The Azimuth Project
Blog articles in progress (Rev #107)

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Contents

Tomi Johnson

Quantum network theory (part 1)

Quantum network theory (part 2)

Tim van Beek

Your model is verified, but not valid! Huh?

Summary: Critics of climate science in general, and sceptics of the statement that anthropogenic (man-made) global warming is real, often claim that climate models are wrong and cannot be trusted. But what does “wrong” mean here and what can and should be done to make models more trustworthy, from a software engineering viewpoint? In a first step, we’ll introduce some relevant technobabble.

Putting the Earth in a box

Summary: An introduction to energy balance models.

A quantum of warmth

Summary: A closer look at the heat balance of the Earth and its atmosphere. An explanation of downward longwave radiation (DLR), generated by the atmosphere of the earth and why it does not violate the second law of thermodynamics.

Eddy who?

Summary: A short introduction to turbulence: It is all about eddies.

Fluid flows and infinite dimensional manifolds (part 1)

Summary: Fluid flows can be modelled by one parameter subgroups of diffeomorphism groups. How diffeomorphism groups can be seen as infinite dimensional Riemannian manifolds, and how certain nonlinear partial differential arise as geodesic equations.

Fluid flows and infinite dimensional manifolds (part 2)

Summary: Explaining Euler’s equation as a geodesic equation of ideal incompressible fluids.

Fluid flows and infinite dimensional manifolds (part 3)

Summary: Comparing the Euler equation with the Navier-Stokes equation for incompressible fluids.

Fluid flows and infinite dimensional manifolds (part 4)

Summary: Explaining the Jacobi equation and how this can be used to put a bound on weather predictions.

Fluid flows and infinite dimensional manifolds (part 5)

Summary: Revisiting the Burgers equation as the simplest example of a nonlinear partial differential equation

Good vibrations

Summary: Recapitulation of previous blog posts and short tour to molecular quantum mechanics necessary to understand that radiation equations of the atmosphere of the earth.

The color of night

Summary: How big is the effect of downward longwave radiation (DLR) really? What do measurements say? What instruments are used? Are there alternative explanations of the 33 Kelvin gap of the zero dimensional energy balance model?

Increasing the signal-to-noise ratio with more noise

Summary: An introduction to stochastic resonance.

Matteo Smerlak

The mathematical origin of irreversibility

Summary: An introduction to the mathematical origin of irreversibility in nonequilibrium thermodynamics, including results such as the detailed fluctuation theorem and the integral fluctuation theorem.

Frederik De Roo

The log forcing

Summary: explains the logarithmic response of surface temperature to the rise in carbon dioxide

Background profiles in the atmosphere

Summary: some background material necessary for the log forcing, but independent so better separately.

Alastair Jamieson-Lane

Stochastic Cross-impact balance analysis: prediction, probability, and the pursuit of a snappier working title

Summary: An extension of cross impact balance analysis to stochastic model. Hopefully some fun examples.

Ken Webb

Connections: Petri nets, systems biology, and beyond

Summary: A brief but systematic exploration of various types of networks, and how they’re really all the same. Starting with a simple reaction network and corresponding Petri net, I demonstrate how to transform these into systems biology networks, systems of differential equations, Unified Modeling Language (UML) diagrams, mind maps, agent based models, and more. Along the way I introduce third-party tools that are able to process each of the resulting formats. This first part restricts itself to the transformation to differential equations.

Staffan Liljegren

Carbon cycle box models (part 1)

Carbon cycle box models (part 2)

Lorenz Borsche

On peak oil

Summary: An introduction to the topic of peak oil.

Curtis Faith

Curtis Faith on the Azimuth Project

Summary: Curtis Faith introduces himself and talks about why he decided to help out on the Azimuth Project.

Making decisions under uncertainty

Summary: Groups often want to make the right decisions. So they spend a lot of time in the decision process itself. A better approach is to acknowledge when perfect decisions don’t exist and to incorporate the uncertainty itself into your plans.

Cameron Smith

Hierarchical organization and biological evolution (part 1)

Summary: An introduction to hierarchical systems, which asks why evolution favors the development of such systems. See also the old Wiki page: [[Blog - evolution and categories]. This is part of a larger attempt to review some of the literature on major transitions in evolution and multi-level selection, sketch a few connections to concepts in category theory, and discuss the potential for using experimental evolution to investigate and strengthen those connections.

Hierarchical organization and biological evolution (part 2)

Summary: An attempt to review some of the literature on major transitions in evolution and multi-level selection, sketch a few connections to concepts in category theory, and discuss the potential for using experimental evolution to investigate and strengthen those connections.

Hierarchical organization and biological evolution (part 3)

Summary: An attempt to review some of the literature on major transitions in evolution and multi-level selection, sketch a few connections to concepts in category theory, and discuss the potential for using experimental evolution to investigate and strengthen those connections.

David Tanzer

Petri net programming (part 1) – Basic Petri nets, with simulator code

Summary: Introduction to Petri nets, taking a hands-on approach. Contains a brief tutorial, then discussion of the programming approach, and ends with a small program to simulate a Petri net.

