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Theory books with Mathematica: Grassmann algebra

Hello John and hello to everyone

This is how the future of theoretical publishing is getting shaped, Johne Brown is one amazing human being for such tedious labor of love:

Grassmann Algebra

Using that package and documentation (for teaching myself) without ever seeing Grassmann algebra I did this:

YBE solutions over Grassmann Algebra

The results were not only matched the pencil and paper computations of others, but what they did in a year between and advisor and a grad student, I was able to do in less than a week! And taught myself.

I have then done more YBE solutions over finite fields (so what!) but the point is how quickly a non-expert could pick up a new theory teach themselves and then code some solutions and contribute.

Dara

PS this was a snake-charm to convince John write theory books as such :) I am not very subtle

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    edited August 2014

    I've tried to calculate the partition function of a few stochastic master equations. It has some applications, but is a hardcore calculation even in special cases. I wonder, but do not know, if a symbolic computer package could do some of these?

    Similarly, there are a few 'inversion problems' that John suggested to me, which basically are about how the rate constants define what can be thought of as a 'free energy' for Petri nets. Calculating how this pseudo temperature depends on the couplings is also hard-core. I still have my notes on these.

    Finally, there are some hardcore calculations that involve conserved quantities for master equations. Calculating commutators. It's hardcore.

    I don't know what John thinks, but these are areas where I've noticed pen and paper are basically hopeless.

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    edited August 2014

    I'm glad you want me to write more books, Dara. Jacob is helping me finish one book, and I have another half-written book with Derek Wise. That's enough books to keep me busy for now!

    Jacob wrote:

    I don’t know what John thinks, but these are areas where I’ve noticed pen and paper are basically hopeless.

    I think for me the issue of understanding "free energy" and "temperature" in Petri nets is mainly a theoretical task that involves working at a high level of generality and side-stepping nasty calculations. You're reminding me how we stalled out on that. Much later, I was happy when Manoj Gopalkrishnan wrote down a free energy function for complex balanced reaction networks. My comments on that blog article show my attempts to understand that more deeply in the very simple special case of continuous-time Markov chains. (A chemical reaction network where each reaction has just one species as input and one as output is the same as a continuous-time Markov chain.) David Anderson's quick proof that the free energy is a Lyapunov function, also in a comment on that article, is another very nice thing.

    The next step is to push this forwards to include more concepts from thermodynamics, like energy and temperature. I'm too busy for this now but I think it's important.

    Actually Manoj and some other guys are writing a paper on this, so it's probably best to wait and see what they do before marching on.

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    I’m glad you want me to write more books, Dara.

    Of course, however I find the traditional math presentation alien to my learning, simply some even scant lines of symbolic or numeric code could open up my mind to learn fast and learn broad. I am not the only one, lots of others who BTW are your fans are also in the same situations. For example your immense collection of notes on Lie theory could easily be learned as such, I found a tiny pdf with your name on it, years ago, about the transformation group of rotation and I have since duplicated your computation style for any Lie computations.

    Example of Grassmann Algebra was quite accurately depicted, I was elated how quickly we could learn and solve YBE equations over Grassmann algebras. Idea appeared that we could allow the machination part of the algebras to be aided by software thus leave more mental energy for the real creative parts of mathematics.

    My problem is that the sorts of theories I love, are from your backyard, and I am learning them too slowly.

    On the climate front, we could develop educational/training as well as regular book-presentations, however with live code and data. So all the latest and greatest are constantly upgraded into the new revisions of the book.

    Dara

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    edited August 2014

    I find the traditional math presentation alien to my learning, simply some even scant lines of symbolic or numeric code could open up my mind to learn fast and learn broad. I am not the only one, lots of others who BTW are your fans are also in the same situations.

    I can believe that. Unfortunately I'm the opposite way around; code is much harder for me to understand than mathematical prose! For me to include it would be like putting an ingredient I don't like into a meal I'm cooking.

    My taste may be slowly starting to change: I am, after all, including code in some recent blog posts. But this is because analysis of climate data requires computers.

    I've never felt any desire to use computers in my work on physics, or pure math. My usual attitude there is: if I need a computer to figure something out, I either haven't thought about it carefully enough or I'm sinking into messy weeds, not staying on the path of simplicity.

    In some ways, this attitude is just an excuse to avoid computers.

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    edited August 2014

    There are some great guns sharing your opinion, Dara. +Charles Wells of Category theory for computater science and Toposes, triples and theores recently posted on G+ that he's learning Coq and Agda for HoTT.

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    I will look this fellows up thanx Jim.

    I tried to code John's work on categories a few years back and there was a major issue with the function composition associativity as one of the components of the category definition. In real life function compositions are not always associative! In mathematics there is no concept of time everything happens at the same time so f(gh) = (fg)h and no one doubts that.

    In programming systems (fg)h might not even exist + might not always have the same value as f(gh). This is due to the fact that there is a concept of sequential time in these function definitions.

    So I got stuck!

    Best way to understand this is to model the function compositions with Petri Nets and you will see the subtleties in an asynchronous system. Each transition, corresponding to a composition, has 2 places: 1) specific function computing some value, 2) input being available for the said function, only when both places have a token within then the transition fires else it won't. This concept is non-existent in category theory i.e. input is always available for all functions! This is assumed and implied in the human mind when reading category theory, but in reality they might not.

    Then you could make the petri net more complicated and add places and transitions for when the input is not available e.g. Mathematica programming environment allows for that and actually produces a non-associative (fg)h vs. f(gh). For example C does not allow (fg)h it only allows for f(gh).

    This is known as function evaluation in programming languages, and we could add that (via petri nets) to Category theory thus have a category theory with EVALUATION-net yet no associativity assumption and ... most Mathematicians would be pissed.

    D

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    Both the book and the paper are available from Charles Wells' homepage:

    http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf

    I bought the first edition of CTCS and I still don't know what a sketch is and why they seem to have gone out of fashion ie. I've never seen them mentioned again.

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    That's serendipitous or maybe not.

    there was a major issue with the function composition associativity as one of the components of the category definition. In real life function compositions are not always associative!

    I can't remember what I was reading yesterday but it was to do with some non-associative structure and I wondered whether this is called a pre-category. I don't see why that would stop you using it.

    In mathematics there is no concept of time everything happens at the same time so f(gh) = (fg)h and no one doubts that.

    Does temporal logic not count?

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    why they seem to have gone out of fashion ie. I’ve never seen them mentioned again.

    Have you ever conversed with Category guys?

    Back to associativity issue, some mathematicians assumed the associativity but instead think of function composition as another function which maps three functions into one function i.e.

    composition (f, g, h) ---> fgh

    Dara

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