bio | website | fernuni-hagen.de/analysis |
---|---|---|
location | Germany | |
age | 36 | |
visits | member for | 2 years, 5 months |
seen | Jan 25 at 6:02 | |
stats | profile views | 1,378 |
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.
Jan 11 |
comment |
ODE properties true in finite dimension but not in Banach spaces of infinite dimension
there is a nice statement by de giorgi about long-time behaviour of gradient-like systems. i can give you a precise statement if you are interested in this kind of properties. |
Jan 8 |
comment |
Schrodinger equation with magnetic vector potential
What exactly are you interested in? The operator on the RHS is skew-adjoint (provided you have "good" boundary conditions and/or reasonable magnetic potential), hence it generates a unitary group on $L^2$ by Stone's Theorem. What is usually referred to as "Kato's theory" is a collection of much deeper results on admissible scalar potentials. |
Jan 5 |
comment |
How to refer to plural of mathematical symbols - with or without an apostrophe
@KConrad I would say, this use is wrong. PDE's is wrong, PDEs is correct, simply put. |
Jan 3 |
comment |
Frobenius-Perron operator leaving a function sub space invariant
Is $\mathcal S=\mathcal V$? |
Dec 30 |
comment |
density of affine functions in $H_0^1$
If you are interested in the 1D-case, i.e., in proving density in $H^1_0(a,b)$, then the Stone-Weierstraß theorem is enough to prove your statement. |
Dec 19 |
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 $L$. |