ChapterZero

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.

Possibly relevant posts:

• Landscape generation (7/9/2007)
• Markov Random Fields, Wavelets, and Terrain Generation (8/20/2005)
• Physical phenomena (10/30/2006)

This has got to be the most pretentious (auto?)biographical sentence that I’ve seen in a while:

When not immersed in science, he relaxes by searching for Wagner’s leitmotifs, musing over Kandinsky’s chaos, and contemplating Wittgenstein’s inner thoughts.

I can (barely) forgive you for thinking it, but not for writing it!

Source: the “about the author” blurb on an attempt to introduce Wolfram’s New Kind of Science to the benefits of rigorous mathematics.

Possibly relevant posts:

• Winter Reading (12/4/2006)
• The scientific method is not magical (10/9/2008)
• Wolfram and Hart (3/1/2007)

Artificial terrain/landscape generation is one of those topics which perennially fascinates me. Since high school I’ve been intermittently looking at the various algorithms used to generate terrains for games, architectural proofs, and computer-generated art. All of the ones I’ve seen documented have been variations on fractal techniques, or applications of Perlin noise generators– all methods attempting to imitate the appearance of the end product of millions of years of natural processes. Surely someone has attempted to go ahead and simulate the actual processes? The lack of responses to my google searches indicate otherwise.

My meager knowledge of geological processes (gained mostly from watching a few PBS specials about rivers) informs me that the most important factors in the development of a landscape are: tectonic activity, wind erosion and deposition, water activity (precipitation, river sedition and erosion, ocean weathering, etc.), and climate. Sure there are other factors, e.g. human activity, but these are the most important. To simulate these processes accurately is a formidable task, but rough approximations should be sufficient to generate realistic landscapes.

I have a few very rough proposals on how to go about simulating these processes. First, using a mesh/heightfield model would be very inconvenient for simulating all of these processes except for climate and tectonic activity. On the other hand, a volumetric model is amenable to all of these processes. So, one could simulate the first process, the tectonic activity that will provide the unweathered bones of the landscape, using wavelets (an earlier post on the rationale) or some other fractal or multiresolution process. I prefer a multiresolution technique, because at least theoretically, you might be able to start from a random process model for the coefficients of the particular type of landscape you’re aiming for– flat, mountainous, what type of interfaces or faults are present, what type of minerals are in the rocks.

A simple multiresolution/fractal approach is probably insufficient for the incorporation of certain weathering effects — perhaps the climate, which affects the landscape more globally than locally, might be drafted into such a random process model, but it’s stretching credibility to believe that e.g. river flow patterns could be convincingly added into the same model. Maybe you could use a separate model, but then determining how the two models should interact (e.g. ensuring a river flows up and then down a mountain) would be nontrivial.

Instead, maybe a good approach for weathering is to (very coarsely) simulate the physics of the weathering processes. Here you’d have to take into account such things as the type and disposition of the rocks in the landscape, and maybe use various evolution PDEs with tunable parameters to simulate weathering. For example, for the rivers, an initial parameter could be the velocity of the water, a random starting point at a relatively high location would be chosen, and some PDE (one that tracks local geodesics, for example) used to ensure that the river took the path of least resistance to an appropriate lower point (branching if appropriate), and portions of the river’s course could be ‘eroded’ depending upon the hardness of the material present. Subtleties such as the deposition of bars (important because this factors in the formation of meanders and ox-bows) could be handled with auxiliary PDEs. This process could be iterated to simulate the passage of years. Similarly, a prevailing wind pattern could be established, and used to drive a PDE that gouges appropriate amounts of material from the landscape.

An ambitious undertaking, but conceptually simple; devilish only in the design/coarsening process for the PDEs. The main disadvantage of this approach — that it isn’t a one shot process — means that it couldn’t be used in real-time applications, but the advantages: extensibility — more weathering processes could be layered on –, realism, and support for nontrivial geometries such as caves, suggest that it could be useful in off-line applications.

Possibly relevant posts:

• Fractal River Basins (7/12/2007)
• Markov Random Fields, Wavelets, and Terrain Generation (8/20/2005)
• Research in CG? (7/3/2007)

Some of my favorite blogs for getting social and political commentary are written by women, partly because of their perspective from outside of the patriarchal universe of discourse (ok, I just really wanted to drop that classic in), and partly because they cover different issues than your generic sociopolitical commentary site, and are often humorous (read sarcastic). Pandagon’s a good starting point.

Possibly relevant posts:

• Blog Rolling (Rolo-dexing) (2/21/2003)
• “The trouble with Islam” (4/22/2007)
• Self-referential, gratuitous posting (7/26/2001)

Found these two problems @ JD2718:

• Given n 6-sided dice, what’s the most probable sum?
• Consider a system of a circle and a cubic. What’s the largest number of intersections possible? Construct a system that realizes this number.

