just a comment on notation: For a measure μ on the Borel subsets of R – for instance a spectral measure – the notation ∫baxdμ(x) does not make clear whether this should mean ∫[a,b]xdμ(x) or ∫(a,b)xdμ(x), for instance.
Best wishes, Jürgen
Christian Seifert, 2023/01/17 09:56
Dear Jürgen,
Many thanks; you are of course right. We have made this more clear now.
Best regards,
Christian
Delio Mugnolo, 2022/12/16 23:00
Dear lecturers, thanks for another interesting lecture.
I have two small comments regarding the exercises:
* In Exercise 1, you certainly want to assume the graph to be connected, since otherwise positivity of u may fail.
* In Exercise 1 – and even more so in Exercise 2 and Exercise 3 – the wording may sound unlucky. Given a graph (b,x) over (X,m), there is only *the* associated Dirichlet form, right? If not: what is *an* associated Dirichlet form?
Best regards,
Delio
Christian Seifert, 2022/12/21 16:59
Dear Delio,
Many thanks. Concerning Exercise 1, you are right.
Concerning Dirichlet forms on graphs: There may be more than one Dirichlet forms on a graph, see e.g. Lecture 06 (Chapter 04). Do you think about regularity and the form associated with the Dirichlet Laplacian L(D)?
Best,
Christian
Delio Mugnolo, 2023/01/09 16:33
Dear Christian,
thanks for your answer. Now I get what you mean.
Best
Delio
Leon Berghoff-Flüel, 2022/12/15 16:14
Dear lecturers,
thank you for the lecture from the Darmstadt group. Some remarks from our side:
• In the proof of theorem 6.1 on page 96 we were struggling with the sentence “Now, having a minimum at x0 yields u(t0,y)=0 for all y∼x0.”
We suppose, the minimum principle 2.10 is being used here. If this is the case, an explicite reference would help the reader's understanding. Also, if 2.10 is being used, u(t0,y) immediately holds true for all y connected to x0, and “for all y∼x0” is confusingly restrictive.
• In the proof of theorem 6.3 after the first indented equation, “the spectral measure μf for f∈D(L) and μf(σ(L))=∥f∥2” could falsely be interpreted as if μf(σ(L))=∥f∥2 was a requirement on f.
• Seven lines below, the reference to Proposition 3.13 (a) appears to be a mistake. 6.2 (a) or 3.18 would probably suit better.
• In the second half of the lecture we were at several occasions, e.g. on page 100 right after the end of the proof of 6.4, wondering what precisely is meant by “L is a self-adjoint operator arising from a Dirichlet form associated to
a […] graph”. This lecture seems to primarily deal with L(D) respectively Q(D), but isn't the Neumann form in the sense of lecture 6 a “Dirichlet form associated to a graph” as well?
Is the wording unfavorable here, or are we missing something?
• In the definition of the ground state (also on page 100), it states that “we have a unique strictly positive eigenfunction which minimizes the energy”. Uniqueness certainly only holds true when we require normalization in addition, which rises the question, whether the term “ground state” is reserved for normalized functions.
This becomes relevant again in Corollary 6.9, where one should either say that “u is a multiple of the ground state”, or that “u is a ground state”.
Best regards,
Leon
Christian Seifert, 2023/01/02 10:28, 2023/01/19 10:27
Dear Leon,
Many thanks.
Concerning your first point, you are right with the minim principle; however, y∼x0 just means that y is connected to x0 (meaning that y and x0 are neighbors, i.e. connected by an edge). Of course, one can then iterate (as in the proof Theorem 2.10) to obtain the statement for all y in the connected component containing x0.
Concerning your second point, many thanks for making this point.
Concerning your third point, you are right and we adjust accordingly.
Concering your fourth point: Section 6.2 concerns general Hilbert space theory. Lemma 6.4 however makes use of Theorem 6.1, so we need regularity of the Dirichlet form (and we are thus dealing with the form Q(D) and the operator L=L(D)).
Concerning your fifth point, you are right and we adjust accordingly. By ground state we mean the normalized state.
Best,
Christian
Robert Haller, 2023/01/18 17:00
Dear Christian,
coming back to your answer to Leon about the meaning of x∼y, we are again confused after having read the 10th lecture. You write x∼y means that x and y are connected in the graph, however in the proof of the local Harnack inequality, lines 4 and 5 on page 120, it is again used in the sense of x and y being neighbours, which was also the meaning we had in mind from earlier lectures, cf. the post above from Leon.
Best, Robert
Christian Seifert, 2023/01/19 10:30, 2023/01/22 20:09
Dear Robert,
Many thanks. My answer was indeed somewhat vague. (Sorry for that!). What we mean by x∼y is that x and y are neighbors, i.e. b(x,y)>0, or put differently, x and y are connected by an \emph{edge}. Concerning Leon's post, indeed one can directly reason from Theorem 2.10 that one gets u(t0,y)=0 for all u in the connected component of x0.
Sorry for the confusion!
Best,
Christian
Hendrik Vogt, 2023/01/20 22:32
Dear Robert, dear Christian,
today, in Bremen we were trying to understand the proof of Theorem 6.1. Unfortunately we couldn't quite follow how one can appy Theorem 2.10.
However, Fritz suggested a different proof, namely by application of Proposition 1.24 to the connected finite graph Kn: Given a positive non-trivial f∈ℓ2(X,m) choose n so large that f|Kn is non-trivial; then for strictly positive times the semigroup on ℓ2(Kn,m) applied to f|Kn yields a strictly positive function, and it remains to apply domain monotonicity and positivity of the semigroup.
