Azimuth News (Part 4)

1 January, 2016


Happy New Year!

I’ve been rethinking my approach to life. Nothing major, just some small course corrections.


First, I’ve decided to stop posting so much on Google+, and post more here on Azimuth. I explain why here:

New Year’s Resolution.

I want a vibrant, lively online environment where people talk about things in serious, almost obsessive ways. At times this blog has been like that. So was the n-Category Café, back when a core group of people were all focused on roughly the same thing.

Lately I’ve been posting very little here on Azimuth. Instead, I’ve been working hard with a team of grad students to figure out how network theory can organize our understanding of circuits, control theory and nonequilibrium thermodynamics. My more general posts on ecological issues and science in general have been going to Google+. I now suspect it’s bad to split up my blogging that way. If I write more stuff in the same place, I hope more people will come here to talk.


Second, I have slowly come to realize that my talents lie in highly theoretical work—for example, network theory—rather than data-driven work like climate science, or the practical task of figuring out what to do about global warming. I still think that adapting to the Anthropocene and developing an ‘ecotechnic civilization’ is the challenge of the century for
civilization as a whole and scientists in particular. But I think the more urgent aspects of this—the ones that need to be done really soon—are not the ones I’m good at. The stuff I’m good at will help later, if at all. And yet, it doesn’t pay for me to do things that other people are already doing better.

There’s been a big change since I started this blog in 2010! Back then, it seemed only a few people knew quite how serious global warming would be, so I felt the need to shout an alarm. This December, thousands of politicians from around the world met in Paris to do something about it—and while they haven’t done enough yet, they all know the basic facts that had me so worried: for example, that we need to leave most of the world’s fossil fuels unburnt in the ground, in part because if we don’t, the effects will last for thousands of years:

More generally, there’s been a huge shift towards recognizing that we have to change our habits, quickly. In 2013, Copenhagen announced it will try to go carbon-neutral by 2025. In October of this year, California passed a law requiring the state to obtain 50% of its electricity from renewables by 2035. In December, San Diego became the largest city in the USA to require that all of the city’s power to come from renewables—again, by 2035.

And so on. The battle has not been won, but it has been joined. Countries, localities, cities and individuals around the world are volunteering to tackle this problem.

I can serve as a cheerleader for this trend, but I’m not really a politician, an engineer or even an experimentalist. Mathematicians, too, have their part to play. So, this year I’ll keep talking about network theory, and push it toward the ‘green mathematics’ I’ve been dreaming about, by getting serious about its applications to biology and ecology.

More on that soon!

Azimuth News (Part 3)

6 February, 2015

post by David Tanzer

Here are some notes from the back offices of the Azimuth project. After a long and productive stay as the Azimuth tech guy, Andrew Stacey is moving along and passing the baton to me. As part of this change, we’ve relocated the servers to a new Azimuth hosted account, and updated the forum software.

The forum is now at a new location:

This is where we collaborate on writing wiki and blog articles, on research and education projects, and on software development and systems issues. It’s also a fun place to chat with other professionals in a wide range of science-related fields.

So come on down to the forum! If you want to post, just apply for an account there. Acceptance criteria are minimal. A sincere desire to help goes a long way.

Important:  please use your full name, using “camel case” capitalization e.g. DavidTanzer, as your userid.  I will then put the spaces into your user ID.  (We want the spaces, but the registration form blocks them.)  The point is that we want to present ourselves as we really are.

The Stochastic Resonance Program (Part 1)

10 May, 2014

guest post by David Tanzer

At the Azimuth Code Project, we are aiming to produce educational software that is relevant to the Earth sciences and the study of climate. Our present software takes the form of interactive web pages, which allow you to experiment with the parameters of models and view their outputs. But to fully understand the meaning of a program, we need to know about the concepts and theories that inform it. So we will be writing articles to explain both the programs themselves and the math and science behind them.

In this two-part series, I’ll explain this program:

Stochastic resonance.

Check it out—it runs on your browser! It was created by Allan Erskine and Glyn Adgie. In the Azimuth blog article Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek give a nice explanation of the idea of stochastic resonance, which includes some clear and exciting graphs.

My goal today is give a compact, developer-oriented introduction to stochastic resonance, which will set the context for the next blog article, where I’ll dissect the program itself. By way of introduction, I am a software developer with research training in computer science. It’s a new area for me, and any clarifications will be welcome!