Stumped on the first– the straightforward approach I can see involves counting the number of ways to write an integer as sums of 1 through 6, or the integer partition function, which doesn’t have a closed form that I’m aware of. But even if it has a closed form, that’d be way beyond a reasonable level of knowledge for this problem.

My current, stalled approach is to let and find a relationship between by considering ‘one-step transformations’ that take an element in one and give elements (the plurality is a beotch) in the other. That’s tedious and I’m not sure it’s worth it.

I console myself with the thought that this is a nonlinear integer optimization problem, so it’s not unreasonable to expect it not to have an easy, or even closed-form, solution.

As for the second, I’m thinking of how to approach it — I think I knew how to do it once upon a time — but in the meantime, my answer’s provisionally 5.

Possibly relevant posts:

• Equivalence relations (3/27/2005)
• Problem solving for non-geniuses (12/18/2004)
• Lattice Problem (2/16/2005)

I’m not a big fan of religion, and not at all of fundamentalism, but today I heard a quote on NPR from a Salafist that resonated with me: “Islam is about virtue, not violence.” Quite on point: that’s a sentiment that every religion should follow.

Of course, here’s the thing: just as the Salafi movement gave rise to jihadist groups, including Al Qaeda, and fundamentalism has given rise to some of the nastiest American domestic and foreign politics, it’s a slippery slope from believing yourself virtuous to justifying using any means to spread that virtue. And this descent into self-righteousness isn’t limited to the realm of religion; hence we have: extraordinary rendition and domestic wire-tapping and secret courts and the ‘Patriot Act’ in the War on Terror, gutted sex-ed programs.

Possibly relevant posts:

• on my arrogance (12/8/2001)
• The NannieBot Story (3/23/2004)
• More in the tradition of the Patriot Act (3/2/2007)

I’m depressed, because I faced the fact that I have no fucking life. I spent yesterday eating a tub of ice cream while reading online fantasy stories; happy 4th to me, huh? Damn it, I need something more.

All I can say is that my life is pretty plain
I like watchin' the puddles gather rain
...
And I don't understand why I sleep all day
And I start to complain that there's no rain
And all I can do is read a book to stay awake
And it rips my life away, but it's a great escape
escape......escape......escape......
All I can say is that my life is pretty plain
ya don't like my point of view
ya think I'm insane
Its not sane......it's not sane

Possibly relevant posts:

• my toe’s in trouble (12/30/2001)
• Had a Bad Day Again (3/24/2002)
• Modest Mouse “Bukowski” (6/14/2008)

One possible avenue of research that looks promising is computer graphics. In particular, our multiresolution group looks to be doing some interesting work. I know from my undergraduate experience that I would be comfortable working with wavelets, splines, and such in a mathematical setting, but I’m not so sure about the discrete setting. However, I’ve always been interested in computer graphics– in the sense of reading about the algorithms used :)– and this group seems to have a strong investment in mathematically rigorous techniques, like the discrete differential geometry approach. The most attractive aspect of working with this group is that I’d get to indulge my interests in both multiresolution methods and differential geometry. So, after familiarizing myself with some of their work, I’ll contact the lead professor and see if he’s accepting grad students– my goal is to have this done by the end of next week.

Hmmm. Looks like I was conflating the multiresolution group with the Applied Geometry group, which is even more right up my alley.

Possibly relevant posts:

• Advisorship request sent out (8/31/2007)
• Winter term courses (11/19/2006)
• l₁ Magic (3/16/2006)

I mentioned this problem in the last post, and thought it might be fun to work through:

If for is continuous in the space of bases of an -dimensional space and is the isomorphism that maps , then is continuous.

Intuitively, this has to be true, because close implies are close, so you’d expect that the mappings that take and are close (because they’re linear– clearly this isn’t true for arbitrary mappings). Translating to mathese, we want small when close.

To establish this, note that since they’re isomorphisms, , so if we can make the last term arbitrarily small by controlling , we’re good. Let’s do some estimation: let , , , then

Now we can use some geometrical reasoning (or plug and chug into some inequalities) to see that knowing how much expands each of the basis vectors gives a bound on the norm of the operator:
for some .

Putting it together, and can be made arbitrarily small by making close, so is continuous. Note that is the condition number of the operator .

Now here’s another question. Does a curve through basis space give a curve through the general linear group? I.e. if is differentiable, then is also differentiable?

Possibly relevant posts:

• The Gelfand representation theorem (7/4/2008)
• Kvetches about Janich’s book (7/2/2007)
• Hilbert space class (7/7/2005)