1,233 reputation
317
bio website fernuni-hagen.de/analysis
location Germany
age 36
visits member for 2 years, 3 months
seen 19 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.


1d
comment How do these two extensions of Sobolev spaces relate to each other?
What you are actually considering are Sobolev spaces over a compact manifold without boundary, namely the one-dimensional torus. That these two constructions agree is standard, see e.g. the book by Adams-Fournier.
Dec
18
comment significance of the Fučík spectrum
Answer to myself: I have just discovered an interesting relation between Fučík spectrum and spectral minimal partitions in a 2005 article by Conti-Terracini-Verzini.
Dec
17
comment Speed of Approach to Invariant Measure
I don't think it's particularly relevant that the generator's kernel agrees with the space of constant functions, and in fact there exist example of Markov(-type) processes that converge towards non-uniform distributions.
Dec
17
answered What are examples of good toy models in mathematics?
Dec
12
comment Speed of Approach to Invariant Measure
@ Nate Eldredge I don't see how to deduce discreteness of the spectrum from the mere fact that the constants form the kernel of N, but anyway this property has been proved (for a large class of Ornstein-Uhlenbeck operators) by Metafune, Pallara and Priola in their 2002 JFA article "Spectrum of Ornstein–Uhlenbeck operators in $L^p$ spaces with respect to invariant measures". Now, you only need to apply the spectral theorem to get the conclusion, as already suggested by Nate Eldredge.
Dec
12
comment Is the heat kernel more spread out with a smaller metric?
@Paul Siegel: I am surprised by your assertion. The heat semigroup typically has infinite speed of propagation, that is, as soon as the initial data are positive but not identically 0, the solution of the heat equation is instanteneously strictly larger than 0 at any point of the domain/manifold. This can be e.g. proved by irreducibility of the semigroup, see e.g. Ouhabaz' 2005 book. Am I missing something?
Dec
10
comment Local fractional Sobolev inequality
The trace operator is bounded from $H^{\frac{1}{2}+\epsilon}(\Omega)$ to $L^2(\partial \Omega)$, but your point evaluation is not, unless $n=1$, so in general your inequality will likely not hold (I am sure about the case $n=2$). Roughly speaking, $\{0\}$ is "too small" (google for "capacity of a set") to matter.
Dec
9
comment a variation on the theory of equitable partitions for graphs
I have just seen that the end of Chapter 4 in the old book by Cvetkovic-Doob-Sachs is devoted to exactly this question. An algorithm for factoring (certain classes of) graphs is also presented.
Dec
9
accepted almost equitable partitions and spectra
Dec
9
comment Do perfect matching(s) have signatures in the graph eigenvalues?
The quoted result still leaves open the question for the case of $r$-regular graphs, for $r=3,4$ (the case of $r=2$ being of course trivial).
Dec
8
comment Generalizations of the Four-Color theorem
well, it was not meant as a generalization. while it is known that the chromatic number gives a lower bound for the choosability number, the choosability number cannot be bounded from above by a function of the chromatic number, so they are indeed two different objects. i mentioned thomassen's theorem simply as a related result.
Dec
8
comment Generalizations of the Four-Color theorem
...who found it in 1996, when she was 19 years old.
Dec
8
answered Generalizations of the Four-Color theorem
Dec
2
comment Are these three different notions of a graph Laplacian?
I see. So when you talked about the "spectrum of the line graph" you were actually thinking of the adjacency spectrum of the line graph? I was always wondering whether there is a natural interpretation of the Laplacian of the line graphh for non-regular graphs. Anyway, the result you mention did certainly appear already in Harary's book (Thm. 13.3) and in Biggs' book (Lemma 3.6).
Dec
1
comment interpretation of generalized eigenvalue/vectors in spectral graph theory
I would imagine that this very much depends on the null space of $L$.
Dec
1
comment Are these three different notions of a graph Laplacian?
I was not aware of the correspondence in your last sentence. Can you give me a reference?
Dec
1
revised Are these three different notions of a graph Laplacian?
fixed two typos
Nov
30
answered Are these three different notions of a graph Laplacian?
Nov
30
comment significance of the Fučík spectrum
Upvote, but I'm not completely sure that that paper by Drabek & co. is actually devoted to the Fucik spectrum. In my eyes, it's rather like a parameter-dependent semilinear beam equation that, when one focuses on its stationary version, can be seen as a very particular kind of secular equation for the Fucik spectrum. No special derivation for the stationary version is provided, though, and it is not even clear whether the beam equation is converging towards a stationary state. Given the lack of a Fucik pendant of the spectral theorem, I really don't see a clear connection.
Nov
13
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