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.

Click to read more ...

Aphorismically Consistent

Oscar Wilde Action FigureOne of the wonderful things about words and sentences is the ability to create paradoxical statements that are more clever than they are troubling. No, verbal paradoxes aren't typically offensive in the same way that mathematical ones are. Mathematical statements are usually not labeled "paradoxical;" instead, they are described as contradictions, or, even worse, inconsistent statements. (Try to think of a paradoxical mathematical statement that isn't a contradiction - go ahead, I dare you)

Mathematical inconsistencies are often nasty, unwanted, theorem-killers, and their presence is usually a sign of things gone awry. Actually, inconsistency and/or incompleteness is built into all of the mathematics we do - sort of. After all, Gödel showed that mathematics cannot be both consistent and complete.

Ahh, but words are a different story. Words are the atoms of sentences, and if somehow the sentence doesn't make "sense", well, consider metal atoms that are frozen into a glassy state, but in fact want to be in a nice, regular matrix. These metallic glasses are then "inconsistencies."

Click to read more ...

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:

Click to read more ...

Flaming Symmetric Fractals

Apophysis Fractal Flame (click to enlarge)I've seen the flames - fractal flames - and they are amazing. Originally an outgrowth of the ideas developed in Symmetry in Chaos by Field and Golubitsky, fractal flames are related to the Chaos Game because they are created by tracking the iterates of starting points as they are mapped into other points. As in the Chaos Game, the iterates reveal, over time, the structure of the attractor associated with the map. Flame fractals use a combination of non-linear maps and very inventive approaches to coloring/visualization.

Flame mathematics is interesting enough on its own, but the intrigue of flames is the ability to generate images of surreal organic beauty. In the following excerpt from The Fractal Flame Algorithm by Scott Draves for the Cosmic Recursive Fractal Flames site, aesthetics are essential:

Click to read more ...

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.

Click to read more ...

Turbulence in Space

Turbulence in SpaceTrying to model and understand turbulence is one of the main thrusts of chaos theory. So it may be a good thing or bad, depending on where you are chaotically, that more turbulence has been found - this time in deep space.

As reported in APS Physics News for 2006:

If you think chaos is complicated in the case of simple objects (such as our inability to predict the long-term velocities and positions of planets owing to their nonlinear interactions with the sun and other planets) it's far worse for systems with essentially an infinite number of degrees of freedom such as fluids or plasmas under the stress of nonlinear forces. Then the word turbulence is fully justified. Turbulence can be studied on Earth easily by mapping such things as the density or velocity of fluids in a tank. In space, however, where we expect turbulence to occur in such settings as solar wind, interstellar space, and the accretion disks around black holes, it's not so easy to measure fluids in time and space. Now, a suite of four plasma-watching satellites, referred to as Cluster, has provided the first definitive study of turbulence in space. The fluid in question is the wind of particles streaming toward the Earth from the sun, while the location in question is the region just upstream of Earth's bow shock, the place where the solar wind gets disturbed and passes by the Earth's magnetosphere. The waves in the shock-upstream plasma, pushed around by complex magnetic fields, are observed to behave a lot like fluid turbulence on Earth. One of the Cluster researchers, Yasuhito Narita (y.narita@tu-bs.de) of the Institute of Geophysics and Extraterrestrial Physics in Braunschweig, Germany, says that the data is primarily in accord with the leading theory of fluid turbulence, the so called Kolmogorov's model of turbulence. (Narita et al., Physical Review Letters, 10 November)

Kolmogorov is one of the famous trio Kolmogorov - Arnold - Moser. after whom the KAM theorem is named. Ironically, the KAM theorem shows the existence of quasi-periodic orbits in a chaotic solar system. The idea of stability within turbulence is an archetypal chaos construct.

Modeling Pandemic Strategies

Spanish Flu in SpokaneModeling how a disease progresses in a pandemic, and the related modeling of the effects of different strategies on stopping the pandemic, are, next to perhaps nuclear attack modeling, some of the most sensitive mathematics being done today.

Consider the difficulties of determining pandemic-containment strategies by looking to past pandemics.

Efforts of several cities to halt the spread of the 1918 Spanish flu have now been analyzed and modeled by several research teams. One technique that appears promising is "social distancing" - referred to a s a non-pharmaceutical intervention (NPI) - a fairly obvious strategy of reducing the potential contact between members of the community by closing schools, churches, stores, etc.

