The SPAMS Schedule - Spring 1999

Abstract: I will explain a simplifying way of solving the Riemann problem. Namely Roe Linearization, where one can replace the nonlinear equations with linear ones for faster solution of the Riemann problem. This is the method of choice for fast efficent Riemann problem solution, and is implemented in most codes. I will remind everyone of the Riemann problem and go through the various derivations of a Roe Solver for the isothermal equations (an 2 by 2 example of a hyperbolic system) I will also attempt to explain some of the fine points that can result when a nonlinear system is replaced with a linear one, and what fixes are needed to "preserve" the non-linearity, in the cases considered.

abstract: For the standard discrete random walk, the probability distribution of displacement from the origin has long been understood to be Gaussian. In $1+1$ dimensions, the area or integral of the graph of a random walk is also readily seen to be Gaussian (pretty exciting huh?).

I'll outline a calculation of the form of the joint probability distribution for displacement and the area. This distribution is also found to be Gaussian with a simple form for the mean and a nasty expression for the variance is obtained in closed form. The result is based on the observation of a connection to the theory of restricted partitions.v

Along the way there will be much discussion involving words such as restricted, partitions, generating, functions, descent, and steepest.

In the case of a first return to the origin, I'll discuss the joint distribution for area and number of steps taken. This last result is significant in the study of river networks (which is significant in the creation of my thesis).

Abstract: The purpose of this talk is to explore numerical Hamiltonian problems where symplectic methods can be used. The physical model that we chose to study was Kepler's problem since it does have a Hamiltonian framework and is well understood. Various numerical methods are applied to this problem, some symplectic and others not, in order to learn about the strengths and weaknesses of symplectic methods in general.

The enveloping nutshell: We will survey the history and proof of the four color theorem, following the excellent presentation of R. Thomas in the Notices last year.

Abstract: We discuss the classical game of Nim, and its known winning strategy (Bouton, 1901). Then we introduce a generalisation by placing the piles on the vertices of a simplicial complex. We allow a move to affect the piles on any set of vertices that forms a face of the complex. Under certain conditions on the complex we present a winning strategy. This is plagiarised from the paper by Ehrenborg and Steingrimsson (EJC, 1996).

On solving the dozen most well hard problems in fields as diverse as mathematical physics, fluid dynamics, combinatorial optimisation, combinatorics, computer science and why the morning edition of Neighbours lasts 20 minutes but the afternoon version takes 25.

System identification is a collection of mathematical techniques used to model complex/hidden physical systems in engineering and other fields. I will give an introduction to linear system ID, show Bussgang's Theorem, and make a digression on Toeplitz matrices.

Spabstract: Numerous methods have been introduced over the last few hundred years for attaching sums to divergent series. We'll begin by quickly mentioning two of these, due to Cesaro and Borel, and describing an extension of them using eigenvalues and eigenfunctions of operators. The main part of the talk will then discuss how these relate to convergence of Fourier series. We'll describe a famous theorem of Fejer that shows that the convergence behaviour of Fourier series is much better when we use Cesaro summation than just ordinary summation. But we'll then outline a computation suggesting that in some ways Borel summation may be even better.

Abstract: This talk is basically about the idea and practice of reduction, drawn from several different aspects and examples about how we reduce complicated model or math to Simple Person Appled Mathematics. They can almost all be said to be some kind of generalize "model reduction". I will talk about what I have been spamming and maybe also what I want further. I will plan to cover four types of reduction(not neccesarily disajoint): 1. Physical reduction: with the example of the density functional theory and what they can do now; 2. Computational reduction:with the example of the so called model order reduction in control system, from linear to nonlinear; 3. Object reduction: with the example of classical problem of geometric probability; and 4. Combinatorial reduction: with the example of Ising models and long range correlation and some further questions on it. I will try to convey the impression of what can simple minded models and argueing achive amazingly sucess and how should we look at them. And if time permit I may also talk about Compoite Fermion approach for Quantum Hall Effect as an example of reality reduction and also complification.

