibly. I mean I haven't actually been taking tests but very soon after I finished classes at one point I went to have crowns done and right before that I was given some blood pressure test and the result was a bit high
Okay so, I don't know exactly when he plans on doing it, but last class we already defined the degree of a map, for example, so my preliminary guess is that we're probably gonna be at Morse theory by 4th or 5th week
It seems right now the progression of the class is: Basics of manifolds and technical stuff -> degree and transversality (we're there right now) -> Morse theory -> De Rham cohomology -> Lusternik-Schnirelmann and/or characteristic classes if we have time
I've heard there's a more AT-ish way to approach characteristic classes? Based on this + he is doing cohomology, we may do that stuff, though I don't exactly know what Neves has in mind
One recipe has Portuguese chorizo in it (plus pork shoulder plus clams), the other doesn't have chorizo. I'm inclined to use chorizo, though. What do you think?
But think of it that way. Parametrize the unit circle, and then make the unit normal make a $180º$ rotation as you go around, then use your $t$ variable.
Well, there's no question you can compute something with that parameterization. The question is what it means. You mentioned the stuff on the remaining line segments ...
Keep the surface fixed. Which way is the water flowing? left-to-right or right-to-left ... to decide a sign on the flux, you need to assign a direction for the outward-pointing normal.
You only can make sense of a sign on flux if you have an orientation. You can sense the presence of non-zero flux, I suppose, but you can't decide if it's positive or negative.
So you can make sense of flux across a non-orientable surface by choosing an orientation (arbitrarily?) when you remove a set of measure 0. But does Stokes's Theorem make sense when you do so? For example.
what on earth is happening, I'm looking through the Lie algebra notes and I see representations of $\mathfrak{sl}(2)$ and it looks just like the algebraic representations of $SL_2$
Hi everyone, I was trying to verify the claims at the end of the page 125 of Hatcher's AT, namely that "(delta_1)^n - (delta_2)^n in the first group corresponds to the cycle (delta_1)^n" part, but I had (delta_1)^n - const. Now, I am trying to show that const is in the boundary and showed it for odd n (namely it's the image of the "upper" const map) but couldn't show it for even n. Can someone help with that sentence from the book and also give me a hint to show that const is always a boundary?
Hey, i'm having trouble in understanding how to approach calculating weak derivative of a function involving multiple absolute values, e.g. f(x)=2|3x-|3x-2||. Where could I find some solved examples?
Just a follow up to the following two questions:
Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.
Theorem 3.18, Rudin's functional analysis
The second question has almost all the proof, I'm trying to understand the second part.
...
@AkivaWeinberger, I am stuck with this Artin Algebra exercise: Write the matrix $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$ as a product of elementary matrices, using as few as you can and prove that your expression is as short as possible.
@AkivaWeinberger Thank you very much! $\begin{pmatrix}1&0\\ 3&1\end{pmatrix}\begin{pmatrix}1&0\\ 0&-2\end{pmatrix}\begin{pmatrix}1&2\\ 0&1\end{pmatrix}$. Is there any $2\times2$ invertible matrix that can't be expressed as product of no less than four elementary matrices?
I am asking this because Artin asks to find least number $n$ such that any invertible 2*2 matrix can be expressed as product of no more than n elementary matrices. And i know n=4 works.
It's neat how we can interpret the statement$$\begin{pmatrix}1&0\\0&-2\end{pmatrix} \begin{pmatrix}1&2\\0&1\end{pmatrix} = \begin{pmatrix}1&2\\0&-2\end{pmatrix}$$in two different ways
If we multiply the second row of (1 2 \\ 0 1) by -2 we get (1 2 \\ 0 -2)
and if we add twice the first column of (1 0 \\ 0 -2) to the second we get (1 2 \\ 0 -2)
(Like, either see the first matrix as a recipe for manipulating the rows of the second, or see the second matrix as a recipe for manipulating the columns of the first)
@Silent I think the same algorithm works. First kill the bottom-left entry
(unless the top-left entry is 0)
(but in that case we can swap the rows)
and then kill the top-right entry and/or turn the bottom-right entry to 1 (depending on your perspective)
Let $V$ be a vector space with $[-,-]$ being an alternating bilinear form. For $x \in V$ let $ad_x(y) = [x,y]$. Then the Jacobi identity says $[ad_x,ad_y] = ad_{[x,y]}$
let A be a (non-commutative) algebra over K, if A is semisimple then every module over A is semisimple.
every semisimple module over a p-group and field of char p has trivial action from the group
if N is a normal subgroup of G and V decomposes as V1 + ... + Vn as a KN-module then its factors as a KG-module are of the form g1Vi + g2Vi + ... + gkVi where G/N = {g1N, ..., gkN}
I’ve only done the first problem sheet so don’t expect much from me at this stage
@KonformistLiberal Maybe what you are looking for is the fact that $H_k(X, Y)$ fits into a long exact sequence in terms of $H_*(X)$ and $H_*(Y)$, but just as well with $\hat H_*(X)$ and $\hat H_*(Y)$, those being reduced homology.
Just a follow up to the following two questions:
Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.
Theorem 3.18, Rudin's functional analysis
The second question has almost all the proof, I'm trying to understand the second part.
...
