(Dirichlet is like beta but on an n-simplex, where the whole mass is concentrated somewhere on the interior of the face, in a way that the induced mass on the faces aka the various conditionals are also Dirichlet)
The pdf is $\text{Beta}(\alpha_1, \cdots, \alpha_{k+1}) x_1^{\alpha_1} \cdots x_k^{\alpha_k} (1 - \sum_{i = 1}^k x_i)^{\alpha_{k+1}}$ if you want to think about it supported on the right angled $k$-simplex in $\Bbb R^k$
where that big beta is the multibeta function Gamma(blah + ... + blah)/Gamma(blah)...Gamma(blah)
Right so what I was wondering is if $f_t$ is a small homotopy through functions which are negative somewhere can you also choose metrics $g_t$ with $f_t = \text{scal} g_t$ such that $g_t$ is a small homotopy through metrics
The first step is that since $f(p)<0$, we can in fact assume that $f$ is close to $-1$ on a "good portion" of $M$, that is, $f$ is $\varepsilon$-close to $-1$ in $L^p$.
I think you can just fix a background metric for that to make sense
I didn't specify in my thesis
But you do this by taking $p$ to be the inf of $f$ and using a diffeomorphism to blow this up a nbhd of this point to be a good portion of $M$.
since the problem is diff invariant, this is fine
now there's a cool trick for getting a background metric with scalar curvature identically $-1$ on $M$
now the idea is that you can take this background metric and prescribe its scalar curvature in an $L^p$ nbhd
this follows from the inverse function theorem and elliptic theory
(you need to work in a Sobolev space)
so you just need to check that the inverse function theorem gives you a curve of solutions
Can U and V be negative? I know D can be negative and since V is computed with D (and U is computed with V) it would make sense that U and V can be negative.
$V_{temp} = \frac{V_{i+1}^2+DU_{i+1}^2}{2} mod C$ I know normally the square only applies to $U_{i+1}$ (order of operation) but in this situation is it somehow different?
No, it's not an absolute value, the numbers are just taken to be positive normally, of course you can write $-4$ mod $C$ but it's the same as $C - 4 \bmod C$
You can keep adding multiples of C till you get to the "other side" of zero: $-5 \pmod{7} \equiv (-5+7)\pmod{7} = 2\pmod{7}$ and $-14\pmod{3} = (-14+3(5))\pmod{3} = 1\pmod{3}$ etc.
Let $(X,d)$ be a metric space whose metric topology is discrete. Is the reason why that $(X,d)$ is not Geodesic essentially due to the fact that there are no continuous injections from $[0,1]$ to $X$?
