ChapterZero

I haven’t done too much this weekend. I read the first of four sections in the chapter on tensors for the geometry course, and predictably got lost in a welter of details and notation. Did I mention the horridness of the official course text? I’m going to do the homework for that section tomorrow– if I find myself lost, I remember a book in the library that covers exactly the same material in a more friendly manner, so I can go reference that.

After some conversations with one of my roomies, an aerospace engineering grad student, I decided to take a look at continuum mechanics, and see if that’s a good choice for my required course external to the applied math department. There are only a few subjects I think would be worth the time and effort at this point in my career, when I should be focusing mainly on my research: analytical mechanics, continuum mechanics and fluid dynamics, relativity, and quantum mechanics. With the exception of relativity and quantum mechanics, these areas are tied to applied mathematics at the hip– a bunch of techniques and the attendant physical intuitions from fluid dynamics have made their way into the collective psyche of applied mathematics, and it’s next to impossible to go a couple of days in our department without hearing about a lagrangian or hamiltonian.

In fact, given the way physics terms– viscosity, stress tensor, lagrangian, boundary layers, advection, shocks, streamlines etc.– (a weak selection, but this is just off the top of my head) are tossed about by our professors, I’m surprised we don’t have a comprehensive physics for applied mathematics course in our department. There’s probably a mathematics of physics course somewhere on campus, but the emphasis in such courses is on applying mathematics to physics problems, as opposed to seeing how physical problems have motivated mathematical techniques and how artificial embeddings of problems in a physical context can assist tremendously in developing helpful intuitions. A subtle but important difference.

As for relativity and quantum mechanics, I just think they’re natural subjects to study. You know, being the two most important physical theories developed in the past century or so. Relativity is to me, one of the sexiest branches of physics, and quantum mechanics seems to be of growing importance in applied mathematics– Schrodinger’s equation seems to be becoming a motivational force in the same way, if to a lesser extent, that the Navier Stokes equations indisputably are. I’ll admit that I base this entirely on my person experiences, seeing a load of papers that somehow trace their motivation back to Schrodinger’s equation. I imagine that someday we’ll teach sympletic integration in our introductory courses on the numerical solution of ODEs.

All of which brings me, in a roundabout manner, to the original reason for this post: a link to the awesomeness that is the National Committee for Fluid Mechanics Films. I’ve only viewed part of the cavitation video (the segment showing the bubbles spun off of a propeller is achingly beautiful), but already I feel like pestering someone in the aero- or mechanical engineering departments to let me see a live hydrodynamics experiment.

Today was a fruitful day, in that I found two books that I’m really excited about. The first, Mechanics by Scheck, is apparently exactly the type of physics book I’ve been looking for: it starts off with Newtonian physics, develops the Lagrangian and Hamiltonian formulations and the whole idea of variational formulations of mechanics, and introduces manifolds and lie groups, etc. Of course, there are many books that purport to cover the same material, but this is the first that I feel is readable for someone who has only seen basic statics and kinetics, a couple of years ago; additionally, it has a reassuringly earthy feel to it. If it measures up to my first impression, this will be the next physics book I buy.

The second is a bit more unusual. I visited the geology library today (for the first time), to find material which might help in designing a landscape evolution system such as the one that I mentioned before. The amount of subjects that revealed themselves as relevant was daunting: structural geology, hydrology, soil mechanics, mineralogy, these are just a few of the geological subdisciplines that you’d need to be familiar with to construct a reasonably representative model of the physical phenomena involved in landscape generation. I wandered around hoping to find an introductory survey book that covered the broad strokes of these areas, but no luck. However, I did run across Fractal River Basins: Chance and Self-Organization, which contains some neat ideas on fractal/network models of river basins. The focus of the book (if I understand it correctly) is more on coming up with models that tell you something useful about the underlying processes, but the illustrations of artificial landscapes generated with some of the models are beautiful. Here’s a book review that appeared in Nature.

