Hello,

More of my philosophy of what is the kind of math that i can do..

I am a white arab from Morocco, and i think i am smart since i have also
invented many scalable algorithms and algorithms..

Archimedes Plutonium has just asked me if i can do math,
and i replied that i can do math by giving a logical proof
of it by giving my following thoughts of how i can do math,
read it again carefully in the following link:

https://groups.google.com/g/soc.culture.quebec/c/uCF55Jczyak

And as you have just noticed in the above link that
i am also using Markov chains in mathematics,
here is also why i need Markov chains in mathematics:

Yet more precision about the invariants of a system..

I was just thinking about Petri nets , and i have studied more Petri
nets, they are useful for parallel programming, and what i have noticed
by studying them, is that there is two methods to prove that there is no
deadlock in the system, there is the structural analysis with place
invariants that you have to mathematically find, or you can use the
reachability tree, but we have to notice that the structural analysis of
Petri nets learns you more, because it permits you to prove that there
is no deadlock in the system, and the place invariants are
mathematically calculated by the following system of the given
Petri net:

Transpose(vector) * Incidence matrix = 0

So you apply the Gaussian Elimination or the Farkas algorithm to the
incidence matrix to find the Place invariants, and as you will notice
those place invariants calculations of the Petri nets look like Markov
chains in mathematics, with there vector of probabilities and there
transition matrix of probabilities, and you can, using Markov chains
mathematically calculate where the vector of probabilities
will "stabilize", and it gives you a very important information, and you
can do it by solving the following mathematical system:

Unknown vector1 of probabilities * transition matrix of probabilities =
Unknown vector1 of probabilities.

Solving this system of equations is very important in economics and
other fields, and you can notice that it is like calculating the
invariants , because the invariant in the system above is the vector1 of
probabilities that is obtained, and this invariant, like in the
invariants of the structural analysis of Petri nets, gives
you a very important information about the system, like where market
shares will stabilize that is calculated this way in economics. About
reachability analysis of a Petri net.. As you have noticed in my Petri
nets tutorial example (read below), i am analysing the liveness of the
Petri net, because there is a rule that says:

If a Petri net is live, that means that it is deadlock-free.

And here is my tutorial that shows my methodology of analysing and
detecting deadlocks in parallel applications with Petri Nets, my
methodology is more sophisticated because it is a generalization and it
modelizes with Petri Nets the broader range of synchronization objects,
and in my tutorial i will add soon other synchronization objects, you
have to look at it, here it is:

https://sites.google.com/site/scalable68/how-to-analyse-parallel-applications-with-petri-nets

Thank you,
Amine Moulay Ramdane.