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This page is a blog article in progress, written by John Baez and David Tanzer. To discuss this article while it’s being written, visit the Azimuth Forum.

contribution to the MPE 2013 blog, by John Baez and David A. Tanzer

Green mathematics for the era of environmental jeopardy

It is increasingly clear that we are witnessing the beginning of a series of unfortunate environmental events. The problems include global warming, ice melting, permafrost melting, sea level rise, extreme weather events, wars, peak oil, loss of habitats, mass extinctions, deforestation, soil erosion, acid rain, air pollution, famines, pathogen mutations, antibiotic resistant diseases, epidemics, ocean acidification, ocean dead zones, radioactive waste, ozone depletion, the water crisis, and the accumulation of toxins.

Deep changes will be required to stabilize and regenerate the environment, but, unfortunately, the political will for such difficult changes may not be mustered until things get significantly worse. Whether it comes sooner, or later, the recovery of an injured biosphere will be recognized as the top priority for continued social development. This will – and does – pose a great challenge for science and its twin, mathematics. At the Azimuth project, we want to get started now to help with the mathematical foundations for this science project, which is bound to become socially urgent as things progress.

Historically, major transformations in mathematics have been linked with major transformations in society. For example, the birth of written numerals in the Middle East was closely intertwined with the agricultural revolution and the development of trade. For convenience, contracts were represented by little clay tokens, stored in sealed envelopes.

Later, these tokens were represented by marks on the outsides of the envelopes. Eventually these marks evolved into the Babylonian numeral system. Many Centuries centuries later, the Industrial Revolution was deeply connected to breakthroughs in mechanics and calculus. Now, as the 21st Century unfolds, mathematics will become increasingly driven by the great social need tounderstand the biosphere and our role within it.

We refer to the social enterprise of developing mathematics suitable for understanding the biosphere as green mathematics.

Since the biosphere is a massive network of interconnected elements, it is plausible that network theory will play an important role in green mathematics. Network theory is a sprawling field of investigation, just starting to become organized, which combines ideas from graph theory, systems theory, biology, ecology and sociology. Networks such as the biosphere display massive complexity . Hence network theory finds an a important partner in the the science of complexity itself, which can now be seen in the particular forms like such as computational complexity theory, chaos theory, and organizational complexity theory. Computation itself is a partner to network theory, for it is both a network-theoretic structure – think e.g. for instance of computations being that are defined by networks of logic gates – and the means by which network dynamics can besimulated and therefore approached in an experimental manner.

One major application of network theory is the study of “tipping points,” through which there is an abrupt passage from one dynamic regime to another. It is critical for environmental scientists to be able to identify nearby tipping points, in order to inform policy makers and guide their decisions. in This turn, is scientists a may mathematical call topic. on mathematicians to make in-depth analysis of environmental time-series data.

Another key area is the study of shocks to systems. When is can it possible for a system, or an organism, to recover from a major blow to one of its subsystems? What are the repair processes that take place when a tree, or a worm, is cut?

Next we assert that the network theory is not just another name for biology, etc., because in it we are see seeing the outlines of newmathematical terrains . Here We we illustrate describe with two recent developments developments. in mathematical network theory.

First, consider a leaf. In The Formation of a Tree Leaf , by Qinglan Xia, we see what might could be the secret key to one of Nature’s algorithms – the formation growth of the vein veins patterns in a tree leaf. The system vein of system, veins, which is a transport network for nutrients and other substances, is modeled by a directed graph – a “tree” in this case – where the nodes are cells, and the edges are the “pipes” that connect the them. cells. Each Every cell generates a “revenue” of energy energy, from and photosynthesis, but it also incurs the cost of transporting substances between it from and the base of the leaf leaf. (along with flows in the opposite direction). The net revenue from a cell is the excess of its energy revenue over its transport cost.

The total transport cost of fluid transportation depends on the network structure. There is a transport cost for each pipe, and a cost for the turning of the fluid around the bends in the network. For each pipe, the cost equals Length * (CrossSectionalArea ^ Alpha) * Flow, where Alpha is a model parameter, and Flow is proportional to the total number of cells that get “fed” through this pipe. Alpha captures the cost savings from using a thicker pipe to transport materials together in parallel. Similarly, it is more efficient to pack many letters onto a single mail truck, and transport them together along the common part of their paths. The There explanation of the cost for the bends, which is more another involved, parameter, uses Beta, a parameter for the penalization cost due of to turning bending. the fluid.

