like, the intro says you can skip chapter 0, which is supposed to just give you a sense of the flavour of the subject, but sometimes he'll refer you back there for a definition
which is weird because he'll redefine stuff other times
@Ted: Good thing I got that book last year for christmas
Yup, I've recently learned the value of examples. Looking in hindsight at courses where I knew a lot of examples compared to those where I didn't, while I often got the same grade the difference in understanding of the material is huge.
I try to get the following habit: every time I learn a new definition, I should come up with one example where the definition holds and another in which it does not hold.
Excellent practice, @AndrewT. When I took point-set topology, Munkres was big on checking out all our various examples spaces for each property as it came along. Sometimes tedious, but really a good pedagogical thing.
Even in grad courses I used to try to assign lots of examples/computations for homework, as opposed to just abstract things.
Yeah, actually. The book was in xeroxed form and appeared about a year and a half later. I also had Guillemin for Differential Topology, about a year and a half before that book appeared, and Mike Artin as he was starting to write his :) Now you know why I'm such a mess. :)
I'm going to TA elementary algebra next semester. I mostly give 'challenging' theoretical exercises, as they already have pre-assigned a bunch of computations.
@DanielFischer do you know how I can get my stackoverflow account unbanned ? I'm surprised it even happened in the first place, I have a positive upvote-downvote ratio.
@dREaM You have two deleted and downvoted questions. Together with the two non-deleted downvoted questions, and the zero-score questions, that puts you below the threshold. But according to a very educated guess, if you polish your not well-received questions - as mentioned in the help centre page - and get an upvote or two that would probably get you unbanned. [Maybe you need more than two upvotes, nobody knows without reading the source.]
I just released version 0.1.0 of MathEdit http://kasperpeulen.github.io/mathedit/
Some features are: * Doesn't conflict with markdown syntax anymore. So you can safely type markdown and mathjax together (like in stackexchange) * It doesn't flicker (ever!) the live preview if you type LaTeX. * Saves your work to the local browser cache
@user159870 By that you mean you have smooth $c_1, c_2 \colon I \to \mathbb{R}^3$ and wonder whether $t \mapsto c_1(t) \times c_2(t)$ and $t \mapsto c_1(t)\cdot c_2(t)$ are also smooth? They are.
I would like to prove $r(X) = r(P_{X}) = \text{tr}(P_X)$, $r$ denoting the rank, $X \in M_{n \times p}(\mathbb{R})$, and $$P_{X} = X(X^{T}X)^{-}X^{T}$$
where $(X^{T}X)^{-}$ is a generalized inverse of $X^{T}X$.
I don't believe I know the machinery to prove this. But here's what I do know that mi...
Hello, let R = k[X_1,...,X_n] be the ring of polynomials of n variables over the field k and let f: R -> R be an isomorphism that is identity on k. How to see that f preserves total degree of polynomials?
I've been trying to figure out the following problem for weeks, and while I've made progress, I'm ultimately stumped.
Consider a probability distribution in $x$ and $y$ which is in fact uniformly distributed on a circle, centered at the origin, or radius $r$.
What's the marginal distribution of just $x$?
I can do this by writing the distribution as a uniform distribution in the angle $\theta$.
One then uses $x = r \cos(\theta)$ to get $dx = -r \sin(\theta)d\theta$ to find $P(x) \propto 1 / \sqrt{1 - (x/r)^2}$.
That's all well and good, but is there some way to write the 2D distribution in such a way that we can find the distribution of $x$ by integrating out the $y$ variable?
@Clarinet: I thought we had already discussed your question on rank of the projection matrix. Certainly when the columns of $X$ are linearly independent, you know what to do.
The question intrigued me. I was puzzled that in my all-too-long life I'd never run into it, if it were really true. And then I thought for about 5 seconds.
We have that $N$ is perpendicular to the tangent vector $t=\gamma'$ and to the unit normal $n$. To find the relation of $N$ with the binormal $b$ do we do the following?
$n,b,\gamma'$ consist the orthonormal basis of $\mathbb{R}^2$, so
$$N=c_1 n+c_2 b +c_3 \gamma' \Rightarrow N\cdot N=c_1 n\cdot N +c_2 b\cdot N +c_3 \gamma'\cdot N \Rightarrow 1=c_2 b\cdot N \Rightarrow N \parallel c_2b \Rightarrow N \parallel b$$
Knowing that $N$ is perpendicular to the tangent vector $t=\gamma'$ and to the unit normal $n$ and that $b$ is also perpendicular to the tangent vector $t=\gamma'$ and to the unit normal $n$, we conclude that $N\|b$.
@Clarinet: Officially you want SVD, but think about replacing $X$ with basis vectors in the first $r$ columns and then zeroes in the remaining columns. Then you should know pretty easily what the pseudoinverse of $X^\top X$ will be.
It doesn't look like I'm going to get feet and feet of snow and blizzards as in my previous visits to Michigan.
yeah, I'm also thinking about supporting handwritten math/pictures/diagrams there are tools to export handwritten stuff (with bamboo/ surface pro) to svg on the fly
@MikeMiller: Maybe I actually have an idea. If $n\alpha$ is the given homology class in $H_2(\Bbb{CP}^2)$, consider the Poincare dual $\gamma$ in $H^2(\Bbb{CP}^2)$. One can pick a map $f : \Bbb{CP}^2 \to \Bbb{CP}^\infty$ representing $\gamma$ upto homotopy. Cellular approximate to get a map $g : \Bbb{CP}^2 \to \Bbb{CP}^2$ in the homotopy class. We know (by Yoneda) that $\gamma \in H^2(\Bbb{CP}^2)$ is $g^*(\hat{\alpha})$ where $\hat{\alpha}$ is the Poincare dual of $\alpha \in H_2(\Bbb{CP}^2)$.
So if I can just choose $g$ to be smooth, I can homotope it to be transverse to $\Bbb{CP}^1 \subset \Bbb{CP}^2$ (which represents $\alpha$), and $g^{-1}(\Bbb{CP}^1)$ is then going to be a manifold, and the fundamental class would be Poincare dual to $\gamma$ which is Poincare dual to $n\alpha$, so $g^{-1}(\Bbb{CP}^1)$ represents $n\alpha$.
But I am not sure if I can choose $g$ to be smooth. Also, I do not know how to compute genus of $g^{-1}(CP^1)$.