The concept of stochastic resonance

Stochastic resonance is a phenomenon, occurring under certain circumstances, in which a noise source may amplify the effect of a weak signal. This concept was used in an early hypothesis about the timing of ice-age cycles, and has since been applied to a wide range of phenomena, including neuronal detection mechanisms and patterns of traffic congestion.

Suppose we have a signal detector whose internal, analog state is driven by an input signal, and suppose the analog states are partitioned into two regions, called “on” and “off” — this is a digital state, abstracted from the analog state. With a light switch, we could take the force as the input signal, the angle as the analog state, and the up/down classification of the angle as the digital state.

Consider the effect of a periodic input signal on the digital state. Suppose the wave amplitude is not large enough to change the digital state, yet large enough to drive the analog state close to the digital state boundary. Then, a bit of random noise, occurring near the peak of an input cycle, may “tap” the system over to the other digital state. So we will see a probability of state-transitions that is synchronized with the input signal. In a complex way, the noise has amplified the input signal.

But it’s a pretty funky amplifier! Here is a picture from the Azimuth library article on stochastic resonance:

Stochastic resonance has been found in the signal detection mechanisms of neurons. There are, for example, cells in the tails of crayfish that are tuned to low-frequency signals in the water caused by predator motions. These signals are too weak to cross the firing threshold for the neurons, but with the right amount of noise, they do trigger the neurons.


Stochastic resonance, Azimuth Library.

Stochastic resonance in neurobiology, David Lyttle.

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally formulated in terms of systems that are bistable — where each digital state is the basin of attraction of a stable equilibrium.

An early application of stochastic resonance was to a hypothesis, within the framework of bistable climate dynamics, about the timing of the ice-age cycles. Although it has not been confirmed, it remains of interest (1) historically, (2) because the timing of ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current research.

In the bistable model, the climate states are a cold, “snowball” Earth and a hot, iceless Earth. The snowball Earth is stable because it is white, and hence reflects solar energy, which keeps it frozen. The iceless Earth is stable because it is dark, and hence absorbs solar energy, which keeps it melted.

The Milankovitch hypothesis states that the drivers of climate state change are long-duration cycles in the insolation — the solar energy received in the northern latitudes — caused by periodic changes in the Earth’s orbital parameters. The north is significant because that is where the glaciers are concentrated, and so a sufficient “pulse” in northern temperatures could initiate a state change.

Three relevant astronomical cycles have been identified:

• Changing of the eccentricity of the Earth’s elliptical orbit, with a period of 100 kiloyears

• Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears

• Precession (swiveling) of the Earth’s axis, with a period of 23 kiloyears

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce climate state changes. In more recent Milankovitch theories, a deterministic forcing mechanism is used. In a theory by Didier Paillard, the climate is modeled with three states, called interglacial, mild glacial and full glacial, and the state changes depend on the volume of ice as well as the insolation.


Milankovitch cycle, Azimuth Library.

Mathematics of the environment (part 10), John Baez. This gives an exposition of Paillard’s theory.

Bistable systems defined by a potential function

Any smooth function with two local minima can be used to define a bistable system. For instance, consider the function V(x) = x^4/4 - x^2/2:

To define the bistable system, construct a differential equation where the time derivative of x is set to the negative of the derivative of the potential at x:

dx/dt = -V'(x) = -x^3 + x = x(1 - x^2)

So, for instance, where the potential graph is sloping upward as x increases, -V'(x) is negative, and this sends X(t) ‘downhill’ towards the minimum.

The roots of V'(x) yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The latter separates the basins of attraction for the stable equilibria.

Discrete stochastic resonance

Now let’s look at a discrete-time model which exhibits stochastic resonance. This is the model used in the Azimuth demo program.

We construct the discrete-time derivative, using the potential function, a sampled sine wave, and a normally distributed random number:

\Delta X_t = -V'(X_t) * \Delta t + \mathrm{Wave}(t) + \mathrm{Noise}(t) =
X_t (1 - X_t^2) \Delta t + \alpha * \sin(\omega t) + \beta * \mathrm{GaussianSample}(t)

where \Delta t is a constant and t is restricted to multiples of \Delta t.

This equation is the discrete-time counterpart to a continuous-time stochastic differential equation.

Next time, we will look into the Azimuth demo program itself.

2014 on Azimuth

31 December, 2013


Happy New Year! We’ve got some fun guest posts lined up for next year, including:

Marc Harper, Relative entropy in evolutionary dynamics.

