I see three major divisions for a Petri net initiative: math, science and computation.
Starting from their definition, Petri nets can be treated as purely mathematical entities. On the one hand, there is the classical mathematics which studies th…
General note: John got stretched too thin with research at UCR, graduate students, and the blog, so at the moment he is not actively present at the forum. He said this could change along with circumstances. In any case he made clear that he is su…
Diagnosis: since I was fishing for something not yet clearly formulated - and which opens a big topic - to keep things organized I should have asked my technical question in a separate discussion off the strategy thread. So yes the bug was on my …
Ex 6. Show that the hom functor \(A^{\rightarrow} = hom_X(A,.): X \rightarrow Set\) preserves monics, that is, if \(\alpha: C \rightarrow D\) is monic in X, then \(A^{\rightarrow}(\alpha): hom_X(A,C) \rightarrow hom_X(A,D)\) is also monic.
Ex 7. …
Meeting tomorrow, 2-4 pm EDT
Zoom ID: 807-569-857
Please review pages 43-56 and continue onto page 57. Do exercises 6, 7 and 23.
This week, we will be covering the prerequisites for understanding the Yoneda Lemma. To that end, please read Tai-Dan…
That's a fair point. I see now that my wording of the question didn't really express what I was trying to get at. I tried again with the parable of the software engineer at the party, but still that wasn't clear enough. That's on me, sorry for…
Another indicator to look at regarding the scope of comments is the category of a discussion. This discussion is "Petri nets at Azimuth in a new context," in category Strategy.
Aided and abetted by each other, we have led the discussion rather out…
@WebHubTel wrote:
"I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium."
Distinction perhaps between e…
@DavidTanzer wrote:
I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.
...
But the charge against thos…
Can we put a stop here please. You clearly know a lot more about equilibrium than I do. In this thread I am trying to achieve some progress towards getting a Petri net thing going for us at Azimuth. I started with a basic question, perhaps I did…
But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context.
Your point is that the 'first order' L-V model, which doesn't take into …
I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.
Perhaps that's why the scientists are all mystified by this, as they may be deeply attached to the mathematical idealism of eigenvalue-based solutions.
I propose a lasting accord -- call it the treaty of Azimuth -- between those of us who are m…
I can discuss this aspect of natural vs forced response all day.
This are very interesting point you are making! And I am interested to further discuss it. But it is veering away from the topic of this discussion, which is how we can get sta…
But \(f: 2^X \rightarrow 2^Y\) and \(g: 2^Y \rightarrow 2^X\) are not just any pair of order-reversing functions, they have an additional property which classifies them as a Galois connection:
\[f(x) \supseteq y \iff x \subseteq g(y)\]
Using Will…
These derivation operators have some nice order-theoretic structure.
First, note the powersets \(2^X\) and \(2^Y\) are of course partially ordered by the inclusion relation.
Next, it is not hard to see that \(f: 2^X \rightarrow 2^Y\) and \(g: 2^Y …
Exercise: prove (or refute) that this is equivalent to the definition I gave in comment #3:
That is, it is a subset X of the objects along with a subset Y of the attributes, such that every object in X has all the attributes in Y, and X is maxim…
Then a formal concept is defined as a pair (x,y), where f(x) = y and g(y) = x.
Or, using our other notation, (x,y) is a formal concept means that x' = y and y' = x.
The derivation operator \(g: 2^Y \rightarrow 2^X\) is defined symmetrically.
Let \(y \subseteq Y\) be a subset of the attributes.
Then define \(y' = g(y) \subseteq X\) as the set of objects which have every attribute in \(Y\).
Let X be all the objects, and Y all the attributes.
Then using standard notation, the powerset \(2^X\) consists of all subsets of \(X\), i.e., all sets of objects. And \(2^Y\) consists of all subsets of \(Y\), all sets of attributes.
Define func…
You could also search for tutorials, exercises and labs on "reaction networks," which are equivalent to Petri nets. This is fundamental to chemistry, which gives another route to finding such materials.
@DanielGeisler I recommend that we find some specific, simple examples of Petri nets to put on the table -- i.e., to post to the forum to talk about. That's why I suggested in comment #1 above to find references to tutorials, or courses, that have …
Now that we have some examples to inform our intuition, let's look at the standard technical definition of a formal concept, which is based on what are called "derivation operators."
Op. cit., p.14
The first formal concept is, perhaps, the concept of citrus fruits. The second formal concept is that of a purple vegetable. There are plenty of these: red cabbage, purple cauliflower, purple carrots, purple asparagus, endive, and…
This context could be derived by analyzing a small text, such as "Today I would like to get either an orange fruit, a green fruit, or a purple vegetable."
In a simple model, natural language processing tells us that an attribute/adjective X applie…
Op. cit., p.13
Formal concepts are illustrated with a toy example, where:
Objects O = {fruit, vegetable}
Attributes A = {green, yellow, purple}
The formal context given is \(\lbrace (fruit,orange), (fruit,green), (vegetable,purple) \rbrace\).
…
A formal context T is a relation between the objects O and attributes A. It indicates which attributes apply to which objects. It is represented by a bipartite graph.
A formal concept C is a special kind of sub-relation of T, corresponding I bel…
Hi @DanielGeisler,
We're a pretty freewheeling group. The homepage states our general concept. Otherwise, it's just us, posting ideas about math, science and projects. The only formal document is the moderation guidelines, which are only intend…
Formal concept analysis, Wikipedia
Formal concept analysis (FCA) is a principled way of deriving a concept hierarchy or formal ontology from a collection of objects and their properties. Each concept in the hierarchy represents the objects sha…