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Blog - Petri net programming (part 3)

Subtitle: The role of differential equations

In this article, I explain some of the mathematical context for differential equations to programmers who have not worked on scientific applications. This should help to orient them as we approach the next programming goal, which is to write an Euler-method simulator for the dynamics of a reaction network.

I picture a reader who is technically advanced, is capable of abstraction, and is motivated to learn, but has only been exposed to the typical applications which involve limited use of mathematics. They will have skills in discrete mathematics, in the analysis of process logic, in making complex deductions about the behavior of complex mechanisms, in data and process abstraction, and in the practical application of the scientific method to the experiments involved in fault diagnosis. But their work will not have led to topics in, for example, calculus or abstract algebra. Yet there is no impermeable membrane here. For instance, data abstraction falls under the heading of abstract algebra, and algorithms are both objects and vehicles of mathematical study.

Any reader with these credentials can be considered to be a *virtual apprentice to science*. And there are lots of them! Here it is estimated that there are 40 million programmers in the world; elsewhere, I saw an estimate of 2 million scientists. Further, they are human and so we can assume that the bulk of them are concerned about the planet.

*These considerations must have strategic implications*, both for the Azimuth project and for science itself.

The world's problems, as presented to science, are accumulating far faster than they are being solved. Our ship is in danger -- all hands on deck! We need more scientists, and we need broader public participation at the interface between science and society. This calls for an outreach effort from science to the broader public. To my mind, this either already is, or deserves to be, one of the core planks of the Azimuth strategy, because it is a vital way for scientists to contribute to the saving the planet. Whatever form they take, as programmers or otherwise, the virtual apprentices to science represent a potential base for the influx of people, energy and ideas into the Azimuth project. It is plausible that this kind of positive feedback could lead to an acceleration in the growth of planetary scientific awareness.

This blog article is an item of scientific outreach literature. Within the confines of a short and informal presentation, I attempt a rough explanation from the ground up of a fundamental mathematical idea. I try to keep it implicitly accurate, in the sense that with more space it could be elaborated to a rigorous presentation. I also aim to convey some of the "cultural flavor" of the subject, especially towards the end.

The article is ready for review. Thanks.

## Comments

Great! Your rousing call to action could even be a nice part of a blog article!

I looked at your article here and it looks ready to go. I fixed a few typos, and I might catch some more typos while I change it to the Wordpress format, but I see no bigger problems.

Does anyone else have comments? Or should I go ahead and post this?

`Great! Your rousing call to action could even be a nice part of a blog article! I looked at your article here and it looks ready to go. I fixed a few typos, and I might catch some more typos while I change it to the Wordpress format, but I see no bigger problems. Does anyone else have comments? Or should I go ahead and post this?`

From the article:

Is that intended to be "Not all..." ?

Otherwise, it looks good.

`From the article: > None of x, y or z can be zero. Is that intended to be "Not all..." ? Otherwise, it looks good.`

Thanks Graham.

That was a glitch. I changed the sentence to: They can't all be zero, can't all be positive, can't all be negative, can't all be even, and can't all be odd.

`Thanks Graham. That was a glitch. I changed the sentence to: They can't all be zero, can't all be positive, can't all be negative, can't all be even, and can't all be odd.`

Grammar fix to the last paragraph in the section "Computational approach."

`Grammar fix to the last paragraph in the section "Computational approach."`

Let me know when you feel ready for me to put the article on the blog, David! If you're itching to make last-minute changes, now is the last minute.

`Let me know when you feel ready for me to put the article on the blog, David! If you're itching to make last-minute changes, now is the last minute. <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif" alt = ""/>`

Okay, it's posted here!

Thanks a lot, David! I'll try to get some people on G+ interested in it, and I hope they have questions and comments, and I hope you can respond to them.

By the way, I made a couple of small edits at the last minute:

In TeX the exponential function is

`$\exp$`

, not`$exp$`

. This produces $\exp$, which is better than $e x p$. Same goes for all our favorite functions. (On the Wiki and Forum, we are lulled into a false sense of security by Jacques Distler's decision to make multi-letter strings come out roman rather than italic in math mode. So on the Wiki you'll have seen`$exp$`

give $exp$, but that doesn't work most places, and I really wish this "feature" had been omitted.)It's best whenever possible to use HTML instead of Markdown on articles that are going to become blog posts, since Wordpress uses HTML.

