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.

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