Blog - prospects for a green mathematics (Rev #6, changes)

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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, blog by John Baez and David A. TanzerJohn Baez and David Tanzer*

It is increasingly clear that we are witnessing the~~ beginning~~ start of a series of unfortunate environmental events. The problems include habitat loss, an increased rate of extinction, global warming, the melting of ice~~ melting,~~ caps~~ permafrost~~ and~~ melting,~~ permafrost,~~ sea~~ an~~ level~~ increase~~ rise,~~ in extreme weather events,~~ wars,~~ gradually~~ peak~~ rising~~ oil,~~ sea~~ loss~~ levels,~~ 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~~ spread~~ crisis,~~~~ and~~~~ the~~~~ accumulation~~ of~~ toxins.~~ oceanic “dead zones”, a depletion of natural resources, and ensuing social strife.

These are not separate problems. They all come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as the norm. Deep changes will~~ be~~ occur~~ required~~ as~~ to~~ these~~ stabilize~~ simplified~~ and~~ views~~ regenerate~~ bring us crashing into the~~ environment,~~ brick~~ but,~~ wall~~ unfortunately,~~ of reality. If we do not muster the~~ political~~ will~~ for~~ to~~ such~~ take~~ difficult~~ action~~ changes~~ before~~ may~~~~ not~~~~ be~~~~ mustered~~~~ until~~ things get significantly~~ worse.~~ worse,~~ Whether~~ we will need to so later. While we may plead that it~~ comes~~ is~~ sooner,~~ “too difficult” or~~ later,~~ “too~~ the~~ late”,~~ recovery~~ this~~ of~~ won’t~~ an~~ matter:~~ injured~~~~ biosphere~~~~ will~~~~ be~~~~ recognized~~~~ as~~~~ the~~~~ top~~~~ priority~~~~ for~~~~ continued~~~~ social~~~~ development.~~~~ This~~~~ will~~~~ –~~~~ and~~~~ does~~~~ –~~~~ pose~~ a~~ great~~ transformation~~ challenge~~ is~~ for~~ inevitable.~~ science~~ All~~ and~~~~ its~~~~ twin,~~~~ mathematics.~~~~ At~~~~ the~~~~ Azimuth~~~~ project,~~ we~~ want~~ can do is start where we find ourselves, and begin adapting to~~ get~~ life~~ started~~ on a finite-sized planet.*now*~~ to help with the mathematical foundations for this science project, which is bound to become socially urgent as things progress.~~

~~ Historically,~~ Where does mathematics fit into all this? While the biggest issues facing us are cultural, major transformations in~~ mathematics~~ society have always caused and been~~ linked~~ helped~~ with~~ along by major transformations in~~ society.~~ mathematics.~~ For~~ Starting~~ example,~~ near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals~~ in~~~~ the~~~~ Middle~~~~ East~~~~ was~~~~ intertwined~~~~ with~~~~ the~~~~ agricultural~~~~ revolution~~ and geometry. Centuries later, the~~ development~~ Industrial Revolution brought us calculus, and eventually a flowering of~~ trade.~~ mathematics~~ For~~ unlike~~ convenience,~~ any~~ contracts~~ before.~~ were~~ Now,~~ represented~~ as the 21st century unfolds, mathematics will become increasingly driven by~~ little~~ our~~ clay~~ need~~ tokens,~~ to~~ stored~~ understand~~ in~~ the~~ sealed~~ biosphere~~ envelopes.~~ and our role within it.

We refer to mathematics suitable for understanding the biosphere as *green mathematics*. It is just being born, but we can already speculate on what it will be like.