Marc Harper uses ideas from information theory in his work on bioinformatics and evolutionary game theory. This article explains some of his new work. And as a warmup, it explains how relative entropy can serve as a Lyapunov function in evolution!

This includes answering the question:

“What is a Lyapunov function, and why should I care?”

The brief answer, in case you’re eager to know, is this. A Lyapunov function is something that always increases—or always decreases—as time goes on. Examples include entropy and free energy. So, a Lyapunov function can be a way of making the 2nd law of thermodynamics mathematically precise! It’s also a way to show things are approaching equilibrium.

The overall goal here is applying entropy and information theory to better understand the behavior of biological and ecological systems. And in April 2015, Marc Harper and I are helping run a workshop on this topic! We’re doing this with John Harte, an ecologist who uses maximum entropy methods to predict the distribution, abundance and energy usage of species. It should be really interesting!

But back to blog articles:

Manoj Gopalkrishnan, Lyapunov functions for complex-balanced systems.

Manoj Gopalkrishnan is a mathematician at the Tata Institute of Fundamental Research in Mumbai who works on problems coming from chemistry and biology. This post will explain his recent paper on a Lyapunov function for chemical reactions. This function is closely related to free energy, a concept from thermodynamics. So again, one of the overall goals is to apply entropy to better understand living systems.

Since some evolutionary games are isomorphic to chemical reaction networks, this post should be connected to Marc’s. But there’s some mental work left to make the connection—for me, at least. It should be really cool when it all fits together!

Alastair Jamieson-Lane, Stochastic cross impact balance analysis.

Alastair Jamieson-Lane is a mathematician in the master’s program at the University of British Columbia. Very roughly, this post is about a method for determining which economic scenarios are more likely. The likely scenarios get fed into things like the IPCC climate models, so this is important.

This blog article has an interesting origin. Vanessa Schweizer has a bachelor’s degree in physics, a masters in environmental studies, and a PhD in engineering and public policy. She now works at the University of Waterloo on long-term decision-making problems.

A while back, I met Vanessa at a workshop called What Is Climate Change and What To Do About It?, at the Balsillie School of International Affairs, which is in Waterloo. She described her work with Alastair Jamieson-Lane and the physicist Matteo Smerlak on stochastic cross impact balance analysis. It sounded really interesting, something I’d like to work on. So I solicited some blog articles from them. I hope this is just the first!

So: Happy New Year, and good reading!

Also: we’re always looking for good guest posts here on Azimuth, and we have a system for helping you write them. So, if you know something interesting about environmental or energy issues, ecology, biology or chemistry, consider giving it a try!

If you read some posts here, especially guest posts, you’ll get an idea of what we’re looking for. David Tanzer, a software developer in New York who is very active in the Azimuth Project these days, made an organized list of Azimuth blog posts here:

Azimuth Blog Overview.

You can see the guest posts listed by author. This overview is also great for catching up on old posts!

Azimuth Blog Overview

6 September, 2013

We’ve got lots of series of articles on this blog. Some people say it’s a bit overwhelming. So David Tanzer of the Azimuth Project had a good idea: create an organized list of the articles on this blog, to make them easier to find. Here it is:

Azimuth Blog overview.

You can also find a link to this on top of the “ALSO READ THESE” list at the right-hand side of this blog!

Needless to say, this could be improved in many ways. Don’t say how: just do it!

What To Do? (Part 2)

28 August, 2013

Dear John,

If you could do anything to change the world what would you do? Many people haven’t had the opportunity to ponder that question because they have been busy studying what could be possible within a particular set of resource constraints. However, what if we push the limits? If all the barriers were removed, then what would you do?

The XXXXXXXXX Foundation has an open, aggressive, and entrepreneurial approach to philanthropy. Our goal is to produce substantial, widespread and lasting changes to society that will maximize opportunity and minimize injustice. We tap into the minds of fearless thinkers who have big, bold, transformational ideas, and work with them to invest in strategies designed to solve persistent problems.

Our team is reaching out to you because we believe you are the type of innovative thinker with ideas that just might change the world. While this is not a promise of grant funding, it is an invitation to share your ideas. You can learn more about the XXXXXXXXX Foundation by visiting our website. Thank you for your interest and I look forward to hearing your ideas.


I got this email yesterday. While I have some ideas, I really want to make the most of this chance. So: what would you do if you got this sort of opportunity? To keep things simple, let’s assume this is a legitimate email from a truly well-meaning organization—I’m checking that, and it seems to be so. Assume they could spend 1-10 million dollars on a really good idea, and assume you really want to help the world. What idea would you suggest?