I wrote "fairly obvious" - but is it? There are so many contingencies that affected each city that it is hard to draw conclusions. Consider the report of the studies as described by Maryn McKenna of the Center for Infectious Disease Research & Policy at U. Minnesota

But while NPIs make intuitive sense, actual evidence for their ability to block or slow flu transmission has been limited. An Institute of Medicine report released last December concluded that the measures might help in a pandemic but should not be oversold.

"It is almost impossible to say that any of the community interventions have been proven ineffective," the report said. "However, it is also almost impossible to say that the interventions, either individually or in combination, will be effective in mitigating an influenza pandemic."

Click to read more ...

Mathematics Reveals the Artistry

Brueghel's Fall of IcarusDaniel Rockmore writes about the mathematical analysis of art in the June 2006 Chronicle (The Style of Numbers Behind a Number of Styles). In the essay Rockmore describes Richard Taylor's work in analyzing Jackson Pollock pieces, which may be forgeries. ( See my post on this topic)

The Pollock intro is a lead in to a description of stylometry - the mathematical/scientific analysis of literary texts that attempts to address issues of authorship. (See Bookish Math, an excellent intro to stylometry by Erica Klarreich for Science News Online.) Rockmore than describes a method he developed with co-workers Siwei Lyu and Hany Farid that uses wavelet analysis to determine unique "signatures" of different artists - in effect a stylometry for visual images.

The actual mathematics of the wavelet approach can be found in A Digital Technique for Art Authentication. Here the authors use examples of Pieter Bruegel and Perugino to test their model. They claim that their "techniques, in collaboration with existing physical authentication, to play an important role in the field of art forensics."

The wavelet technique is different from Taylor's fractal analysis of Pollock's works, but both are examples of stylometry applied to visual information. Both Taylor and Rockmore are attempting to quantify art, an activity that Rockmore admits is unsettling/impossible to some. According to Taylor, this quantification should be expected: "Both mathematics and art are all about pattern...it would be unusual that you would not apply mathematical analysis to the question."

Rockmore is more explicit about what mathematical categorization of art analysis does not do: "Fractal analysis doesn't diminish Pollock's athleticism and movement, nature and turbulence, chaos and beauty; it reveals and amplifies it."

For more on this topic, see Can Mathematical Tools Illuminate Artistic Style?, by Sara Robinson for SIAM.

A Myth of Gaussian Proportions

When I was a freshman in high school, my home room teacher gave us a very nasty assignment during an after-school detention session - to calculate 35 to the 35th power!

This assignment was particularly cruel and unusual punishment because there were no such things as calculators back in 1967.

What I really needed was something I didn't know about until college: a closed-form solution.

Finding closed form expressions for partial sums is a standard calculus exercise. The ur-example of this type of problem is the sum of the first n integers, which is easily shown to be n(n+1)/2.

This closed form expression collapses (n-1) operations into three. Because it yields an exact answer, it is not really a predictor, but, in a sense, it is a model of a process.

When this example is done in a calculus class, a typical accompanying story is how young Gauss solved this problem in record time, totally showing up the teacher who had given out the onerous task of adding the first 100 integers. (The version I always heard was that this was a punishment because the students had been particularly noisy that day. The sadistic mathematical punishments of my high school teacher certainly lends credence to this tale.)

Click to read more ...

On the Mathematical Nature of the World

I had one of those very rewarding teaching moments yesterday in my General Physics lab.  Some students and I got into a discussion of just what is physics, what is the connection between mathematics and physics, and whether the world is itself mathematical.

The nature of the world as mathematical is a common theme in the Chaos and Fractals course.  I have developed a seminar module  that has one main reading, and I keep finding supplemental readings every time I teach the course. I list a few of these here in order to collect them in one place - for future renditions of the course, and as a post that will hopefully generate some debate from interested readers.

The main reading is John Barrow's The Mathematical Universe in which he poses the question "The orderliness of nature can be expressed mathematically. Why?"  This article is an excellent summary of the main schools of mathematical philosophy - realism, inventionism, formalism, and constructivism.

Anyone interested in deeper views of  mathematical realism must read The Unreasonable Effectiveness of Mathematics in the Natural Sciences.  This 1960 article by Eugene Wigner is definitely the  "mother-of-all mathematics and the world"  arguments for realism.  In it he describes the uncanny connection between mathematics developed as part of pure, formal, abstract systems and physical observation, which naturally leads to ontological questions on the  mathematical nature of the world:

Click to read more ...