The ‘Alcubierre wave effect‘ was mentioned in Vacuum Diagrams– Stephen Baxter’s fictional history of the universe and the future of humanity, which I’m finding good reading so far. Looking it up on Wikipedia, I discovered that at least some serious physicists have given the topic of FTL serious consideration. It seems that a real warp-drive may be theoretically and practically impossible: most theories require the use of exotic matter to generate negative energy densities, and mind-boggling amounts of energy input, violate certain generalized uncertainty principles (the main sticking point), and don’t supply any hint of how one would apply them to the engineering of a working FTL propulsion system. It’s still exciting though! Enough to make me want to learn QM and GR.

Since we’re on the topic, I’d like to know what the science says on the feasibility of blackholes as methods of transport, ‘wormholes’. This is such a prevalent science fiction device that I often find myself nodding in familiarity when it’s mentioned in a book, but when I stop to think about it, I feel uncomfortable with their ubiquity. Even with my inferior popular science background in physics, I can identify some problems beside the obvious question of whether you can join two black holes (or a black hole and a white hole– do these things even theoretically exist?). For starters, there’re the tidal effects that would probably rip apart any vessel that entered the event horizon of the wormhole. Then there’s the question of how to stop the wormhole from eating everything in range. I wonder if there are any non-condescending popular science books on exotic applications of blackhole physics? Probably.

If you were ever wondering why H is used for entropy instead of say E, here’s one theory courtesy of Tom Carter’s lecture notes:

“The enthalpy is [often] written U. V is the volume, and Z is the partition function. P and Q are the position and momentum of a particle. R is the gas constant, and of course T is temperature. W is the number of ways of configuring our system (the number of states), and we have to keep X and Y in case we need more variables. Going back to the first half of the alphabet, A, F, and G are all different kinds of free energies (the last named for Gibbs). B is a virial coefficient or a magnetic field. I will be used as a symbol for information; J and L are angular momenta. K is Kelvin, which is the proper unit of T. M is magnetization, and N is a number, possibly Avogadro’s, and O is too easily confused with 0. This leaves S . . .” and H. In Spikes they also eliminate H (e.g., as the Hamiltonian). I, on the other hand, along with Shannon and others, prefer to honor Hartley. Thus, H for entropy . . .

Apparently Japanese scientists have worked out a way to store data in the DNA of a bacteria. They’re really on the ball when it comes to genetic engineering: they’ve also modified lettuce to produce miraculin, which may be the next great diet craze. Look for it in a store near you!

But seriously, imagine the possibilities that the informational bacteria open up. Assuming that the encoding and decoding processes can be streamlined sufficiently– it’s dubious that this will ever be feasible as a consumer technology, but one can hope– you could store information virtually invisibly, and diffuse it effortlessly. About a fourth of a megabyte of information can be geneered into a single bacteria.

This raises a lot of interesting math questions: if I wanted to ensure that the entirety of the Encylcopedia Brittanica was freely available within a square foot everywhere on the surface of the Earth (assuming the Earth is spherical, and that the bacteria could and would flourish on every surface, and each breed of bacteria carrying different information could coexist harmoniously with the other breeds, and that you addressed the problem of genetic drift corrupting the encoded information), how many geneered species of bacteria would I need, and given a population model for the bacteria, how would I perform optimum seeding? Or, if you’re into hard core stochastics, imagine coming up with models that could model the effects of genetic drift closely enough to facilitate the recovery of information corrupted by evolution/mutation.

Lots more questions I could think up, but the point is that this is so cool… and now I have to run home within 7 minutes to catch Heroes.

Schoolchildren from Caversham have become the first to learn a brand new theory that dividing by zero is possible using a new number - ‘nullity’.
…

Dr James Anderson, from the University of Reading’s computer science department, says his new theorem solves an extremely important problem - the problem of nothing.

“Imagine you’re landing on an aeroplane and the automatic pilot’s working,” he suggests. “If it divides by zero and the computer stops working - you’re in big trouble. If your heart pacemaker divides by zero, you’re dead.”

Nope, this is not an excerpt from the Onion, it’s from a BBC article.

There’s a wide range of physical phenomena out there, barring even quantum and relativistic effects, that are utterly fascinating. Take cavitation, for example– imagine that under non-esoteric conditions you can form bubbles that when they collapse radiate with black body termperature equal to the sun’s! Almost enough to make you want to be a physicist, or a mechanical engineer. But then, you realize you’d rather just daydream about cool ways to exploit those phenomena than do the grunt work of gathering data and formulating theories to explain and properly harness them.

