@Ted: Of course, it depends who you are and what you're doing. Some of these people who are truly exceptional at finding a differential on the third page of a spectral sequence don't always think geometrically.
At the level of Hatcher I firmly agree.
Someone patient should write a careful answer for this guy that doesn't just say "read Hatcher", which his question runs the risk of.
One can still give a description of what's going on and why one would develop the machine of the fundamental group, in the same sense that someone asking a naive question about irrotational and conservative vector fields being the same deserves some honest discussion about topology without telling them to read a book.
I am working on the following problem:
Let $L$ be a closed, rectifiable simple curve, traversed
counterclockwise. Let $f(z)$ be a differentiable function on a domain
$G$ where $L \cup E(L)$ ($E(L)$ is the exterior of $L$, $I(L)$ is
the interior of $L$) is contained in $G$. (That is, $f...
Let us define the free strict monoidal category $\Sigma(\textbf{Pos})$ on the category of posets $\textbf{Pos}$. How may we prove that $\Sigma(\textbf{Pos})$ is a noetherian category if we know that $\textbf{Pos}$ is noetherian?
@MikeMiller If someone is asking a question like this,I don't think they likely need any motivation for why one would develop the machinery of the fundamental group.
What are the possibilities for the number components of a link of round unknots of the same diameter with all 2 component sublinks equivalent and a pair of 3-component sublinks non-equivalent?
I'm in calculus and I keep getting hung up on remembering to apply formulas, especially when the math lets me go down one path until it becomes a dead end. Students generally take 1/2 - 1 hour per problem whereas it takes me up to 3 hours. An example of what I'm talking about would be integrating $x^3/6 + x/2$. I didn't have the wisdom to magically auto-recognize that I needed to combine the terms there; I thought that would make a more difficult problem (of probably partial fraction decomposition), but I was wrong. I later got stuck trying to factor a nasty polynomial. Is there any specifi…
The point of mathematical notation is clarity. While you could make an argument that your expression isn't self-contradictory, I could make a better argument that your expression is dumb.
More precisely what I'm after is whether or not the conditions of a piecewise function are to be evaluated strictly in their listed order, or if that's not well defined.
i flagged this question earlier today but i guess it's okay to be asking the pizza hut problems at this hour since the pizza hut page has thousands of answers already? math.stackexchange.com/questions/1697186/…
although it's not clear if anyone got the third one right since it asks for all of the possible numbers of rings
obviously 3 rings works but i haven't figured out any other possibilities yet
oh wait there is an update on the pizza hut page! OPTION C: YET TO BE SOLVED, No one has gotten this one exactly right yet! Hint: It helps to show your work!
i assume it requires a list of some numbers. maybe they are all less than 100, if so i have a 2^-100 chance of free pizza by guessing
well no fair talking about it here either i guess
but i answered my own question when i checked the blog, B) is solved so the MSE question should stay open
i actually woke up this morning and started working on A, 10 minutes later i was still debugging my program which had 32 bit integers in it. then i reloaded and realized there were hundreds of correct answers for it already.
I just made a quick Python program to compute $5^1 \mod {503}$, $5^2 \mod {503}$, etc., and then it found that $5^{502} \pmod {503} = 1$. So that's why I say I don't know why it is, I just know that it is.
@GGG well we know that $5^{502} \equiv 1 \pmod {503}$, so then $5^n \equiv 5^{n \mod {502}} \pmod {503}$. Then if we want to find the value $5^{2009^{1492}} \mod {503}$ all we need to to is find $2009^{1492} \mod {502}$
So what is that value?
It turns out that $2009^2 \equiv 1 \pmod {502}$.
So then $2009^{1492} \equiv 2009^{1492 \mod 2} \equiv 2009^0 \equiv 1 \pmod {502}$, so then $2009^{1492} \equiv 1 \pmod {502}$.
I get it now after thinking about it the reason isn't as I stated above
but because of the fact that if $\alpha$ is a cycle and we send it somewhere else, which is a cycle and define the map as the equivalence class, so that map would make sense.
