@0celo7 sure, would really love to know wtf a spectral sequence really is, especially using the language of obstructions
"Mayer Vietoris says that to work out the obstructions in AuB, that's like taking disjoint copies of A and B and looking at their defects, and then counting the ones you counted twice from their intersection that you ignored: it is the inclusion exclusion principle. The snake lemma tells you that if you were looking at n-dim holes, then you next need to take into account the n-1 dim holes - or the boundaries of the n-dim holes." https://www.physicsforums.com/threads/exact-sequences.164213/
"There are three different ways in which mathematicians have reached the subject now called algebraic topology: Classification, properties and invariants. One looks for ways of distinguishing topological spaces. Obstructions. What are the difficulties in getting certain types of maps between spaces. Categorical. What is the structure of the category of topological spaces or some subcategories of it? *As one realises during the study of algebraic topology all these are different ways of stating the same question--and the answer is to apply the methods of algebraic topology.*"
@0celo7 its very difficult understand based on the way its written and has some sentences that say nothing meaningful (maybe most poorly exposited would be better). Anyway I won't be baited by personal insults because I am criticizing someones internet posts.
so . . . I was messing around with some customized vectors in this project i'm working with . . . and made a function I'm calling Vector.excite and I refuse to believe there isn't already a name for this function
the function takes the summation of values in the vector and than multiplies the normalized vector by this sum
use is application. I find it useful- because it implies every vector is the 'excited' form of another vector, and there fore each vector belongs to a set of recursive uses of the excite function.
@punkerplunk (v.norm*v.sum).mag=abs(v.sum) (which is totally not how math people would notate it btw), is completely trivial
applying something to itself does not make it useful
application means solving a problem that could have been encountered independently, or deriving an interesting result that can be understood independently
I'm not exactly sure it'll ever be useful, but it does yield an series when applied recursively to a vector, and maybe one day someone will find it useful? idk, I suppose i just ask my question for posterity's sake?
@bolbteppa I agree with PVAL that this particular paragraph (not the whole thread; I think the defect explanation is standard but computationally and conceptually useless) is close to being incoherent. I think anyone familiar with these kind of topology would agree.
I don't think it's "confusing"; the OP's trying to say that inclusion-exclusion of Euler char is a result of Mayer-Vietoris but that provides no insight on what M-V is.
And he's trying to fit it in the context in a weird way using "defects" - those aren't a thing. Topological spaces in itself don't have "defects".
I should say that I am also surprised that you're accusing a topologist about not understanding topology, even though you're not one.
But doesn't vector orientation matter here? The example dots:
2 . 2
3 1
But isn't this invalid? A (2x1) . (2x1) operation isn't allowed?
Well this is bad, with my reasoning even the length of a vector is invalid because any vector $u$ cannot be multiplied by itself since it doesn't fit matrix multiplication requirements.
Okay. I suppose that there is a difference between the row/column vector of a matrix and a vector in the sense that they are using them.
Because the only way that I can reconcile the dot product of a vector with itself is to write it as: $v^{2} = v \cdot v^{T}$ if v is a horizontal vector. And $v^{T} \cdot v$ if it's a vertical one.
It's with rules regarding dot product, and how it seems to differ when concerning things like dotting a vector with itself. Something not allowed when I consider the dimensions.
Are Euclidean vectors always supposed to be "vertical" ?
Never mind
Well I can reconcile this with frobenius norm then. It's defined as the square root of the sum of all elements squared in a matrix. For a vector, it works the same way. The square root of the sum of all elements squared.
I have a small normal distribution question. If I wanted to find the probability at a point, like x = 50, would the integral look like this? $$\lim\limits_{\substack{a\rightarrow 50 \\ b\rightarrow 50}} \int^{b}_{a} p(x)dx$$
@BalarkaSen I'm not surprised, was expecting this from you, I've given you professors giving explanations of this and articles by professors explaining there are 3 ways to look at algebraic topology, and all that is thrown out the window by you guys for no reason with emotional claims, that should say it all
Well your doubts are the same doubts calling that thread incoherent despite the fact they illustrate one of three ways of viewing the subject, we can go with the students or the experts on this one
I didn't call the thread incoherent: just that particular paragraph.
But I do believe that although thinking of homology as measuring defect is standard, it's not helpful. There are more geometrically transparent (and computationally useful) ways of thinking of it.
@bolbteppa Suppose $X$ is a manifold, $A$ a subspace inside. The snake map $H_n(X, A) \to H_{n-1}(A)$ is essentially sending an embedded manifold with boundary (call it $M$) inside $X$ with boundary $\partial M \subset A$ to it's boundary $\partial M$ in $A$.
