Cabbage Leaves and Temporal Fractals

Fractal tumor on Wild Cabbage LeafI have always considered fractals in time to be related to self-similar music (such as a nested fugue), or just a plain-old self-similar time-series, such as stock market fluctuations, or the corn price fluctuations at the Chicago Mercantile Exchange, whose fractal nature was first noted by Mandelbrot.

Now there's a different way to consider time-fractals - proposed by Carlos Escudero and colleagues of the Institute for Mathematics and Fundamental Physics in Madrid, in their Dynamic Scaling of Non-Euclidean Interfaces

Escudero "performs calculations of the dynamic scaling (how a surface changes in space and over time at several different scales) of growing structures, such as the kind of semiconductor films used in the microchip industry where, even under the most carefully controlled of conditions, rough (non-Euclidean) geometries can exist. He found that the moment-by-moment behavior of the surfaces are strongly effected by the fractal geometry."

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An Absorbing Collision

CO₂-collision/absorbtionBelief in global warming, and especially the causes of GW (if one believes the data) depend crucially on modeling. The physics of atmospheric gases-solar radiation interactions, especially those involving carbon-dioxide molecules, is of major importance because the increase of CO₂ is often quoted as a correlate to warming. The story is basically that CO₂ absorbs some of the infrared radiation (IR) streaming to the earth from the sun, and reflects the rest back.

Just how much is absorbed? The answer to this question is a crucial one. Until recently, the basic physics of light absorption by gas molecules, though pretty well understood, doesn't get the amounts of IR absorption correct for atmospheric CO₂. Is this a failure of physics, or the model used?

Get serious. Of course it's not the physics. To paraphrase, it's the model , stupid. Physicists decide what goes into a model, and then the physics (in the form of fundamental laws) takes over, yielding the model prediction.

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Systems Chemistry

The B-Z reaction underway...Recently, R. Frederick Ludlow and Sijbren Otto, both at Cambridge, published a paper in Chemical Society Reviews calling for a new type of chemistry. Titled Systems Chemistry, their approach "deals with the emergent properties of interacting chemical systems or networks. In other words, properties that result from the interaction between the components in a network, rather than any one species acting individually."

Systems Chemistry is a different way of looking at patterns that emerge in space and time because of the complex interplay among/between constituent reactants and reactions. (Not surprisingly, the Belousov-Zhabotinsky reaction is a canonical example of the complex results of a complex system.)

Complex chem systems are either under thermodynamic (equilibrium) or kinetic control.

The authors point out that chemical systems are good models for certain biological systems, and make a rather bold prediction:

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DNA in Series and Parallel

quattro-colored dna pastaModeling biological systems sure seems to be radically different from modelling something such as a chain of balls on springs. For balls on springs, Newton's 2nd law is written for each mass, yielding a pretty straightforward system of differential equations. The positions and velocities of the masses in time are the solutions to the system. Each variable x(t) in any of the differential equations refers directly to the actual position of a specific mass. For bio models, however, modeling is done more at a meta-level using a systems approach. For example, you wouldn't normally see Newton II applied in pharmokinetic modeling; instead a compartment model where the compartments are body systems such as blood stream, gut, etc. is typically used, often with great predictive power.

What about population modeling, and especially modelling of interacting species? Is this closer to balls on springs, or a compartment model? The differential equations that are typically used to predict the behavior of these populations employ mathematical expressions of the interactions chosen to produce a desired population behavior. Predator-Prey, mutual competition, and cooperation models are really the same with just minor changes to the terms in a differential equation system. I then think of this type of modeling more like compartments - the interaction terms are plugged into the differential equations in a manner analogous to building models with compartments.

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Sci-Fi Economics and Intelligent Design

Over two years ago John Allen Paulos, Temple mathematician famous for his book Innumeracy, wrote a provocative op-ed that pointed out the analogies between evolution of organisms and the evolution of the free-market system. His point is that the complexity of a free market economy is not designed by anyone, and ism, in fact reasonably described by Adam Smith's Invisible Hand model. Moreover, in a delicious irony, it is often the adherents of such a free-market system that deny the possibility of biological evolution without Intelligent design.

Paulos does point out the limitations of the analogy, but his piece raises an intriguing question for me: starting from scratch, how would one re-design the free-market system? Is there any other imaginable way to do it? I ask this because, as a science fiction reader, I often read about alternate life forms that may exist on other planets. (e.g. silcon-based) While the fundamental molecular bases of life as we know it seems to lend itself to another structure, I can't imagine a different free-market economy. What would it look like?

Of course, the fact that I can't imagine an alternate free-market economy may only be a result of my woeful lack of knowledge of economic theory and models.

Or a lack of science fiction writing that focuses on the economics of alien civilizations, which is a perfect excuse for a new Ursula K. LeGuin novel to be titled The Invisible Left Hand of Darkness...

