Categories are equivalent if there are functors $F, G$ such that $F \circ G$ and $G \circ F$ are naturally equivalent to the relevant identity functors?
do you say naturally equivalent or naturally isomorphic lmao
"Wir ignorieren die mengentheoretischen Probleme die da auftauchen, aber falls Sie Interesse haben können Sie sich ein dickes Buch holen und die Probleme da durchlesen"
> Tarski's ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II.
> Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy [in Warsaw University] which Tarski hoped to fill. Oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities.
@Ted: did your answer to the triple tube intersection involve no integration whatsoever? I had to compute $\int_{1/\sqrt3}^1\left(1-z^2\right)\,\mathrm{d}z$.
I did a different integral for the integration approach (natural region of symmetry is easy in cylindrical coordinates), but it's morally equivalent to yours. Yes, a completely integral-free approach, as Archimedes figured the volume of a sphere. If you're interested, I'll give you the link to my MAA slides.
@loch: Well, much to the chagrin of the room, I haven't disappeared yet. :) Hanging in there. Antique enough — and lucky enough — that I've actually been vaccinated.
@Astyx: Well, it's only because the former president who thinks he won in a landslide is gone that things are moving. But still way behind on people of color, teachers, and slogging through the over-65 folks.
Oh, our society (including the military and medical personnel) is full of people who refuse to be vaccinated. I suspect most of them are doing this in solidarity with Trompolini. We require all sorts of vaccinations to allow kids to go to school or young adults to go to university. I think things have to be mandatory.
But, yeah, people without computers or cell phones (and sometimes cars) are disadvantaged for sure.
@Astyx I'm tryingn to make more visualizations like yours, how did you get the text display which also included numerical expressions involving the variables?
It's named "transpose" (you can see it in the tab of the webpage) and I'm trying to use your thing as a base to make other illustrations so I'd like to be able to change that
uhh I wanted to make a point but that backfired I think lol. I switched up parity I think. One case is trivial, the other leads to 0. Yeah I meant even dimensions
boundary of odd-dimensional simplex is 0 so I don't immediately get that the even-dimensional constant simplex is null in homology
The point of all this is proving something completely different anyways. I want to show that for spaces $X,Y$ with chosen points $x_0,y_0$ such that $(X,x_0)$ and $(Y,y_0)$ form good pairs, $p_{X}^{*} \oplus p_{Y}^{*}: H^{k}(X ; R) \oplus H^{k}(Y ; R) \rightarrow H^{k}(X \vee Y ; R)$ is an isomorphism, where $p_X$, $p_Y$ are the collapse maps from $X \vee Y$ to $X$ resp. $Y$.
The problem is basically that an isomorphism in the other direction is easy to get via the LES of a pair, and in principle I just gotta concatenate this with the sum of the collapses and see that it yields the identit…
MV is also a possibility to get from $X \vee Y$ to the sum, maybe easier to compute
Our lecturer gave the following "proof" for connected CW complexes
I should probably also assume connectedness. Also, $k \geq 1$
comments you expect to make when grading a physics quiz problem: "You assumed the acceleration was zero when it's not." "You didn't account for the weight." "You included a spurious force in your diagram."
comment you don't expect to make: "There are 60 minutes in an hour, not 60 seconds."
What are ways to prove that the euler-characteristic of the boundary of a 2n-dimensional manifold is zero, other than this one: Use that the LES of the pair $(M, \partial M)$ with $\Bbb Z_2$ coefficients is a sequence of vector spaces so alternating sum of dimensions is zero, then use that $\dim_{\Bbb Z_2}(H_k(M;\Bbb Z_2)) = \dim_{\Bbb Z_2}(H_{2n-k}(M, \partial M;\Bbb Z_2))$
I mean, it's sensible since the LES relates boundary and manifold itself, and the last result helps us get rid of the relative homologies
still feels kinda random
ah I guess this needs compactness to ensure finiteness of dimensions
I Admit I havent opened math in a while, I know how to take partial derivatives but regarding this video youtu.be/4b4MUYve_U8?t=1749 why the partial derivative of theta-j Xj will be zero except to the theta-J, Can you point me what do I need to read
Take $k \in \Bbb N$ intersecting American footballs and configure them inside a unit sphere such that each football touches two opposite ends of the sphere. Each of the shapes are spaced evenly apart.
In the case $k\to \infty$ the volume should be equal to the volume of a sphere.
Related Problem ...