Petri net programming (part 2) – Introduction to stochastic Petri nets

Summary: Introduces the concept of stochastic Petri nets. Starts with a general discussion of reaction network dynamics, followed by an exposition of the mass action kinetics (and its limitations). This is followed by an exploration of the continuous deterministic limit of the stochastic model. Finally, I calculate equilibrium solutions for simple reaction networks.

Petri net programming (part 3) – The rate equation

Petri net programming (part 4) – Limits of the deterministic model

Petri net programming (part 5) – Markovian dynamics

Petri net programming (part 6) – Stochastic simulation algorithms

Blake Stacey

Invasion fitness in moment-closure treatments

Summary: Moment closures are a way of forgetting information about a system in a controlled fashion, in the hope that an incomplete, fairly heavily “coarse-grained” picture of the system will still be useful in figuring out what will happen to it. Sometimes, this is a justifiable hope, but in other cases, we are right to wonder whether all the algebra it generates actually leads us to any insights. Here, we’ll be concerned with a particular application of this technology: studying the vulnerability of an ecosystem to invasion. We shall find expressions for invasion fitness, the expected relative growth rate of an initially-rare species or variety.

David Tweed

Worried about the environment? You’re seeing things!

Summary: Thoughts about the disproportionate impact of pictures on human psychology.

This Week’s Finds

Week 309

Summary: Another application of Hopf bifurcations with noise, this time to predator-prey systems.

Week 314

Summary: The first part of an interview with Thomas Fischbacher.

Week 317

Summary: A sketchy introduction to glacial cycles and Milankovich cycles.

Week 318

Summary: A bit more detail on Milankovitch cycles.

Interview with Didier Paillard

Summary: John Baez interviewing Didier Paillard about his work on the glacial cycles.

Other

Azimuth Project news

Summary: What’s new on the Azimuth Project?

Stabilization wedges (part 5)

Summary: Pacala’s 2008 followup on the original Pacala-Socolow Stabilization wedges paper.

Mathematics of the Environment (part 3)

Summary: a discussion of the greenhouse effect, including the Schwarzschild equation and Beer-Lambert law.

Network theory

Network theory (part 1)

Summary: Networks, and diagrams of networks, show up in many branches of science. It would be nice to find a unified framework for these ideas.

Network theory (part 2)

Summary: One can adapt ideas from quantum field theory to describe the theory of stochastic Petri nets. The master equation versus the rate equation.

Network theory (part 3)

Summary: The rate equation of a stochastic Petri net, and applications to chemistry and infectious disease.

Network theory (part 4)

Summary: The master equation of a stochastic Petri net, and analogies to quantum field theory.

Network theory (part 5)

Summary: Analogies between quantum theory and probability theory.

Network theory (part 6)

Summary: Writing the master equation using annihilation and creation operators.

Network theory (part 7)

Summary: An example: the stochastic version of the logistic equation in terms of annihilation and creation operators, and how to obtain an equilibrium Poisson distribution.

Network theory (part 8)

Summary: A review of our work so far: how to get the rate equation and the master equation from a stochastic Petri net. A bit about Feynman diagrams.

Network theory (part 9)

Summary: A quantum field theory proof of the most exciting theorem in D. F. Anderson, G. Craciun and T. G. Kurtz’s paper Product-form stationary distributions for deficiency zero chemical reaction networks.

Network theory (part 10)

Summary: An example of a simple reversible reaction.

Network theory (part 11)

Summary: Brendan Fong proves a version of Noether’s theorem for Markov processes.

Network theory (part 12)

Summary: A comparison of quantum mechanics and stochastic mechanics.

Network theory (part 13)

Summary: A comparison of the quantum and stochastic versions of Noether’s theorem.

Network theory (part 14)

Summary: The Desargues graph and its role in chemistry.

Network theory (part 15)

Summary: More on the Desargues graph; graph Laplacians and how they generate Markov processes and also quantum processes.

Network theory (part 16)

Network theory (part 17)

Network theory (part 18)

Summary: The rate equation for a diatomic gas: an example of many of the concepts we’ve seen.

Network theory (part 19)

Summary: The master equation for a diatomic gas: an example of many of the concepts we’ve seen.

Network theory (part 20)

Summary: A discussion of the Perron-Frobenius theorem and its role in understanding Dirichlet operators.

Network theory (part 21)

Summary: A deeper look at the concept of ‘deficiency’, in preparation for proving the deficiency zero theorem.

Network theory (part 22)

Summary: A reformulation of the rate equation, in preparation for proving the deficiency zero theorem.

Network theory (part 23)

Summary: A study of Markov processes, in preparation for proving the deficiency zero theorem.

Network theory (part 24)

Summary: Proof of the deficiency zero theorem.

Network theory (Biamonte guest posts)

Summary: Jacob Biamonte on stochastic Petri nets and chemical reaction networks: material that might go into posts on the Azimuth blog.

Network theory (Fong guest posts)

Summary: Brendan Fong on stochastic Petri nets and chemical reaction networks: material that might go into posts on the Azimuth blog.

Contributions to external blogs

Prospects for a green mathematics

category: blog, meta