Modeling The Universe: The History of Cosmology

So much of the history of mathematics and science is encapsulated in the study of the heavens. One can argue that the first modeling may have been early views of the universe, and the planets riding on their celestial spheres in a cascade of epicycles. The American Institute of Physics has set up a wonderful site devoted to the history of cosmology that is a terrific resource for learning more about these models, and how our ideas of the solar system and universe have matured.

Titled Cosmic Journey: A History of Scientific Cosmology, the site is ddivided into two broad , complementary areas - History (e.g. The Greek Worldview, The Mechanical Universe, Big Bang) and Tools (The Naked Eye, The First Telescopes, Spectroscopy).

For good reason, the site devotes ample space to Harlow Shapley, whose pioneering work in 1916 on globular clusters and the real size of the heavens exploded our view of the universe and caused us to reappraise our position in it. Shapley write eloquently about how his discoveries, and the work of all of those before him, have necessarily changed our position as observers within the physical universe, and hence the way we model the universe and ourselves:

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Do E-fields Cause or Hinder Cancer?

Field near a TNT molecule. Click to enlargeWith the ongoing debate about potential health hazards of living near radio/TV towers, or the dangers of extensive cell-phone use, it is reassuring to see the latest news on the efficacy of electric fields in slowing down some cancers.  (Of course a skeptic might claim that the news is a spin job by cell operators to get activists from fighting the placement of towers and cell sites near population centers.)

Nevertheless, the study is intriguing because it describes a set of potential reasons why the E-fields might be the causal agent for the stunted cancer growth.

In vitro, the electric fields were seen to have two effects on the tumor cells.

First, they slowed down cell division. Cells that ordinarily took less than an hour to divide were still not completely divided after three hours of exposure to an electrical field of 200 kHz...

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Modeling Acts of God, Part 2 - Earthquakes

This is #2 in a series. (Click here for #1 on modeling hurricanes)

Earthquake modeling seems to generate a wide range of emotions, vitriol, successes, and failures - so wide as to need a logarithmic scale, a la Richter.

The Parkfield Earthquake Experiment, now running for over 22 years, has been the development test bed and experimental "lab" for US Geological Survey/State of California efforts to develop  physical models of earthquakes that will lead to viable predictions. The USGS site contains a wealth of information on the experiment, and good background on the history of earthquake prediction, which is still highly hit or miss. An interesting excerpt from the site neatly illuminates the need for prediction based on an understanding of physical causes as opposed to one based on statistical correlation only:

Early scientific efforts toward earthquake prediction in the U.S. were directed primarily toward the measurement of physical parameters in areas where earthquakes occur, including seismicity, crustal structure, heat flow, geomagnetism, electrical potential and conductivity, gas chemistry. Central to these efforts was the concept that a precursor might be observed in one or more of these measurements. However, the connection between a commonly accepted precursor and the earthquake was often speculative and uncertain. A coherent physical model was lacking.

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Constructal Theory of Everything?

A very interesting text that describes what appears to be a bold new approach to modeling many possible systems - even human ones - is about to be released. Titled Constructal Theory of Social Dynamics and  edited by Adrian Bejan and Gilbert W. Merkx, the text "brings together for the first time social scientists and engineers to develop a predictive theory of social organization, as a conglomerate of mating flows that morph in time to flow more easily (people, goods, money, energy, information). These flows have objectives (e.g., minimization of effort, travel time, cost), and the objectives clash with global constraints (space, time, resources). The result is organization (flow architecture) derived from one principle of configuration evolution in time (the constructal law): "for a flow system to persist in time, its configuration must morph such that it provides easier access to its streams."

Begin and Merkx are from Duke. The Duke press release  is very enticing - this is certainly a text that will serve as a reference for the chaos and fractals course, or perhaps a primary source.

Some tantalizing tidbits:

Why does a railway network look like a river? Why do the streets of old Rome look like a leaf? Because whether their shape is determined by the interactions of molecules or the choices made by individual humans, all of these systems of flow are governed by a relatively simple new principle of thermodynamics.

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In Search of the Fastest Rubik's Quark in the World

Magnetic/Acrylic Rubik by NTronicsThere's a great scene in the Pursuit of Happyness in which Christopher, the protagonist played by Will Smith, becomes infatuated with his son's Rubik's Cube, and ultimately goes on to be a Cube Solver. In a chance encounter with a stockbroker he is trying to impress, Chris/Will solves the cube - a feat that blows away the broker, and which leads to an interview, and then...I won't go on here - rent the movie, it is a good one, and based on a true story.

Back to the Cube. Jessica Fridrich of SUNY Binghamton, who completed a Ph.D. in non-linear dynamics (Removing observational uncertainty from orbits of nonlinear dynamical systems) is something of a cube speed freak. Consider that she won the First Czechoslovak Championship in Rubik's Cube in 1982. At the top of her game she " routinely solved the cube in an average time of 17 seconds...actively using more than 100 algorithms." You'll find some of her solution techniques and algorithms here. You'll also find links to other speeders, including sage advice on how to grease your cube.

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