`Okay, it's posted here! * [Petri net programming (part 3)](http://johncarlosbaez.wordpress.com/2013/04/19/petri-net-programming-part-3/), Azimuth Blog. Thanks a lot, David! I'll try to get some people on G+ interested in it, and I hope they have questions and comments, and I hope you can respond to them. By the way, I made a couple of small edits at the last minute: * In TeX the exponential function is `$\exp$`, not `$exp$`. This produces $\exp$, which is better than $e x p$. Same goes for all our favorite functions. (On the Wiki and Forum, we are lulled into a false sense of security by Jacques Distler's decision to make multi-letter strings come out roman rather than italic in math mode. So on the Wiki you'll have seen `$exp$` give $exp$, but that doesn't work most places, and I really wish this "feature" had been omitted.) * It's best whenever possible to use HTML instead of Markdown on articles that are going to become blog posts, since Wordpress uses HTML.`

John, that's great, thanks a lot for publishing it. Viva la Blog!

`John, that's great, thanks a lot for publishing it. Viva la Blog!`

Hi John,

Also thanks for editing and fixing the formulas.

Two editing requests. Can you repair my grammar error in this sentence, at the beginning of the third paragraph of the section called formula-based approach:

Just nix the "to".

Also, at the end of that section, I give the computation for e, but when I view it the line is too wide for the browser, so the ending of it " \approx 2.71828" isn't visible. Can you make it all visible, by splitting the equation across two lines? I'd rather have it look awkward on two lines, than lose the concrete number that results.

Thanks!

`Hi John, Also thanks for editing and fixing the formulas. Two editing requests. Can you repair my grammar error in this sentence, at the beginning of the third paragraph of the section called formula-based approach: > The picture changes, however, if we let the formulas to contain an infinite number of operations. Just nix the "to". Also, at the end of that section, I give the computation for e, but when I view it the line is too wide for the browser, so the ending of it " \approx 2.71828" isn't visible. Can you make it all visible, by splitting the equation across two lines? I'd rather have it look awkward on two lines, than lose the concrete number that results. Thanks!`

Hi, I fixed those mistakes. I hadn't even noticed this one:

because I'm getting so used to Germanic English.

I fixed your equation in a different way than you suggested, by deleting $e^1 = \exp(1)$. I think writing $e = e^1 = \exp(1) = X(1)$ is only interesting for the tiny minority of people who are maniacally concerned with formalisms and how different formalism are related (like me). Most people will consider it either confusing, or boringly lengthy throat-clearing.

I can reinsert these extra equations and add an extra line to the equation if you want.

`Hi, I fixed those mistakes. I hadn't even noticed this one: > The picture changes, however, if we let the formulas to contain an infinite number of operations. because I'm getting so used to Germanic English. I fixed your equation in a different way than you suggested, by deleting $e^1 = \exp(1)$. I think writing $e = e^1 = \exp(1) = X(1)$ is only interesting for the tiny minority of people who are maniacally concerned with formalisms and how different formalism are related (like me). Most people will consider it either confusing, or boringly lengthy throat-clearing. I can reinsert these extra equations and add an extra line to the equation if you want.`

By the way, David, here's a suggestion that only occurred to me too late. Your post is very clear, but if you want to provoke more discussion it's good to include something a bit more "stressful" - like a question, or a challenge, or something shocking or hard to understand. It seems that if everything is easy to understand, fewer people seem motivated to say anything.

I like discussions, in part just to know that people are paying attention. So I deliberately include a bit more gnarly material. You can put it at the end, and mark it as such, if you don't want to scare away people who find it scary.

`By the way, David, here's a suggestion that only occurred to me too late. Your post is very clear, but if you want to provoke more discussion it's good to include something a bit more "stressful" - like a question, or a challenge, or something shocking or hard to understand. It seems that if everything is easy to understand, fewer people seem motivated to say anything. I like discussions, in part just to know that people are paying attention. So I deliberately include a bit more gnarly material. You can put it at the end, and mark it as such, if you don't want to scare away people who find it scary.`

Hi John, thanks for fixing those issues on the blog -- your version looks good. Also thanks for posting the note on Google+.

You idea about adding questions / challenges / problems is good. I will think about it as I am writing the article on the rate equation.

`Hi John, thanks for fixing those issues on the blog -- your version looks good. Also thanks for posting the note on Google+. You idea about adding questions / challenges / problems is good. I will think about it as I am writing the article on the rate equation.`