~~ Later,~~ Since~~ these~~~~ tokens~~~~ were~~~~ represented~~~~ by~~~~ marks~~~~ on~~ the~~ outsides~~ biosphere is a massive network of~~ the~~ interconnected~~ envelopes.~~ elements,~~ Eventually~~ it~~ these~~ is~~ marks~~ plausible~~ evolved~~ that~~ into~~~~ the~~~~ Babylonian~~~~ numeral~~~~ system.~~~~ 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~~~~ to~~~~ understand~~ network~~ the~~ theory~~ biosphere~~~~ and~~~~ our~~~~ role~~~~ within~~~~ it~~~~ .~~ will play an important role in green mathematics. Network theory is a sprawling field of investigation, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays a specially important role, for it is both a network-theoretic structure — e.g. computations that are defined by networks of logic gates — and the means by which network dynamics can be*simulated* and studied.

~~ We~~ One~~ refer~~ application~~ to~~~~ the~~~~ social~~~~ enterprise~~ of~~ developing~~ network~~ mathematics~~ theory~~ suitable~~ is~~ for~~~~ understanding~~ the~~ biosphere~~ study~~ as~~ oftipping points*green mathematics*~~ .~~ , through which a system abruptly passes from one regime to another. It is critical for scientists to identify nearby tipping points in the biosphere, to inform policy makers and guide their decisions. Scientists need mathematicians and statisticians to develop ways of analyzing data to detect incipient tipping points, and find the best ways to head off catastrophic changes. Another key area is the study of shocks and resilience. When can a system recover from a major blow to one of its subsystems?

~~ Since~~ We~~ the~~ claim~~ biosphere~~ that~~ is~~~~ a~~~~ massive~~ network~~ of~~ theory~~ interconnected~~~~ elements,~~~~ it~~ is~~ plausible~~ not~~ that~~ just another name for biology, ecology, or any other existing science, because in it we see the outlines of~~ network~~ new~~ theory~~ mathematical terrains~~ ~~ .~~ will~~ We~~ play~~ illustrate~~ an~~ with~~ important~~ two~~ role~~ recent~~ in~~ developments.~~ 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*~~. The study of network complexity, and the related study of the computational complexity of network simulations, are therefore important aspects of network theory. Computation itself is also partner to network theory, for it is both a network-theoretic structure – e.g. computations that are defined by networks of logic gates – and the means by which network dynamics can be ~~*simulated*~~ and therefore approached in an experimental manner.~~

~~ One~~ First,~~ major~~ consider~~ 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~~~~ identify~~~~ nearby~~~~ tipping~~~~ points,~~~~ in~~~~ order~~~~ to~~~~ inform~~~~ policy~~~~ makers~~~~ and~~~~ guide~~~~ their~~~~ decisions.~~~~ In~~~~ turn,~~~~ scientists~~~~ need~~~~ mathematicians~~~~ and~~~~ statisticians~~~~ to~~~~ analyze~~~~ environmental~~~~ data.~~~~ Another~~~~ key~~~~ area~~~~ is~~~~ the~~~~ study~~~~ of~~~~ shocks~~~~ to~~~~ systems.~~~~ When~~~~ can~~ a~~ system,~~ leaf.~~ or~~ In~~ an~~~~ organism,~~~~ recover~~~~ from~~~~ a~~~~ major~~~~ blow~~~~ to~~~~ one~~~~ of~~~~ its~~~~ subsystems?~~The Formation of a Tree Leaf by Qinglan Xia, we see what could be the secret key to one of Nature’s algorithms: the growth of the veins in a leaf. The vein system, which is a network for transporting nutrients and other substances, is modeled by a directed graph—a “tree” in this case—where nodes are cells, and edges are “pipes” that connect them. Each cell generates a “revenue” of energy, and it incurs the cost of transporting substances between it and the base of the leaf.

