5
votes
Accepted
Generation of strict contraction semigroups
Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in
Engel, Klaus-Jochen; Nagel, Rainer, ...
- 4,729
3
votes
Gradient flows: convex potential vs. contractive flow?
It should be noticed that already on $R^d$ equipped with a non-Euclidean norm $\|.\|$ the answer to your question is no. Ohta-Sturm [1] proved the following: let $\lambda\in R$ and consider the ...
- 660
3
votes
Accepted
Gradient flows: convex potential vs. contractive flow?
Doesn't this follow from dependence on initial data?
Consider the flow mapping $\Phi(t,X)$ which solves
$$ \frac{d}{dt}\Phi(t,X) = - \nabla V(\Phi(t,X)) $$
so taking the derivative in $X$ we have
...
- 39k
2
votes
Accepted
Small contraction for Hyperkähler Varieties
Let $f:X\to Y$ be a birational contraction where $X$ is hyperkähler, then $K_X\sim 0$ and $K_Y=f_*K_X\sim 0$, and hence $K_X=f^*K_Y$. In particular, this means that $Y$ has canonical singularities. ...
- 1,164
1
vote
Accepted
Is this a contraction mapping for small $T$?
$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\De}{\Delta}\newcommand\R{\mathbb R}$Edit: This answer is insufficient, even though (almost) all the reasoning appears relevant to the ...
- 128k
1
vote
Set operations over iterated function systems
If $F = f_1, \ldots, f_n$ is an IFS, then the attractor of any subset of F is a subset of the attractor of F. This is readily seen through coding points in the attractor (see Barnsley's book for the ...
- 4,990
1
vote
Small contraction for Hyperkähler Varieties
Another way to see it is the following. Let $f\colon X\to Y$ be any birational contraction from a projective hyperkähler manifold $X$ onto a normal projective variety $Y$. The exceptional locus of $f$ ...
- 115
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