1,170 reputation
316
bio website fernuni-hagen.de/analysis
location Germany
age 36
visits member for 2 years, 1 month
seen 11 hours ago

I work on evolution equations, sometimes on their applications too. I am interested in PDEs and functional analysis. I am fond of algebraic graph theory.


Sep
24
comment Examples for Markov generators with pure point spectrum
I don't quite understand what you mean by saying that "the only three generators that can arise are the Ornstein-Uhlenbeck, Laguerre and Beta generator". If you consider the Laplacian as a (very) special case of the Ornstein-Uhlenbeck operator, then I agree. In any case, $\Delta$ on $L^2(\Omega)$, $|\Omega|=1$, with Neumann boundary conditions seems to be what you need: It generates a Markov semigroup under very mild regularity assumption on $\partial\Omega$ and its spectrum can be almost arbitrarily complicated, the main constraint being the (dimension dependent) Weyl asyptotic formula.
Sep
15
answered Results true in a dimension and false for higher dimensions
Sep
13
comment Another kind of the positivity of matrices
a matrix fulfilling your notion of positivity is called "positive definite" in linear algebra - that is, in each linear algebra textbook. you can look for answers there.
Sep
1
awarded  Revival
Aug
29
awarded  Yearling
Jul
2
awarded  Curious
Jun
27
revised Find multiple non-adjacent paths in a graph
added 121 characters in body
Jun
27
answered Find multiple non-adjacent paths in a graph
Jun
16
revised Continuity with values in L^2
added 1 character in body
Jun
10
comment $H^s(\mathbb T)$ is a Banach algebra for $s>1/2$
But 7.57 is an imbedding result for Sobolev spaces on domains of $\mathbb R^n$! How does this implies the claimed property? I have checked a bit in the literature: It seems that Thm. 3.5 in Sobolev spaces on Riemannian manifolds by E. Hebey settles my question.
Jun
7
comment $H^s(\mathbb T)$ is a Banach algebra for $s>1/2$
Can you provide a reference for your last assertion? Does one have that $W^{s,p}(M)\hookrightarrow L^\infty(M)$ if $sp>{\rm dim } M$?
Jun
5
accepted doubly-stochastic isomorphisms of graphs
Jun
3
comment How to model a time-discrete heat equation on a graph?
I see. But then why won't you use a time-continuous model? Then the solution would be given by the semigroup generated by the Laplacian (with negative sign if you want to converge towards equilibrium).
Jun
2
revised How to model a time-discrete heat equation on a graph?
added 10 characters in body
Jun
2
answered How to model a time-discrete heat equation on a graph?
May
31
comment Heat equation with graph laplacian
is there a reason for deleting the very same question (which had already received some comments) and repost it here?
May
29
comment Time-inhomogeneous Markov Chains
I would like to add that in the field of differential equations on Banach spaces (which contain time continuous Markov chains as special cases) transition matrices that can vary over time become time-dependent operators. There is a well-developed (if not very elementary) theory for this kind of problems, which starts by replacing $C_0$-semigroups by objects called "evolution families". And yes, much is known the long time behavior of this kind of problems: see e.g. www.math.kit.edu/iana3/~schnaubelt/media/survey1.pdf
May
21
comment Should a theorem be numbered by where it is first stated or where it is proven?
I completely agree with Trevor Wilson. You give a theorem a number for the sake of later reference, but if a later reference is already easy and possible, why bother?
May
20
awarded  Nice Question
May
20
comment doubly-stochastic isomorphisms of graphs
thanks, now i understand.