@L33ter There ought to be a simpler reason. The snake lemma is completely unrelated to topology and is basically just a statement about abelian categories
Trying to prove that $M=\Bbb CP^2\#\Bbb CP^2$ is not the boundary of an orientable 5-manifold, @MikeMiller. So I computed the signature of $M$ but unfortunately I got that it's zero (is that correct?). Tried playing around with Euler-characteristic then but didn't get me anywhere, and now I'm out of ideas.
What kind of meaning does "$x_1, x_2$ are strict relative minima of $f$" have? Google is not helpful, and (because of the word strict) this appears to be stronger than simply $x_1, x_2$ being local minima of $f$.
@JC574 that part is definitely true, and it explains the word strict. Relative is still a bit vague though, from the text it looks like I am looking at simply $2$ strict local minma
$M$ manifold, $\alpha$ 2-form on $M$ and $X, Y, Z$ vector fields: Is the following always true: $$\mathcal{L}_X(\alpha(Y,Z)) = (\mathcal{L}_X \alpha)(Y,Z) + \alpha( \mathcal{L}_X Y, Z) + \alpha(Y, \mathcal{L}_X Z)$$
@RandomVariable I was looking at the second integral Cody put here .. :O I opened the thing inside the integral into a series but I am having a hard time justifying interchange of sum and integral .. maybe you can help?
@Krijn: In practice you don't need to memorize whatever hundred rules you're deriving right now. The most important one is Cartan's Magic formula which is easy to remember. The rest, meh, if you need them again find them.
i'll say, whenever i see those kinds of 'symbol manipulation' issues, it makes me me sympathetic to things like Penrose notation
especially since, when one does physics calculations in that vein, things tend to be represented in index notation and therefore chock-full of dummy indices
@RandomVariable Also can you help me with evaluating $\displaystyle \int_0^{\pi/2} \frac{\sin^2 ax}{\sin^2 x}\,dx$? :-) ($a$ is a positive real number)
Mathematics is best done by the method of reasoning. When that fails one has to resort to the method of repeating and iterating. And when that fails one has to resort to the method of prolonged staring. And when all else fails one has to resort to the method of shouting.
@MikeMiller Could you help me with another question? I think I'm oversimplifying things and need a sanity check
For which $k$ is this true: Let $x_1, \ldots, x_k$ and $y_1, \ldots, y_k$ be any pair of ordered sets consisting of $k$ distinct points of $M$. There exists a symplectomorphism $\phi$ of $(M, \omega)$ such that $\phi(x_i) = y_i$ for all $i$
I claim that this is not true when $k>1$ because, take $y_1 = x_2$ and $y_2 = x_1$ then we would have $\omega(x_1, x_2) = \omega( \phi(x_1), \phi(x_2) ) = \omega(x_2, x_1)$ but $\omega$ is anti-symmetric
Link, $105=7*5*3$: we must choose one of the $3$ numbers between 3 and 5 for the hundreds place. The $5$ then looks like choosing one of the 5 numbers between 3 and 7 for the tens place. But I'm not sure where the $7$ comes from...
Mary, yes by Lagrange: a $p$-Sylow group is a group of maximum order $p^k$, and since $N_G(P)\leq G$, we know that the prime factorization of $N_G(P)$ has no more than $k$ factors of $p$.
@RandomVariable correct me if I am wrong, for this we need to consider $f(z) = \frac{\exp \left(\frac{i}{z\pm ia}\right)}{z}$ and the contribution from the indentations at the essential singularities $z = \pm ia$ are $2\pi i e^{\pm 1/a}$ right?
I am working on the following problem:
Let $L$ be a closed, rectifiable simple curve, traversed
counterclockwise. Let $f(z)$ be a differentiable function on a domain
$G$ where $L \cup E(L)$ ($E(L)$ is the exterior of $L$, $I(L)$ is
the interior of $L$) is contained in $G$. (That is, $f...
Let us define the free strict monoidal category $\Sigma(\textbf{Pos})$ on the category of posets $\textbf{Pos}$. How may we prove that $\Sigma(\textbf{Pos})$ is a noetherian category if we know that $\textbf{Pos}$ is noetherian?