Does it make sense what I wrote in the previous paragraph?
Best wishes, Hendrik
EL-Houcine OUALI, 2022/12/12 16:59
Thank you very much for this lecture.
discussion/lecture08.txt · Last modified: 2022/11/15 18:11 by matcs
Discussion on Lecture 08
Dear virtual lecturers,
just a comment on notation: For a measure μ on the Borel subsets of R – for instance a spectral measure – the notation ∫baxdμ(x) does not make clear whether this should mean ∫[a,b]xdμ(x) or ∫(a,b)xdμ(x), for instance.
Best wishes, Jürgen
Dear Jürgen,
Many thanks; you are of course right. We have made this more clear now.
Best regards, Christian
Dear lecturers, thanks for another interesting lecture.
I have two small comments regarding the exercises:
* In Exercise 1, you certainly want to assume the graph to be connected, since otherwise positivity of u may fail.
* In Exercise 1 – and even more so in Exercise 2 and Exercise 3 – the wording may sound unlucky. Given a graph (b,x) over (X,m), there is only *the* associated Dirichlet form, right? If not: what is *an* associated Dirichlet form?
Best regards, Delio
Dear Delio,
Many thanks. Concerning Exercise 1, you are right.
Concerning Dirichlet forms on graphs: There may be more than one Dirichlet forms on a graph, see e.g. Lecture 06 (Chapter 04). Do you think about regularity and the form associated with the Dirichlet Laplacian L(D)?
Best, Christian
Dear Christian,
thanks for your answer. Now I get what you mean.
Best Delio
Dear lecturers, thank you for the lecture from the Darmstadt group. Some remarks from our side:
• In the proof of theorem 6.1 on page 96 we were struggling with the sentence “Now, having a minimum at x0 yields u(t0,y)=0 for all y∼x0.”
We suppose, the minimum principle 2.10 is being used here. If this is the case, an explicite reference would help the reader's understanding. Also, if 2.10 is being used, u(t0,y) immediately holds true for all y connected to x0, and “for all y∼x0” is confusingly restrictive.
• In the proof of theorem 6.3 after the first indented equation, “the spectral measure μf for f∈D(L) and μf(σ(L))=∥f∥2” could falsely be interpreted as if μf(σ(L))=∥f∥2 was a requirement on f.
• Seven lines below, the reference to Proposition 3.13 (a) appears to be a mistake. 6.2 (a) or 3.18 would probably suit better.
• In the second half of the lecture we were at several occasions, e.g. on page 100 right after the end of the proof of 6.4, wondering what precisely is meant by “L is a self-adjoint operator arising from a Dirichlet form associated to a […] graph”. This lecture seems to primarily deal with L(D) respectively Q(D), but isn't the Neumann form in the sense of lecture 6 a “Dirichlet form associated to a graph” as well? Is the wording unfavorable here, or are we missing something?
• In the definition of the ground state (also on page 100), it states that “we have a unique strictly positive eigenfunction which minimizes the energy”. Uniqueness certainly only holds true when we require normalization in addition, which rises the question, whether the term “ground state” is reserved for normalized functions. This becomes relevant again in Corollary 6.9, where one should either say that “u is a multiple of the ground state”, or that “u is a ground state”.
Best regards, Leon
Dear Leon,
Many thanks.
Concerning your first point, you are right with the minim principle; however, y∼x0 just means that y is connected to x0 (meaning that y and x0 are neighbors, i.e. connected by an edge). Of course, one can then iterate (as in the proof Theorem 2.10) to obtain the statement for all y in the connected component containing x0.
Concerning your second point, many thanks for making this point.
Concerning your third point, you are right and we adjust accordingly.
Concering your fourth point: Section 6.2 concerns general Hilbert space theory. Lemma 6.4 however makes use of Theorem 6.1, so we need regularity of the Dirichlet form (and we are thus dealing with the form Q(D) and the operator L=L(D)).
Concerning your fifth point, you are right and we adjust accordingly. By ground state we mean the normalized state.
Best, Christian
Dear Christian,
coming back to your answer to Leon about the meaning of x∼y, we are again confused after having read the 10th lecture. You write x∼y means that x and y are connected in the graph, however in the proof of the local Harnack inequality, lines 4 and 5 on page 120, it is again used in the sense of x and y being neighbours, which was also the meaning we had in mind from earlier lectures, cf. the post above from Leon.
Best, Robert
Dear Robert,
Many thanks. My answer was indeed somewhat vague. (Sorry for that!). What we mean by x∼y is that x and y are neighbors, i.e. b(x,y)>0, or put differently, x and y are connected by an \emph{edge}. Concerning Leon's post, indeed one can directly reason from Theorem 2.10 that one gets u(t0,y)=0 for all u in the connected component of x0.
Sorry for the confusion!
Best, Christian
Dear Robert, dear Christian,
today, in Bremen we were trying to understand the proof of Theorem 6.1. Unfortunately we couldn't quite follow how one can appy Theorem 2.10.
However, Fritz suggested a different proof, namely by application of Proposition 1.24 to the connected finite graph Kn: Given a positive non-trivial f∈ℓ2(X,m) choose n so large that f|Kn is non-trivial; then for strictly positive times the semigroup on ℓ2(Kn,m) applied to f|Kn yields a strictly positive function, and it remains to apply domain monotonicity and positivity of the semigroup.
Does it make sense what I wrote in the previous paragraph?
Best wishes, Hendrik