is there a piece of terminology for the following? suppose i've got a partition of $m$ containing a term $n$. then each partition of $n$ generates a partition of $M$
something along the lines of those partitions being subpartitions of the initial one
I was referring to the statement that $S(X)$ is weak htpy equivalent to $X$. I am not sure if I am being dumb, but I can't see immediately why that is true.
Well, if two points in $S(X)$ can be joined by a path $\gamma$, then such a path runs through the simplices $\Delta_i^n$ of $S(X)$. I can push that path over to $X$ by doing $\sigma_i^n \circ \gamma$ (i.e., push the path simplex-wise), where $\sigma_i^n$ are the singular simplices corresponding to $\Delta_i^n$. This is continuous by the gluing lemma.
Conversely, if there is a path in $X$ joining two points, that's a singular 1-simplex, so corresponds to a 1-simplex in $S(X)$ jointing the preimage points.
@L33ter: To be more specific. The contents of Ch1-3 of hatcher are an absolute baseline for any topologist. The first line of whatever paper I write will be assumptions on the homology of a 3-manifold, and those only suffice because I understand the universal coefficient theorem and Poincare duality. When calculating certain imvariants of certain 3-manifolds I want to know a bunch of what's in chapter 4. I want to use characteristic classes. etc.
There are certainly things I use less than others, but they still see use. Last year I saw someone talking about Massey products as a way of fistinguishing certain spaces associated to 3-manifolds.
A popular trend nowadays is to have the imvariants not be groups, but rather stable homotopy types, and the way you extract info from these is stable homotopy theory. I've seen people talking about how the right way to write down their stuff is using infinity categories. You use whatever you have available.
On the other hand everything I do is predicated on ideas of morse theory and morse homology and I say the word "transverse" every day.
A loop $\gamma$ in $X$ can be realized as map of two singular 1-simplices, which takes the same value at the endpoints. Then that gives me two edges in $S(X)$ which share the same ends too. Going around one edge and then the other edge gives me a loop in $S(X)$ which pushforward to $\gamma$ by the $\pi_1$-level map, I think.
I wonder if it's possible to not feel more excited about something you don't know and which you're actively learning than the things you aren't actively thinking about.
@AndrewThompson T. Shifrin, "Multivariable Mathematics".
Many schools in many countries offer students in your situation (i.e., being in the 10th grade and conversant in elementary algebraic topology) to skip grades in certain subjects.
Not in India, Albas. Didn't you know that the Bourbaki group are actually Indian high school students, the texts being compiled from their exam-answers?
I don't know if you've seen it online, but there are a growing number of people studying algebraic topology and category theory instead of more geometric or analytic disciplines. I'm confident that the strange choice of schooling is why.
No. The Americans decided to attribute Bourbaki to the French, as they feared Indian dominance in mathematics after the growing popularity of the subject after Ramanujan, @Albas.
@MikeMiller: did you read the blog post about the guy who did a PhD in math and five years later doesn't understand his thesis anymore and what this means for maths?
I'm giving a student lecture in homological algebra. In 90 minutes I go from "This is a module." to "This is how you describe Ext in terms of syzygies."
@DanielFischer Also I want to express all the roots of $g=x^4+x+1 \in \mathbb{F}_2[x]$ as also three generators of the multiplicative group $\mathbb{F}_{16}^{\ast}$ as for the basis $\{ 1, b, b^2, b^3 \}$.
In order to express all the roots do we use the fact that if b is a root of an irreducible polynomial then a^q is also a root?
@MikeMiller Injection on $\pi_1$: Take two loops $\sigma_1, \sigma_2 : [0, 1] \to S(X)$. If $p \circ \sigma_1$ and $p \circ \sigma_2$ are homotopic ($p : S(X) \to X$) then I can cut open the square of the homotopy into two simplices. This in turn gives me two simplices between the images of $\sigma_i$, which match up to a homotopy, I think.
@BalarkaSen: I'm going to leave now. I want to hear the words "Simplicial approximation". When you're mapping from S(X) to X, you have NO IDEA what this map does except on the simplices.
Lots of work done with regards to that question, lots of work to be done
@HariPrasad
I guess the answer is, "At least some. Probably a bit more, but we're not sure. Also, there's a bit of structure that it looks like they have but we can't actually prove that they have yet but they probably do have it."
And the primes are connected with the complex numbers (like $\pi+\sqrt 2i$) for some reason.
Eisenstein primes, eh? That guy had some beautiful ideas. I remember trying to follow his proof of quadratic reciprocity, even though I had no business in doing that.
@Evinda What is only true because we started from $\mathbb{F}_2[X]$? It's always true that an element of $\mathbb{F}_q^{\ast}$ has an order dividing $q-1$, and if $x$ has order $q-1$ and $y = x^m$ with $\gcd(m,q-1) = 1$ then $y$ also has order $q-1$.
I got my monthly burst of inspiration last night and developed an algorithm for constructing something which I know exists but that I'm having trouble characterizing
What novel trick am I missing in the space between recurrence relations and finite differences?
@tabchas yes that's what it seems to be and by the design of the slides its clear that its not well drafted (it seems like took from different sources and put together)
@micklh i don't know much about interpolation myself. but you might check out Wilf's short discussion on the corresponding 3-term boundary value problem in Generatingfunctionology
page 10 is the start of the section, and page 11 mentions cubic splines specifically
@DanielFischer Also I want to express all the roots of $g=x^4+x+1 \in \mathbb{F}_2[x]$ as also three generators of the multiplicative group $\mathbb{F}_{16}^{\ast}$ as for the basis $\{ 1, b, b^2, b^3 \}$.
In order to express all the roots do we use the fact that if b is a root of an irreducible polynomial then a^q is also a root?
@Evinda Sorry, I don't understand what you want to do. Find all roots of $g$, probably. By assumption, $b$ is one, it's easy to check that $b+1$ is also one. You can then find the remaining two. But then what?
Hi! I have a seemingly simple question about Gaussian random variables, but would like to know if there is a very simple argument (hopefully a one-liner or few-words-reference) for it
Namely, if u,v are two fixed, orthogonal vectors in $\mathbb{R]^n$
and X is a spherical Gaussian r.v. in $\mathbb{R}^n$; then how to show that <u,X> and <v,X> are independent?
@BalarkaSen I want to re-assure you, I am still working on that software! Nearly all the progress has been in graph algorithms for representing and manipulating parametric manipulations to and representations of ...algorithms. Though visible progress has also been made: You can now communicate a small subset of formal concepts to it using LaTeX directly!