Two competing approaches to system model reduction have been moment-matching via numerically robust orthogonalized Krylov subspace methods, and Truncated Balanced Realization (TBR). TBR produces a near-optimal Hankel-norm approximation, with a known $\L^{\infty}$-transfer function error bound, but is too computationally expensive to use on large problems. Moment matching methods are inexpensive to apply, but often produce unnecessarily high order models.

I'll present a method of computing reduced-order models by projection via the orthogonalized union of the approximate dominant eigenspaces of the system's controllability and observability grammians. The approximate dominant eigenspaces are obtained efficiently using an iterative Lyapunov equation solver, Vector ADI, which requires only linear matrix-vector solves. I prove projection by the dominant eigenspace of either grammian is equivalent to TBR in the special case when the most controllable and most observable modes of the system span the same subspace.

This new method is as inexpensive as Krylov subspace-based moment matching methods, and produces a reduced model with uniformly small error over the entire frequency range, unlike moment matching which guarantees small error only around certain frequency points. The new approach also offers the flexibility of keeping the order of the reduced model low while making higher order approximations, whereas moment matching methods necessarily increase the order of the reduced model in order to gain more accuracy.

Here's an excerpt from his latest cookbook to hit the market "bork, bork, bork and bork mit the bork bork" (i have no control, sorry)

Continuum theory is used to model a suspension of non-colloidal buoyant particles dispersed in a continuous fluid. All particles are assumed to be of equal size and both constituents of the mixture are assumed to be incompressible. Phase separation takes place under the influence of centrifugal or gravitational force fields, resulting in a segmentation of the flow field into regions of pure fluid, mixture and sediment.

A "mixture model" is formulated in terms of volume averaged velocities. The equations of motion resemble a Navier-Stokes system with a generalized stress tensor and are supplemented by a conservation law for the dispersed phase and a heuristic closure for the relative motion between the two phases. A collision model is incorporated in parts of the analysis.

The presentation is focused on the numerical discretization of the mixture model equations. The implementation combines a variety of discretization schemes. Temporal splitting and projection methods are used to enhance the performance. The projection methods have been modified for implicit treatment of the Coriolis term, which is desirable when simulating strongly rotating flows. The Stokes part of the momentum equation is spatially discretized using a Galerkin finite element method and the advective part is discretized using a high-order upwind finite difference scheme. For the numerical treatment of discontinuities (kinematic shocks) associated with interfaces between regions of pure fluid, mixture and sediment, an upwind finite volume scheme has been developed.

If time permits, a few applications of the code will be presented.

The traditional dispersion estimator is non-robust: it is very sensitive to outliers. We propose a new robust estimator which has 50% breakdown point and a finite gross error sensitivity. We analyze the statistical properties of the estimator such as biasness, asymptotic variance and efficiency. We also carry out simulations to test the estimator, both in 1-d case and in multivariate case. At the end, we apply our estimator in time series to estimate autocovariance. We analyze a set of real data and give out the software of the estimator in S-plus as well.

The phenomena of causing a wineglass to emit tones when one rubs a moistened finger around the rim is well known and has undoubtedly been the subject of countless "informal" studies. An obvious question which arises is what are the natural resonant frequencies of such a system? Using an analysis based on energy considerations, a theoretical estimate will be presented based on the fundamental parameters of the problem such as the radius, density and elasticity of the glass. A demonstration will be attempted using appropriate beverages (no free samples).

This talk is based on a paper by A.P. French (1982) of the same title.

Did you ever wonder why many words which begin with f in Italian begin with h in Spanish?

(e.g. figlio = hijo

fico = higo

fare = hacer

fame = hambre )

I've found an analogous phenomenon in combinatorics.

Let P be a naturally labelled poset, and let J(P) be the corresponding lattice of order ideals. The "f-vector" of J(P) counts the numbers of chains of each length in J(P) and the "h-vector" of J(P) counts the number of linear extensions of P, by descents.

After examining several thousand h-vectors and f-vectors of distributive lattices, one comes to the inescapable conclusion that both classes of vectors look a lot alike. Is the h-vector of every distributive lattice the f-vector of another poset?