All Questions
5
questions
5
votes
4
answers
439
views
Confusion on defining uniform distribution on hypersphere and its sampling problem
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ...
- 2,497
2
votes
1
answer
101
views
Extending a probability measure on a manifold to the ambient space
Let $\pi$ be a probability measure defined on $(\mathcal{M}, \mathcal{B}(\mathcal{M}))$, where $\mathcal{M}$ is a smooth manifold and $\mathcal{B}(\mathcal{M})$ is the Borel sigma-algebra on it. Let $\...
- 5,247
1
vote
0
answers
35
views
$L^1$ almost every $t \in [0,1]$
I'm reading about curves and came across the statement that a curve $v: [0,1]\rightarrow S$ is "differentiable at $L^1$-a.e. point $t \in [0,1]$." $S$ is some metric space on which we're ...
- 2,039
2
votes
0
answers
149
views
Isometry embedding
Problem:
Let $(M,g)$ be a compact Riemannian manifold. Then clearly $(M,d_R)$ is a metric space, where $$ d_R(x,y)=\|x-y\| \quad \forall x,y\in M $$
Now let's see the following:
1. The Kuratowski ...
- 21
2
votes
0
answers
88
views
Transformation between two measures
If $\mu$ and $\nu$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ with smooth densities $p_\mu(\mathbb{x})$ and $p_\nu(\mathbf{x})$, does it always ...
- 2,340