Some ideas

Here are some comments from G+ to get your thoughts going. Heather Vandagriff wrote:

Hard core grassroots organization toward political involvement and education on climate issues. 

Jason Holt wrote:

Ideas are cheap.

Borislav Iordanov wrote:

I don’t agree that ideas are cheap. It could take a lifetime to have a really good one. However, one could argue that really good ideas are probably already funded. But if to maximize opportunity and to minimize injustice is the motivation, I say government transparency should be top priority. I can google the answer to almost any technical or scientific question, any historical fact, or pop culture, you name it. But I can’t know what my government is doing. And I’m not talking only, or even mostly, about things that governments hide. I’m talking about mundane day-to-day operations that are potentially not conducted in the best interest of the people, knowingly or unknowingly. I can easily find what are the upcoming concerts or movies, but it’s much harder to find out what, for instance, my local government is currently discussing so I can perhaps stop by the commissioner chamber and have my voice being heard (why aren’t there TV commercials about the public hearing of the next city ordinance?).

I realize this is not a concrete idea, but there are plenty of projects in that direction around the internet. And I don’t think such projects should come only from within government agencies because there is a conflict of interest.

Bottom line is that any sustainable, permanent change towards a better society has to involve the political process in some way, and the best (peaceful!) way to enact change there starts with real and consequential openness. Didn’t expect to write so much, sorry…

John Baez wrote:

Borislav Iordanov wrote:

But if to maximize opportunity and to minimize injustice is the motivation, I say government transparency should be top priority.

That’s a great idea… and in fact, this foundation already has a project to promote government transparency. So, I’ll either need to come up with a specific way to promote it that they haven’t thought about, or come up with something else.

Noon Silk wrote:

I guess the easy answer is some sort of education program; educating people in some way so-as to generate the skills necessary to do the thing that you really want to do. So I don’t know. Perhaps part of it could be some sort campaign to get a few coursera et al courses on climate maths, etc, and building some sort of innovative and exciting program around that.

Richard Lucas wrote:

Use existing corporate law (thanks, Capitalists!) to create collectives (maybe non-profits?) into which people could elect to participate. Participation would be contingent upon adoption of a certain set of standards for behavior impossible in the broader, geographical society in which we are immersed. Participants would enjoy a guaranteed minimum income, health care, etc – the goals of Communism, but in a limited scope, applied to participants who also exist in the general society. It’s just that participants would agree to share time, resources, and expertise with the collective. If collective living can’t be made to work in such an environment, where participation could be relatively selective up front, to include the honest and the committed…. well, then it can’t work. When the right formula is established, and the standard of living for participants is greater than for peers who are not “participants”, then you can expect more people to join. A tipping point would eventually be reached, where the majority of citizens in the broader, geographical society were also participants in an optional, voluntary, supersociety which does not respect geographic or national boundaries.

This is the only way it will work, and the beauty is that Communists and Objectivists equally hate this idea, because it breaks their frames, and because it is legal, and because if the larger society tried to block it, they would then have to justify the ability of crazy UFO cults and religions to do it. So, it can’t be stopped. There’s no theory to defend. You just do it.

Xah Lee wrote:

put the $10M to increase internet penetration, or in other ways enhance communication such as cell phone.

absolutely do not do anything that’s normally considered as good or helpful to humanity. such as help africa, women, the weak, the cripple, poor, vaccine, donation, institutionalized education etc.

even though, i’m still doubtful there’d be any improvement of humanity. $10M is like a peanut for this. One missile is $10M… 

John Baez wrote:

Xah Lee wrote:

even though, i’m still doubtful there’d be any improvement of humanity. $10M is like a peanut for this.

There are certain activities where the benefit is roughly proportional to the amount of money spent – like, giving people bed-netting that repels malaria-carrying mosquitos, or buying textbooks for students. For such activities, $10 million is often not enough to get the job done.

But there are other activities where $10 million is the difference between some good self-perpetuating phenomenon starting to happen, and it not starting to happen. This is the kind of thing I should suggest.

It’s the difference between pushing a large rock up a long hill, and starting a rock rolling down a hill.

By the way, this foundation plans to spend a lot more than $10M in total. I just want to suggest a project that will seem cheap to them, to increase the chance that they actually pursue it.

Piotr Migdal wrote:

I think that the thing that needs a drastic change in the education system. I suggest founding a “hacker university” (or “un-university”).

