I have a $2$-simplex $\Delta = [v_0, v_1, v_2]$ and a homotopy of, say, maps $f_t : \{v_0\} \to X$. I would like to extend it to a homotopy $g_t : \Delta \to X$ with $g_0|[v_0] = f_0$.
tangen vector $\hat t =d\vec{r}(s)/ds = d\vec{r}(\phi)/d\phi d\phi/ds$ Now d\phi/ds = \rho and in polar coordinates d\vec{r}(\phi)/d\phi=r \vec{e}_\phi So it looks like r=\rho, since |t|=1, which is wrong
@BalarkaSen If the retract is small like $B_\epsilon(I) \to I \cup (\Delta \cap B_\epsilon(v_0))$ then I am fine, the map doesn't lose it's property. In other words the collection of maps to $X$ which has this property is "open"
So that's why I can tiny-extend to $B_\epsilon(I)$
@BalarkaSen So now I want to get to $\Delta \cup B_\epsilon(I) \cup B_\epsilon([v_0, v_1] \times I) \cup B_\epsilon([v_0, v_2] \times I)$ but then the maps defined on $B_\epsilon([v_0, v_i] \times I)$ do not match with the map defined on $B_\epsilon(I)$ where they intersect
@AkivaWeinberger it is rather a substitution of the parameter "length s" with "angle phi", and I'm doing a stupid mistake somewhere, but I don't see it
The original problem is the tangent of an ellipse. Sure, I can do the direct calculation but the above one should also lead to $\vec{e}_\phi$ as the tangent vector
i've learned that apparently akiva has a twitter where he just trashes mathematicians left and right so often that he can't keep track of who he makes fun of
one time i got into a long series of replies with someone about birding stuff we had both seen, and i was going to ask if they'd read a book, and was about to hit send when i realized i was talking to the author of the book
Eh. I don’t count Feynman diagrams as visualizations in the traditional sense. Physicists really don’t understand them as depicting space-time processes
ted, did i tell this story? the other day she held out an empty water bottle to my wife and said "do you want to fill this?" my wife said OK, sure, but was in the middle of a task. when my wife did not immediately respond, my daughter said "so do it"
the books say she's probably not quite at the point where she can understand politeness. she can understand do and don't, but "don't say that" probably gets interpreted too literally to be instructive.
she does thank us for stuff from time to time, even out of the blue. last night at dinner she thanked me for helping her with her bath on sunday. we weren't talking about baths so it felt a little bit like manipulation.
she does ask my wife for hugs as a diversionary tactic, to postpone doing something that she doesn't want to do. it's viciously effective.
You'd think $1-\frac12+\frac13-\dotsb=\ln2$, but Riemann rearrangement says I could rearrange it to literally anything, and there's no order in the probability space
so let's call the one you open $X$ and the other envelope $Y$, which has a half chance of being $10X$ and a half chance of being $X/10$
Then $E(Y-X|X=x)>0$ no matter what $x$ is
but $E(Y-X|Y=y)<0$ no matter what $y$ is
and these expectations are defined
so if I hand you these envelopes and tell you what's in the one you chose, you'll want to switch. But if I tell you what's in the other one, you'll not want to switch. No matter what those amounts are
(assuming $x$ and $y$ are possible values I guess)
I can tell my computer to simulate it, but after playing $\gg N$ games you'll notice the computer isn't hitting some values nearly as often as it should
wait
Here's another game. Say you start with $\$x$, $~x>0$. I offer to let you name any number $y$ in the world. I flip a coin. Heads, your money is replaced with $\$y$; tails, you lose all your money and leave with $\$0$.
Clearly playing this game twice is worse than playing it once
but you always have an infinite expected value
so does that mean there's a maximum $\$x$ at which it's no longer worth it to you to play?
