Hello I apologize for the late question I have a problem on the definition of Q(1x,1y)=l(x,y)

Dear Authors,

Thank you for the very pedagogical and excellent written math lectures. It is pretty rare to find this type of exposition in math books.

Anyway, I am curious about referring to $c\left(x\right)$ as the “killing term”.

Could you elaborate on that decision? In your book, you mention that allows you to “capture numerous phenomena without having to look at case distinctions”.

From your naming, all those phenomena seem to involve loss of information (or increased entropy). Have you faced capturing phenomena where a gain of information is involved?

Best Regards,

Paco

Sorry for the late question, but I am revisiting lemma 1.10. The direction (ii) ⇒ (i) for (b) is only hinted at, but I interpret the argument as follows: If (setting the contraction to $C_{[0,1]}$ for concreteness) (ii) holds, then

$$Q\left(C_{[0,1]}f\right)=\frac{1}{2}\sum _{x\ne y}(-l(x,y\left)\right)\left[C_{[0,1]}f\right(x)-C_{[0,1]}f(y){]}^2+\sum _x\left(\sum _{y}l\right(x,y\left)\right)(C_{[0,1]}f\left(x\right){)}^2$$

is less or equal than $Q\left(f\right)$ due to the fact that 1) all coefficients are positive and 2) the definition of a normal contraction.

Here's my question: Doesn't that mean that this argument goes through for all normal contractions (not just the specific choice of $C_{[0,1]}$)? This would mean that (ii) $\Rightarrow$ (i) for all normal contractions.

If there is an (easy) way of showing the reverse direction, one could then completely remove the intermediate step (iii) in Theorem 1.11, by using the equivalence of items (ii) and (iv) (in notation Theorem 1.11).

PS: I guess that there is one good reason not to do that: Dirichlet forms $Q$ are defined by compatibility with the contraction $C_{[0,1]}$, which is easier to check for a specific choice of $Q$ than compatibility with all normal contractions.

Dear all,

in definition 1.7 the form $Q_{b,c}$ associated to a graph is also called energy form. I was wondering if there is an intuition on why the expression on the right hand side resembles energy? When thinking about this, I imagined $f$ to be some kind of temperature distribution on the graph. However, with this notion, I don't see why the temperature differences between connected nodes are related to the total energy.

In the same sense, what is the intuition on energy forms being compatible with contractions (right below def. 1.7). The assumption $|g\left(x\right)|\le |f\left(x\right)|$ seems plausible to me but again the differences seem to play a role as well.

Thanks for the lecture and sorry for the late question, Tobias

Dear ISEM Hello thank you for the course proposed,je is very interesting.

Dear ISEM Team,

thanks a lot for the lecture. Concerning Exercise 1.2, I suggest to add a definition for positivity of forms.

Best regards Sascha

Dear lecturers,

Firstly thank you for the lecture!

I am always confused in this, but why in the proof of Theorem 1.16. it is written $L1\left(x\right)={\sum }_{z\in X}l(x,z)$? Shouldn't it be divided by $m\left(x\right)$ due to bottom of the page 3 ($Lg\left(x\right)=1/m\left(x\right){\sum }_{y\in X}l(x,y)g\left(y\right)$)?

Best, Anna

Dear all,

Thank you very much for this first lecture.

Best wishes,

EL-Houcine

Hello everyone do we have to submit the solutions of the exercises? if yes where and when is the deadline. Thanks

Dear all,

there seems to be a mistake in exercise 4:
If I take a graph consisting of two vertices $X=\{0;1\}$ with the counting measure and no edges $b\equiv 0$. Then $L$ is the zero operator and $f\in C\left(X\right),f\left(0\right):=1,f\left(1\right):=-1$ fulfills ${\sum }_{x\in X}m\left(x\right)f\left(x\right)=0$ but clearly not $f\in \text{ran}L$.
In order to fix this I think one of the following needs to be done:

• assume $X$ to be connected
• use $V:=\{f\in C(X);{\sum }_{x\in C}m(x\left)c\right(X)=0\text{for all connected components C of X}\}$.

Best wishes, Pablo

Dear all,

I found Definition 1.7 (“Form associated to a graph”) rather unexpected. With Definition 1.5 in mind I would have expected: given a graph $(b,c)$, take the matrix with coefficients $l_{b,c}(x,y)$ given by Definition 1.5, and then take the form $Q$ induced by this matrix, cf. page 2. It appears that this is the same as Definition 1.7, i.e., it turns out that $Q_{b,c}=Q$. Somehow this seems to be the contents of Proposition 1.9, but I found all this a bit confusing.