The educational system was designed for preparation of soldiers and factory workers. Now the job market is very different, and one cannot hope to work in one factory for his/her lifetime. Additionally, the optimal skill set is not necessarily the same for everyone (and it changes, as the World changes). But the worst thing is that schools teaches that “take no initiative, just obey” which stops working once one needs to find a job. Plus, for more creative tasks usually the top-down approach is the worst one (contrasting with the coordination tasks).

While changing the whole system may be hard, let’s think about universities; or a… un-university. Instead of attending predefined classes, let’s do the following:
• based on self-learning,
• lectures are because someone is willing to give them,
• everything voluntary (e.g. lectures and classes),
• own projects highly encouraged, starting from day one.

So basically, a collection of people who actually want to learn (!= earn a degree / prestige / position / fame), perform research which they consider the most fascinating (not merely doing science which is currently most fashionable and well-funded or “my advisor/grant/dean told so”) and undertake projects for greater good (startup-like freedom (unexperienceable in the current academia, at least – for the young) for things not necessarily giving monetary profit).

Sure, you may argue that there are more important goals (unemployment, bureaucracy, poverty, wars, ongoing destruction of natural environment – to name only a few in no particular order). But this one can be a nucleus for solving many other problems – wider in education and in general. And such a spark may yield in an unimaginable firestorm (a bad metaphor, it has to be about creation) seed can grow, flourish and make deserts blossom.


By founding I don’t mean paying for administration. Quite opposite – just rent a building, nothing more (so no tuition and no renting cost for students, to make it accessible regardless of the background). Almost all stuff (e.g. admission) in the first years based entirely on voluntary work.

John Baez wrote:

Noon Silk wrote: “I guess the easy answer is some sort of education program…”

That sounds good. The foundation already has a program to improve K-12 education in the United States. So, when it comes to education, I’d either need to give them ideas they haven’t tried in that realm, get them interested in education outside the US, or get them interested in post-secondary education. Piotr Migdal’s idea of a ‘hacker university’ might be one approach. It also seems the potential of free worldwide online courses has not yet been fully exploited. 

Piotr Migdal wrote:

The point is in going well beyond online courses (which, IMHO, are nice but not that revolutionary – textbooks are there for quite a few years; I consider things like Usenet, Wikipedia and StackExchange way more impactful for education) – by gathering a critical mass of intelligent and passionate people. But anyway, it may be the right time for innovations in education (and not only small patches).

Robert Byrne wrote:

Firstly, thanks for sharing this John! Secondly, congratulations on being chosen!

I would look into three aspects of this. 1) Who funds it, and whether you are comfortable with that, 2) do they choose candidates and generally have processes that make use of the experience of similar organizations such as MacArthur?, 3) what limits are there on using the grant — could you design your own prize to solve a problem using these funds?

But you’ve asked for ideas. The biggest problems that can be fixed/improved for $5 million! I’ll stick to education and technology. Here are some areas:
• Education reform in the U.S., think-tanks or writers who can create a model to switch away from municipal public education funding, with the aim of reducing disadvantage,
• Office, factory and home power efficiency technology, anything that needs $1 million to get to prototype,
• Solve the commute/car problem — e.g. how can more people work within the suburb in which they live? How can public transit be useful in sprawling suburbs?

John Baez wrote:

Robert Byrne wrote:

Firstly, thanks for sharing this John! Secondly, congratulations on being chosen!

Thanks! I’ve been chosen to give them ideas.

“I would look into three aspects of this. 1) Who funds it, and whether you are comfortable with that, 2) do they choose candidates and generally have processes that make use of the experience of similar organizations such as MacArthur?, 3) what limits are there on using the grant — could you design your own prize to solve a problem using these funds?”

Thanks – I definitely plant to look the gift horse in the mouth. They didn’t say anything about giving me a grant, except to say “this is not a promise of a grant”.

So, right now I’m treating this as an exercise in coming up with a really good idea that I’m happy to give away and let someone try. Naturally there’s a self-serving part of me that wants to pick an idea where my participation would be required. But knowing me, I’ll actually be happiest if I can catalyze something good in a limited amount of time and then think about other things.

“Solve the commute/car problem — e.g. how can more people work within the suburb in which they live? How can public transit be useful in sprawling suburbs?”

My wife Lisa raised this one. I would love to do something about this. But what can be done for just 1-10 million dollars? To do something good in this field with that amount of money, it seems we’d need to have a really smart idea: something where a small change can initiate some sort of chain reaction. Any specific ideas?

