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Blog - network theory (part 24)

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I hope to actually finish the proof of the deficiency zero theorem. In the book version it shouldn't take as long as it has here, because I'll do some stuff in earlier sections. There's a beautiful picture here, which also involves electrical circuit theory, but I'm sort of making it up as I go, so it's not perfected, and it's coming out in dribs and drabs now.

This section starts with a long review, which I've written... but I haven't written the meat yet!

Comments

  • 1.
    edited August 2012

    a

    Comment Source:a
  • 2.

    I've finished the meat of

    and indeed finished the proof of the deficiency zero theorem. I'm happy to be done with it, but I see very clearly that lots of the lemmas here are really about Markov processes, not stochastic reaction networks... having them all mixed in makes it harder to see what's going on. And my approach to Markov processes is based heavily on graph theory and the 'discrete differential geometry of graphs' that Eric Forgy likes so much... but which I haven't actually explained much in this series!

    So, while I think I've found the really good way to prove the deficiency zero theorem, it's not organized well yet. It needs a little book, which starts out talking about Markov processes and the differential geometry of graphs, and then applies those ideas to stochastic reaction networks.

    I will try to do this reorganization at some point while polishing up the little book that I'm indeed trying to write.

    Comment Source:I've finished the meat of * [[Blog - network theory (part 24)]] and indeed finished the proof of the deficiency zero theorem. I'm happy to be done with it, but I see very clearly that lots of the lemmas here are really about Markov processes, not stochastic reaction networks... having them all mixed in makes it harder to see what's going on. And my approach to Markov processes is based heavily on graph theory and the 'discrete differential geometry of graphs' that Eric Forgy likes so much... but which I haven't actually explained much in this series! So, while I think I've found the really good way to prove the deficiency zero theorem, it's not organized well yet. It needs a little book, which starts out talking about Markov processes and the differential geometry of graphs, and _then_ applies those ideas to stochastic reaction networks. I will try to do this reorganization at some point while polishing up the little book that I'm indeed trying to write.
  • 3.
    edited August 2012

    I added a bunch of extra chat to

    and posted it on the blog.

    Comment Source:I added a bunch of extra chat to * [[Blog - network theory (part 24)]] and posted it [on the blog](http://johncarlosbaez.wordpress.com/2012/08/23/network-theory-part-24/).
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