Best wishes, Hendrik

Dear all,

thank you for the first lecture,

Some minor remarks:

• Middle of page 5, where ${\Phi }_{\left(\alpha \right)}\left(L\right)$ is shown to be the resolvent, one should write brackets around $\alpha$ in the displayed formula for consistency reasons.
• I think operator valued should rather be operator-valued
• I think the definition of path and connectedness needs to be the other way around. Formally one could read the current definition is if path was only defined for connected graphs. This in turn makes it difficult to use the definition of connected component as is.
• In Theorem 1.11, I take it that $Q$ is a symmetric form on $C\left(X\right)$. (I haven't found `symmetric form over a finite set' as a formally defined notion. I might have missed it, though!)
• In the proof of Theorem 1.11, implication (i) to (v), it might make sense to refer to Proposition 1.8 and spell this proposition out in the general situation given in (v) and to include the actual statement of Proposition 1.8 with an “In particular, … ”

Cheers,

Marcus

What does the Laplacian characterize concretely for a given graph. I meant how can we interpret this linear form for a given graph?

Thanks

A pleasure to be part of this internet seminar and learning about this nice topic.

At the end of page 7, definition 1.4 assume that the graph is undirected and there is no connection from one vertex to itself. Do we have an equivalent definition for more general graph?

With best regards

Joel

Dear all,

thank you very much for this nice first lecture!

A few very minor remarks from my side:

• On page 4 where you define the spectral calculus at the bottom, is it intended that there are no brackets around the sets? You could also write $C\left(\sigma \right(L\left)\right)$ on the left-hand side (since it is already defined for finite sets).

• On page 12, in the definition $f_{-}\left(x\right):=-f\left(x\right)\vee 0$ I would suggest to put brackets around $-f\left(x\right)$ to keep it consistent with the rest.

• In the proof right after Proposition 1.14, there is a small typo ('is' instead of 'ist').

I am looking forward to the upcoming lectures.

Best wishes,

Patrizio

Dear all

First of all I would like to thank you for the first lecture of this year's ISem. I have some questions/remarks:

• When you speak about “spectral calculus”, I guess that this is comparable with the “functional calculus” or even the same but with another name or is there another reason why it is called “spectral calculus”?
• Definition 1.4 defines a graph over a finite set $X$ to be a pair of functions $(b,c)$ satisfying some conditions. However, I do not see what the role of $c$ actually is, since there is no interaction between $b$ and $c$, they seem to be distinguished. Is it possible to comment on this?

I am already looking forward to the next lectures.

Best wishes Christian

Dear all,

First I want to thank the organizers for this year's ISEM and the first lecture.

Here I just want to give a minor comment. On page 4 there is no need to mention the self-adjointness as the reason for real eigenvalues. The operator L maps $l^2(X,m)$ to $l^2(X,m)$. As mentioned on page 1 those functions are real-valued. If $\lambda$ would be in $C\setminus R$ then $Le$ would not be real-valued.

Please correct me if I'm wrong.

Best wishes,

Christoph

Dear all,

first of all thank you very much for the lecture.

I think in the middle of page 5 there must be required $-\alpha \notin \sigma \left(L\right)$ instead of $\alpha \notin \sigma \left(L\right)$ twice. One time to obtain $(L+\alpha )$ bijective and the other time for the definition of ${\Phi }_{\left(\alpha \right)}\left(s\right)=\frac{1}{s+\alpha }$, for $s\in \sigma \left(L\right)$.

Best regards,

Annika

Dear all,

thanks a lot to the virtual lecturers for the first lecture!

A few minor remarks:

• Is it intended that the set $X$ is allowed to be empty? (I checked it for a few results, and they all seem to remain formally correct if all formulas are interpreted in an appropriate way; it might be a bit confusing, though).

• There's a small inaccuracy at the top of page 4: in order to get the eigenspace, one needs to also take the vector $0$ in addition to the eigenvectors (otherwise one won't get a vector subspace).

• In the definition of the spectral calculus (bottom of page 4) it might be preferable to explicitly speak of the “spectral calculus of $L$”, to stress the dependence on $L$.

Best wishes,

Jochen