It's essentially taking a manifold w/ boundary and sending it to it's boundary, if I had to say it simply.
Homology classes are best visualized as embedded manifolds.
That sounds pretty cool, I don't see it yet but it sounds cool
@BalarkaSen you call it mumble mumble yet professors in that thread, and people in a thread titled "Mathematically mature way to think about Mayer–Vietoris" mathoverflow.net/q/23175/38721 are fine saying inclusion exclusion, I'm just saying...
@BalarkaSen Unfortunately, I think the answer will be fairly involved. I can sort-of see a way forward from what I've read in Bott & Tu. But it will involve spectral sequences.
@0celo7 Here's my suggestion. $f : X \to X$ be a result of homotoping the identity map slightly. $I(\Gamma_f, \Delta) = I(\Delta, \Delta)$ should hold, agree? Then $I(\Gamma_f, \Delta)$ is the # of fixed points of $f$: by Lefschetz that's $\sum_i (-1)^i \text{tr} H^i(f; \Bbb Q)$.
@HirotoTakahashi A holomorphic square root of $f$ (which needs to be a holomorphic function) is a holomorphic function $g$ that satisfies $g(z)^2 = f(z)$ on the domain.
I am looking at the following generalization of Phragmen-Lindelof theorem:
$S$ be a sector in $\Bbb C$ with a vertex at $0$, forming an angle of $\pi/\beta$. $F$ be a continuous function on $\text{cl} S$ which is holomorphic on the interior, and $|F(z)| \leq 1$ on the boundary. Suppose moreover $|F(z)| \leq C e^{c |z|^\alpha}$ for some constant $c, C > 0$, where $0 < \alpha < \beta$
@DanielF OK, so this is not too hard to prove: essentially an application of the maximum modulus theorem. However, I am thinking of the following conclusion.
$f$ be a holomorphic function on all of $\Bbb C$ with $|f| \leq 1$ on the positive real axis and $|f(z)| \leq Ce^{c|z|^\alpha}$ where $\alpha < 1/2$. Then $f$ satisfies the hypothesis of the theorem with $\beta = 1/2$, so $|f| \leq 1$ uniformly on all of $\Bbb C$. Hence it's constant.
Is there a more direct way to see this?
It strikes strange to me because being of growth rate at most like $e^{|z|^{1/2}}$ sounds too weak for $f$ to be constant.
If $1>a>0$, show that $$\int_0^1 x^{-(1-a)}\log(x)\left(\frac{1}{1-x}-x^{-a^2}\frac{1-x^a+a(1-2a(1-x^a))}{(1-x^a)^2}\right) \textrm{d}x=\frac{1}{a}-\frac{\pi^2}{\sin^2(a\pi)} $$
@DanielFischer Right, but $\cos(z^{1/3})$ e.g. has growth rate with exponent strictly smaller than $1/2$. It's absolute value is also bounded by $1$ on the positive real axis. So what did I do wrong?
Is the conclusion I have drawn for $\beta = 1/2$ wrong?
Sounds like a nice conclusion to me: under the relatively weak assumption on the growth rate (being $|f(z)| \leq C\exp(|z|^{c|z|^\alpha})$ for some $\alpha < 1/2$), any nonconstant entire function has to grow unboundedly along any ray from the origin.
Is this the symmetric version of the Dirichlet eta function that satisfies the functional equation symmetric_eta(z)=symmetric_eta(1-z)? I think it is incorrect.
$p_i : E_i \to X$ be bundles over $X$, $i = 1, 2$. $f : E_1 \to E_2$ is a bundle morphism means it's a continuous map such that the triangular diagram $p_1, p_2, f$ commutes and $f$ sends fibers to fibers, yes? Then a bundle isomorphism is a morphism $f$ such that an inverse bundle-morphism $g$ exists.
Then $f : E_1 \to E_2$ is a continuous map with $g : E_2 \to E_1$ a continuous inverse. That means the total spaces are homeomorphic.
@0celo7 But that's not an issue if you're working with smooth vector bundles?
'Cause then bundle morphisms are automatically smooth, by definition.
@SoumyoB (1) Can you build a torus by gluing edges of a square? (2) Can you build a punctured torus (torus minus a small disk on it) by gluing edges of a pentagon? (3) Can you get a 2-torus out of gluing two punctured torii along the boundary of the puncture?
If you can do all of those, you should also be able to see how to build a 2-torus out of an octagon.