And so on…

In some ways this post is a followup to What To Do (Part 1), so if you haven’t read that, you might want to now.

Prospects for a Green Mathematics

15 February, 2013

contribution to the Mathematics of Planet Earth 2013 blog by John Baez and David Tanzer

It is increasingly clear that we are initiating a sequence of dramatic events across our planet. They include habitat loss, an increased rate of extinction, global warming, the melting of ice caps and permafrost, an increase in extreme weather events, gradually rising sea levels, ocean acidification, the spread of oceanic “dead zones”, a depletion of natural resources, and ensuing social strife.

These events are all connected. They come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as a permanent condition. Deep changes will occur as these idealizations bring us crashing into the brick wall of reality. If we do not muster the will to act before things get significantly worse, we will need to do so later. While we may plead that it is “too difficult” or “too late”, this doesn’t matter: a transformation is inevitable. All we can do is start where we find ourselves, and begin adapting to life on a finite-sized planet.

Where does math fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions in mathematics. Starting near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals and geometry. Centuries later, the Enlightenment and Industrial Revolution brought us calculus and eventually a flowering of mathematics unlike any before. Now, as the 21st century unfolds, mathematics will become increasingly driven by our need to understand the biosphere and our role within it.

We refer to mathematics suitable for understanding the biosphere as green mathematics. Although it is just being born, we can already see some of its outlines.

Since the biosphere is a massive network of interconnected elements, we expect network theory will play an important role in green mathematics. Network theory is a sprawling field, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays an important role here, both because it has a network structure—think of networks of logic gates—and because it provides the means for simulating networks.

One application of network theory is to tipping points, where a system abruptly passes from one regime to another. Scientists need to identify nearby tipping points in the biosphere to help policy makers to head off catastrophic changes. Mathematicians, in turn, are challenged to develop techniques for detecting incipient tipping points. Another application of network theory is the study of shocks and resilience. When can a network recover from a major blow to one of its subsystems?

We claim that network theory is not just another name for biology, ecology, or any other existing science, because in it we can see new mathematical terrains. Here are two examples.

First, consider a leaf. In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it.

The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost.

Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.

A growing network

Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.

A second example comes from stochastic Petri nets, which are a model for networks of reactions. In a stochastic Petri net, entities are designated by “tokens” and entity types by “places” which hold the tokens. “Reactions” remove tokens from their input places and deposit tokens at their output places. The reactions fire probabilistically, in a Markov chain where each reaction rate depends on the number of its input tokens.

A stochastic Petri net

Perhaps surprisingly, many techniques from quantum field theory are transferable to stochastic Petri nets. The key is to represent stochastic states by power series. Monomials represent pure states, which have a definite number of tokens at each place. Each variable in the monomial stands for a place, and its exponent indicates the token count. In a linear combination of monomials, each coefficient represents the probability of being in the associated state.

In quantum field theory, states are representable by power series with complex coefficients. The annihilation and creation of particles are cast as operators on power series. These same operators, when applied to the stochastic states of a Petri net, describe the annihilation and creation of tokens. Remarkably, the commutation relations between annihilation and creation operators, which are often viewed as a hallmark of quantum theory, make perfect sense in this classical probabilistic context.

Each stochastic Petri net has a “Hamiltonian” which gives its probabilistic law of motion. It is built from the annihilation and creation operators. Using this, one can prove many theorems about reaction networks, already known to chemists, in a compact and elegant way. See the Azimuth network theory series for details.

Conclusion: The life of a network, and the networks of life, are brimming with mathematical content.

We are pursuing these subjects in the Azimuth Project, an open collaboration between mathematicians, scientists, engineers and programmers trying to help save the planet. On the Azimuth Wiki and Azimuth Blog we are trying to explain the main environmental and energy problems the world faces today. We are also studying plans of action, network theory, climate cycles, the programming of climate models, and more.

If you would like to help, we need you and your special expertise. You can write articles, contribute information, pose questions, fill in details, write software, help with research, help with writing, and more. Just drop us a line.

This post appeared on the blog for Mathematics of Planet Earth 2013, an international project involving over 100 scientific societies, universities, research institutes, and organizations. They’re trying to have a new blog article every day, and you can submit articles as described here.

Here are a few of their other articles:

The mathematics of extreme climatic events—with links to videos.

From the Joint Mathematics Meetings: Conceptual climate models short course—with links to online course materials.

There will always be a Gulf Stream—and exercise in singular perturbation technique.


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