I just got back from the latest departmental colloqium talk, given this week by Richard Tsai from UT Austin. It was a nice talk, about path planning and visibility optimization, reminiscent more of the style of computer scientists’ or engineers’ talks than mathematicians; that is, he used nice pictures and got his point across with few references to equations for justification. Of course, this was in large part possible because of his subject matter, so I can’t say it was better than the talks before it, which flew over my head… just that I enjoyed it more.

In the course of agonizing over my stochastic processes midterm, I realized that a really good book on probability would be one that approached it from applications to gambling, the way the field originally developed, and took it from the basics all the way up through its modern, starvedly rigorous heights, all the while applying it to the solution of interesting/challenging problems. I.e., it’s slightly useful when you motivate martingales by mentioning their usefulness in modeling fair games, but it’s enlightening when you show me that usefulness. Anyone know of any such book?

One of my friends is going to be working at the Office for Naval Research this summer, using FPGAs to implement a parallel processing system specifically for implementing an algorithm for analyzing hyperspectral images. He’s attempting to read through a paper on the algorithm, “Exploiting Manifold Geometry in Hyperspectral Imagery”. Naturally, I got curious: what exactly is hyperspectral imaging, and what does it have to do with manifolds?

It seems that hyperspectral imaging refers to the gathering of many (hundreds) of remote images of an area representing its reflectance at difference wavelengths. Each pixel in a hyperspectral image has a reflectance spectrum associated with it, and materials can theoretically be identified by their characteristic reflectance spectrums; e.g., the mineral hematite strongly absorbs visible light. These pure spectra are called endmembers. Practically, because of the spatial resolution of the images taken (maybe 20 square meters per pixel when a satellite at an altitude of 20km is used to take the image), each pixel contains a mixture of material, so each pixel represents a mixture of endmembers. If you’re lucky, this is a macroscopic mixture, in which each photon only interacts with one material, so the resulting pixel is a linear combination of endmembers. More often, due to shadowing, atmospheric interference, the spectrum of the illumination source (probably the sun), the orientation of the surface imaged relative to the illumination source, etc., the mixture is intimate (nonlinear).

The problem is, given exemplar endmembers, to accurately classify the materials in the image. I’m not clear on the approaches being used, but it seems they all can be divided into linear and non-linear models, like Principal Components. It seems the manifold model described in the paper falls into the non-linear category, and can be used not only to segment images, but also to compress them. I think that the point at which manifolds come in is that each material can be represented as a cloud of points in some high dimensional space, corresponding to interclass variance, and then these clouds can be modeled as manifolds. Something in that vague direction.

I know I’ve kvetched on this point before, but now I have more hard statistics (source: “Mathematics in Public” opinion piece by Richard Schaar, Notices of the AMS Vol52):

• 53% of incoming college students will take remedial mathematics or English courses; over half will never graduate.
• 56% of engineering Ph.D.’s earned at U.S. universities in 2000 went to foreign nationals.
• Between 1995 and 1999, engineering degrees awarded in China increased 37%; in the US they declined 20%.

I have a personal anecdote to the second statistic: in the grad. Stochastic process class I’m taking, 70% of the class is Indian, 30% is Chinese, and there is one white guy and one black guy. I’m not surprised, at all, by the last one– in fact, I’m surprised the discrepancy isn’t higher. But, more than 50% of our students don’t graduate? That’s ridiculous.

Why is all of this so? Borrowing from Richard Hofstadter, I think there are two reasons, both manifestations of the ever-present American anti-intellectualist tendency: 1) unreflective instrumentalism, and 2) unrestrained hedonism. The first is the idea that abstract ideas are no good, only practical skills are relevant; accordingly, math and science are only interesting (rather, tolerable) at a so-called practical level, which precludes research. The second is the consumer culture mentality that says basically, work is good only for getting money to buy the things you want. So why spend the time and effort getting a technical PhD when you could just go into, say, business?

America’s in a sad state. We’re so damned lazy!