Development proceeds through alternate cycles iterations of growth and network optimization. In a growth iteration, a new layer of cells gets added, consisting of all potential cells, cells which, that when adjoined to the network, would have give a positive net revenue. In During an optimization, optimization step, local adjustments are made to the transport graph, to find a local minimum of the cost function. Remarkably, by varying the two parameters, the simulations can give produce realistic models of various types of natural leaves, leaves. such as oak and maple.

As Unlike compared to approaches such as L-systems which merely use an imaginative model to generate somewhat realistic images of plants, the work by Xia takes a truly biological approach, because which it is based upon on a plausible model of how plants actually work. Moreover, it is anetwork-theoretic approach to a biological subject, and it is mathematics – replete with lemmas, theorems and algorithms – from start to finish.

Here is another illustration, illustration which shows that the dynamics of a network dynamics is an area for basic research of a mathematical character. investigation. It pertains to stochastic Petri nets, which are a simple model for reaction networks networks, that applies to a broad range of systems, reactions. including A chemical network reaction contains networks, “tokens”, predator-prey that networks, represent entities, and vending places, machines. which hold the tokens, and represent entity types. Reactions are represented by process nodes that consume remove tokens at from their input places, and generate deposit tokens at their outputs outputs. Each place represents a type of entity, and each token stands for an instance of that type. The reactions proceed concurrently, and generate a data-flow of tokens tokens. through The reaction events are probabilistically determined, by a Markov chain, in which the net. expected In firing rate of a reaction depends on the number of tokens at its inputs.stochastic Petri net, the reaction events are probabilistically determined by a Markov chain where the expected firing rate of a reaction is a function of the number of tokens at its inputs.

What is new is the discovery, reported in the Azimuth Network Theory series series, of the Azimuth project, that some part of the mathematics quantum mechanics mathematics istransferable to the classical realm of stochastic Petri nets. The key step idea, here inspired from quantum mechanics, is to represent a probabilistic state of a Petri net a distribution which assigns a probability to each potential configuration of a Petri net by a power series. A Here power series representation (with complex coefficients) is also used for quantum states. The quantum particle annihilation and creation operators are represented as operators on these power series. Notably, when these formal operators are applied to the stochastic monomials represent possible states of a the Petri network. net, There they is have one a variable meaning for that each rooted place in the annihilation network, and creation its exponent in the monomial indicates the number of tokens stored there. The coefficient of the monomial gives the probability of being in that state.tokens by the reactions of the network.

For Now a in given quantum Petri mechanics, net, the Hamiltonian states operator are for represented its by Markov chain, which expresses the probabilistic law of motion for the network as an infinitesimal operator on the power series, series is constructed as a composition of functions that include use the complex coefficients. The annihilation and creation operators for the species (places) in the network. The structure of this particles composition are is represented given by as a function of the connectivity structure of the Petri net. Furthermore, this new and more abstract mathematics for Petri nets shows potential as aframeworkoperators , because over some these of power series. The first interesting result is that when these formal operators are applied to the fundamental stochastic theorems states about of network equilibrium states, can be proven in a compact Petri net, they take on a meaning that is rooted in the annihilation and elegant creation way of using this operator math.tokens in the network. There is an annihilation operator, and a creation operator, for each place in the network.

We Next, take these results as further evidence that the life Hamiltonian operator for the Markov chain of a network, Petri net gives the probabilistic law of motion for the network. It is an operator on power series, which is composed using the annihilation and creation operators for the networks places. The structure of life, this are composition brimming reflects with the mathematical connections content. in the Petri net. Moreover, this mathematics shows potential as aframework for network dynamics, because in it some of the basic theorems about network equilibrium states, are proven in a compact and elegant way.

All Conclusion: The life of these a subjects network, are and being actively pursued in the networks of life, are brimming with mathematical content.Azimuth Project, which is an open collaboration between mathematicians, scientists, engineers and programers trying to help save the planet. Our goal is to present clear and accurate information on the relevant issues, and to help people work together on our common problems. On the Azimuth Wiki, we are explaining the main environmental and energy problems the world faces today. We are studying plans of action, network theory and climate cycles. We are investigating climate models and how to program them. Already we have developed some interactive climate models that run in a web browser. See the Azimuth Forum for active discussions on all of these topics.

If These you subjects are interested being pursued in helping the out, that would be great. 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.Azimuth Project, which is an open collaboration between mathematicians, scientists, engineers and programers trying to help save the planet. We aim to present clear and accurate information on relevant issues, and to help people work together on our common problems. On the wiki, we are explaining the main environmental and energy problems the world faces today. We are studying plans of action, network theory, climate cycles, and the programming of climate models. 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.

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