~~ We~~ The~~ claim~~ total~~ now~~ transport~~ that~~ cost depends on the network~~ theory~~ structure. There is~~ not~~ a~~ just~~ cost~~ another~~~~ name~~ for~~ biology,~~ each~~ etc.,~~ pipe,~~ because~~ and a cost for the turning of the fluid around the bends in~~ it~~~~ we~~~~ see~~ the~~ outlines~~ network. For each pipe, the cost is proportional to its length, times its cross-sectional area to some power α, times the number of~~ new~~ cells that get “fed” through this pipe. The exponent α captures the savings from using a thicker pipe to transport materials together in parallel. There is also another parameter β that measures the cost of each bend in a pipe.*mathematical terrains*~~. We illustrate with two recent developments.~~

~~ First,~~ Development~~ consider~~ proceeds~~ a~~ through~~ leaf.~~ cycles of growth and network optimization. In each stage of growth, a new layer of cells gets added, consisting of all potential cells that would give a revenue exceeding the cost of bringing fluid to it. During optimization, 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 give realistic models of various types of natural leaves.~~The Formation of a Tree Leaf~~~~, by Qinglan Xia, we see what could be the secret key to one of Nature’s algorithms – the growth of the veins in a leaf. The vein system, which is a transport network for nutrients and other substances, is modeled by a directed graph – a “tree” in this case – where nodes are cells, and edges are “pipes” that connect them. Each cell generates a “revenue” of energy, and it incurs the cost of transporting substances between it and the base of the leaf.~~

The total transport cost depends on the network structure. There is a 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 parameter, and Flow is proportional to the number of cells that get “fed” through this pipe. Alpha captures the 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. There is another parameter, Beta, for the cost of turning the fluid.

Development proceeds through cycles of growth and network optimization. In a growth iteration, a new layer of cells gets added, consisting of all potential cells that would give a positive net revenue. During optimization, 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 give realistic models of various types of natural leaves.

Unlike approaches~~ such~~ that~~ as~~ merely~~ L-systems~~ create~~ which~~ pretty~~ use~~~~ an~~~~ imaginative~~~~ model~~~~ to~~~~ generate~~ images of plants,~~ Xia~~ Xia’s~~ takes~~ approach~~ a~~~~ biological~~~~ approach,~~~~ which~~ is based on a~~ plausible~~ simple but illuminating model of how plants actually work. Moreover, it is a*network-theoretic* approach to a biological subject, and it is *mathematics*~~ ~~ —replete~~ –~~~~ replete~~ with lemmas, theorems and~~ algorithms~~ algorithms—from~~ –~~~~ from~~ start to finish.

Here is another illustration that network dynamics is an area for mathematical investigation.~~ It~~~~ pertains~~~~ to~~~~ stochastic~~~~ Petri~~~~ nets,~~~~ which~~~~ are~~~~ a~~~~ model~~~~ for~~~~ networks~~~~ of~~~~ reactions.~~ A~~ network~~~~ contains~~~~ “tokens”,~~~~ that~~~~ represent~~~~ entities,~~~~ and~~~~ places,~~~~ which~~~~ hold~~~~ the~~~~ tokens,~~~~ and~~~~ represent~~~~ entity~~~~ types.~~~~ Reactions~~~~ are~~~~ process~~~~ nodes~~~~ that~~~~ remove~~~~ tokens~~~~ from~~~~ their~~~~ input~~~~ places,~~~~ and~~~~ deposit~~~~ tokens~~~~ at~~~~ their~~~~ outputs.~~~~ The~~~~ reactions~~~~ proceed~~~~ concurrently,~~~~ and~~~~ generate~~~~ a~~~~ data-flow~~~~ of~~~~ tokens.~~~~ The~~~~ reaction~~~~ events~~~~ are~~~~ probabilistically~~~~ determined,~~~~ by~~~~ a~~~~ Markov~~~~ chain,~~~~ in~~~~ which~~~~ the~~~~ expected~~~~ firing~~~~ rate~~~~ of~~~~ a~~~~ reaction~~~~ depends~~~~ on~~~~ the~~~~ number~~~~ of~~~~ tokens~~~~ at~~~~ its~~~~ inputs.~~stochastic Petri net is a model for networks of reactions. A stochastic Petri net has “tokens”, which represent entities, and “places” which hold the tokens, and represent types of entities. “Reactions” remove tokens from their input places, and deposit tokens at their output places. The reactions proceed concurrently, and generate a flow of tokens. The reaction events occur probabilistically, by a Markov chain in which the expected firing rate of a reaction depends on the number of tokens at its inputs.

