The SPAMS Schedule - Spring 1998

ABSTRACT: Do you want to live forever? Do you want to get rid of your "nerd" gene? What was once science fiction is now becoming reality with the advent of the human genome project. The rapid sequencing of genes has led to the need for annotation programs that can identify introns (regions that get excised before translation) and exons (regions that get glued together and code for proteins). We will discuss some recent computational approaches to this problem.

Disclaimer: Extended life span is not guaranteed to atendees of the talk. Results may vary. Void where prohibited by state or federal law.

I will attempt to explain the basic idea behind Godonov's method for solving systems of first order hyperbolic equations. The method requires solving what is known as a Riemann problem (a particular initial value problem) many times. This will be explained and demonstrated for the Shallow water equations. Godonov's method is 1st order accurate in smooth regions and if time permits I will explain generalizations of Godonov's method that can achieve 2nd order accuracy on smooth regions of the flow. Hopefully, this talk will be accessible to people with a broad background.

Given a directed graph, we can define a "path polynomial" whose k-th coefficient is the number of k-vertex paths in the graph. We can similarly define a "chain polynomial" for a partially ordered set. In both cases, there seems to be a somewhat curious relationship between the roots of the polynomial and properties of the original combinatorial structure. For instance, it is known that if a poset is a semi-order (i.e. 1+3 free and 2+2 free) then the roots of its chain polynomial are real. It is conjectured that if a poset is a distributive lattice, then again, the roots are real.

By using the eigenvalues of an anti-adjacency matrix, we can create new proofs for such known relationships and hope to prove conjectured relationships.

Salient projection of talk-space: There have been observations of large ocean currents and pools of dense water that form near coast lines over gently sloping topography. These water masses, under certain conditions, can travel along the shelf. My talk will introduce these ideas and discuss a new theoretical model that describes such a scenerio. In addition, I will mention stability results concerning gravity currents and (hopefully) show slides of some radiating eddy solutions.

We study the statistical properties of slow flow through porous media at the pore scale by a combination of numerical simulation and theoretical arguments. The numerical simulations show that the flow in low-porosity media is strongly focused, such that the strong localized flow patterns result in long-tailed, non-Gaussian probability density functions (PDF's) for the velocity. This surprising decay in the velocity distribution has been observed and verified by experiments and seems to be a generic property of flow through complex low-porosity structures. On the other hand, higher-order statistics of the flow are very sensitive to the statistical properties of the micro-structure of the porous medium. Unpublished measurements, by A. H. Thompson, of the microscopic flow field through real sedimentary rocks show that the velocity-velocity correlation functions decay slowly, like a power law, while the same correlation functions obtained from flow through artificial random media decay exponentially. These results indicate that the processes that form sedimentary rocks instill correlations not found in random media. We propose theories that predict the aforementioned phenomenology. Specifically, we argue that a principle of localized equilibrium, described by a stochastic form of Darcy's law, holds for Stokes flow in low-porosity disordered media at the pore scale. In this way, we recover the exponential decay of the velocity PDF's. We also indicate how the velocity-velocity correlation function may be related to the micro-structure of the porous media.

In systems theory there are often linear systems with the following input-output relationship

dx/dt = Ax + Bu

y = Cx + Du

u is the input, y the output, x the state variables.

A is nxn matrix, B nxp, C qxn, D qxp.

n is usually very large. In the case of certain circuit analysis problems, the A matrix can be on the orders of 10,000 or 100,000 squared.

The transfer function, relating the input and output, is G(s) = C(inv(sI-A))B + D.

We want to approximate {A, B, C, D} by a much smaller system, {A', B', C', D'} on the order of a few hundred by few hundred variables, such that, that new transfer function, G', is close to the original G(s).

Hankel norm approximation is one way of doing this model order reduction. There is an exact error bound between the new and original transfer functions, and the approximation is optimal in the hankel norm sense.

I shall outline the steps to hankel norm approximation, and show some preliminary results from circuit analysis.

For $\mu$ a partition of $n$, Garsia and Haiman introduced some bigraded Sn-modules $M_{\mu}(X,Y)$. I will describe these modules and talk about some conjectures related to them. In particular, I will talk about the so called "n! conjecture" and try to explain how these modules are related to some symmetric polynomials, called Macdonald's polynomials, which are polynomials with really nice properties. If time permits, I will talk about the work I have done related to that subject, which is the study of some intersections of modules $M_{\mu}$.

When studying Gauge Quantum Field Theories, it is useful to consider topological properties of field configurations, especially in vacuum. We will discuss topological charge and Yang-Mills instantons, which draws on Chern-Simons theory. And if time permits we will see how the Atiyah-Singer Index Theorem may contribute to our understanding of particle cosmology.

I will ramble on about a selection of observations regarding river network patterns. The first is, and this is my belief, that a few simple geometrical assumptions can be shown to imply all statistical features found in natural networks. Thus, an understanding of these assumptions is the essential requirement for the comprehension of the plethora of network quantities that abide in the literature.

We will then venture on to an extremely simple random network model that exhibits all of the functional forms of the statistical features found in natural networks with only the explicit values of various exponents differing. This is thus a most excellent model to examine. I will discuss the known analytic evaluation of several exponents and point out what else needs to be calculated. This random network model is the rightful limiting case that all theories should satisfy - it is, by a loose analogy, an infinite temperature state. Natural networks clearly have some correlations in them but how to simply identify them is unclear.

If time permits, I will also discuss Australia's abysmal performance in their current cricket tour of India. There's really no need to read this last sentence either.

Robotics suffered from not having good tools for manipulating three-dimensional objects. Vectors, coordinate geometry and trigonometry all have deficiencies. Quaternions can be used to solve many of these problems. A brief introduction to robotics will be given. Then we will see a brief development of properties of quaternions. Examples will then be given of how quaternions are used to simplify robotics.

Wave motions in nature often do not appear as a single train of periodic or almost periodic waves but can assume a much less regular form. This can be thought of as a sum of a large number of Fourier components interacting with one another. During this interaction particular wavelengths satisfying specific criteria can resonate and thus be seen to dominate the spectral distribution as time progresses. Water waves provide a good example where such resonances occur, but the ideas remain fairly general and may be applied to a variety of physical systems.

I will briefly talk about water waves, deriving the dispersion relation for the case where both gravity and surface tension play a role. I will then discuss three-wave resonances and the remarkable distinction which arises when one considers the spectrum to be discrete or continuous. This talk is geared to a general audience (including Mats) so no prior knowledge of fluid mechanics or waves is necessary, nor will it be assumed.

Suppose you gently nudge an infinite string. The resulting motion can (roughly) be described as a superposition of traveling waves. The problem is invariant under the additive group of translations $\mathbb{R}$ of the infinite string, and one can interpret the solution in terms of Fourier analysis on the symmetry group $\mathbb R$.

Consider the related finite question of nudging some collection of $N$ identical point masses (in three space, say) connected to each other by identical springs, and suppose the configuration is invariant under the action of a finite symmetry group $G$. The motion is again a linear combination of stable oscillating modes. Finding the allowable frequencies amounts to introducing coordinates and computing the eigenvalues of a $3N$ dimensional matrix. But this procedure is computationally taxing and masks the underlying symmetry of the problem. So one is led to ask: what is the analog of Fourier analysis on $G$, and what does it tell you about the normal mode frequencies?

It turns out that the structure of $G$ partitions the normal mode frequencies into disjoint subsets and gives formulas for sums of powers of elements in each set. If the configuration is symmetric enough, one obtains complete information without ever having to introduce coordinates or solve complicated polynomials. For instance, on the back of a modestly sized envelope, one can determine all normal mode frequencies of the configuration consisting of 10 masses located at the vertices of an icosahedron with each mass connected to its nearest neighbors by identical springs. (The calculation boils down to solving a few quadratics. This is to be contrasted with finding the roots of the degree 30 characteristic polynomial of the potential matrix.)

The method also has some interesting applications to finding the eigenvalues of the adjacency matrix of a homogeneous graph. Physically, for instance, this leads to an elementary derivation of the Huckel energy levels of the truncated icosahedral hydrocarbon buckyball.

Let us see how combinatorial stuff such as generating functions and the umbral calculus can open an entirely new window to the world of wavelets. Satisfaction is not guaranteed since it is really only a subjective try --- a try to build a wavelet world with the MIT Applied Math. character. (We have been drowned in the combinatorial ocean. So, why not more :)