What is new is the discovery, reported in the Azimuth Network Theory series, that part of the quantum mathematics is *transferable* to the realm of stochastic Petri nets. The key idea, inspired from quantum mechanics, is to represent a probabilistic state by a power series. Here the monomials represent possible states of the network. There is one variable for each place in the network, and 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.

~~ Now~~ None~~ in~~ of this is new. The surprising part is that many techniques from quantum~~ mechanics,~~ field theory can be transferred to the~~ states~~ realm~~ are~~ of~~ represented~~ stochastic Petri nets. The key idea is to represent a stochastic state by a power~~ series~~ series.~~ that~~ Here~~ use~~ the~~ complex~~ monomials~~ coefficients.~~ represent~~ The~~ states~~ annihilation~~ in which there is a definite number of tokens in each place. There is one variable for each place in the network, and~~ creation~~ its exponent in the monomial indicates the number of~~ particles~~ tokens~~ are~~ stored~~ represented~~ there.~~ by~~ In a linear combination of these monomials, the coefficients represent probabilities.*operators*~~ over these power series. The first interesting result is that when these formal operators are applied to the stochastic states of a Petri net, they take on a meaning that is rooted in the annihilation and creation of ~~*tokens*~~ in the network. There is an annihilation operator, and a creation operator, for each place in the network.~~

~~ Next,~~ In~~ the~~ quantum~~ Hamiltonian~~ field~~ operator~~ theory,~~ for~~ states~~ the~~ are~~ Markov~~ often~~ chain~~ represented by power series with complex coefficients. The annihilation and creation of~~ a~~ particles~~ Petri~~ are~~ net~~ described~~ gives~~ by operators on the~~ probabilistic~~ space~~ law~~ of~~ motion~~~~ for~~~~ the~~~~ network.~~~~ It~~~~ is~~~~ an~~~~ operator~~~~ on~~ power~~ series,~~ series.~~ which~~ The first interesting result is~~ composed~~ that~~ using~~ when these operators are applied to the stochastic states of a Petri net, they describe the annihilation and creation~~ operators~~~~ for~~~~ the~~~~ places.~~~~ The~~~~ structure~~ of~~ this~~~~ composition~~~~ reflects~~~~ the~~~~ connections~~~~ in~~~~ the~~~~ Petri~~~~ net.~~~~ Moreover,~~~~ this~~~~ mathematics~~~~ shows~~~~ potential~~~~ as~~~~ a~~~~ framework~~ tokens~~ for~~~~ network~~~~ dynamics,~~~~ because~~ in~~ it~~~~ some~~~~ of~~ the~~ basic~~ network.~~ theorems~~ Remarkably,~~ about~~ the~~ network~~ commutation~~ equilibrium~~ relations~~ states,~~ between~~ are~~ annihilation~~ proven~~~~ in~~~~ a~~~~ compact~~ and~~ elegant~~ creation~~ way.~~ operators, often viewed as a hallmark of quantum theory, make perfect sense in this purely classical, probabilistic context.

Next, each stochastic Petri net gives a “Hamiltonian” describing the probabilistic law of motion for that networks. The Hamiltonian is an operator on power series built from annihilation and creation operators. The precise formula for the this operator depends on the reactions in the Petri net. Moreover, this approach lets us prove many theorems about stochastic Petri nets, already known to chemists, in a compact and elegant way.

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

~~ These~~ We~~ subjects~~ are~~ being~~ pursuing~~ pursued~~ these subjects in theAzimuth Project ,**Azimuth Project**~~ which~~~~ is~~ an open collaboration between mathematicians, scientists, engineers and~~ programers~~ programmers trying to help save the planet. We aim to present clear and accurate information on relevant~~ issues,~~ 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.~~Azimuth Wiki 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, either here or on theAzimuth Blog.

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