Sharing Results and Insight, Mexican and U.S. Researchers Meet in Pachuca, Mexico

September 26, 2004

Pedro González-Casanova, Humberto Madrid, and Roger Knobel

Continuing an annual tradition begun in 2000, the Mexico Section of SIAM---MexSIAM---coordinated a joint Sociedad Matematica Mexicana-SIAM session at the 2003 SMM National Congress, held in Pachuca, Mexico, October 16-17. By bringing together mathematicians from Mexico and the U.S., the joint sessions are designed to promote research links between the two countries.

Students, teachers, and researchers from throughout Mexico gather at the National Congress, which is the major event of SMM. Research and education are intertwined at this active meeting through a broad spectrum of special sessions and lectures. SIAM and SMM have developed a growing partnership, due in large part to the formation of MexSIAM in 2000 and to the efforts of Barbara Lee Keyfitz, a co-organizer of all four joint sessions to date. In addition to Keyfitz, the co-organizers of the 2003 SMM-SIAM session were Pedro González-Casanova, Humberto Madrid, and Roger Knobel.

Pachuca, the site of the SMM National Congress, is a city with a rich history, located at the base of the Sierra de Pachuca mountain range. Participants flying into Mexico City were treated to a leisurely 40-mile drive, passing near the famous pyramids of Teotihuacan and the silver mines of central Mexico as they made their way to Pachuca.

The eight speakers in the SMM-SIAM session, four from Mexico and four from the U.S., discussed a variety of interesting topics in the course of the two-day program. Zeferino Parada of ITAM opened the session with a talk on the disconnectivity of cubic splines, in particular the problem of approximating functions with zero curvature via cubic splines in finance. After defining a family of cubic splines that contains the natural, complete, and so-called financial splines, he demonstrated that the financial and natural splines belong to one connected region of this family, while the complete spline is in a separate connected region. Because of the finite precision of computers, this disconnectivity is not fully appreciated in practice. This result explains the tendency among practitioners to discard complete splines in favor of zero-curvature splines.

Next to speak was Jean Taylor, a visiting professor at NYU's Courant Institute and emeritus professor at Rutgers University. In a talk titled "Five Little Crystals and How They Grow," Taylor gave the audience a glimpse of a family of motions that can be formulated and analyzed within the same general framework, based in part on surface energy similar to that of soap bubbles (but varying with normal direction) and inner products determining the kinetics. Taylor showed computer animations from simulations to demonstrate the different motions.

Following with a talk on radial basis functions in approximation theory, Pedro González-Casanova described a new type of radial quasi-interpolant, capable of approximating a data function at any point in a given open bounded domain with a convex polygonal boundary. The approximation of a data function at points close to or on the boundary of a bounded domain has been an active research area for some time. González-Casanova presented a new approach to the problem, including error estimates in Sobolev norms for this scheme. He also discussed the use of the algorithm to build a domain decomposition technique capable of handling large amounts of data. He presented numerical results for both continuous and discontinuous data functions.

In a talk titled "Scattering of Microwaves from Soils," Fernando Brambila-Paz of the Facultad de Ciencias of UNAM used a combination of laboratory experiments and computer simulations to show that microwaves reflected from and transmitted through soils have a fractal dimension correlated with that of the soil's hierarchic permittivity network. He also described a mathematical model relating ground-penetrating radar data to the mass fractal dimension of soil structure. The fractal signature of the scattered microwaves was seen to correlate well with some of the physical and mechanical properties of soils.

José E. Castillo of San Diego State University began the Friday session with a talk on higher-order mimetic discretizations for continuum mechanics. Numerical solutions of partial differential equations based on mimetic finite difference schemes satisfy discrete versions of conservation laws and are thus more likely to produce physically meaningful results, Castillo said. He presented techniques for constructing fourth-order approximations on or near the boundary---a difficult task, even in one dimension on a uniform grid.

Héctor Juárez V. of Universidad Autónoma Metropolitana-Iztapalapa presented a method for the numerical simulation of viscous incompressible flows modeled by the Navier-Stokes equations. By using a combination of finite element methods for the space variables and operator splitting for time integration, the method treats the various operators associated with the computational model separately. The resulting discrete set of equations presents only one serious numerical difficulty at each step, as a result of the splitting technique. Benefits of this approach include a reduction in memory requirements and a decrease in the computational time. In addition, a significant reduction in the complexity of the numerical algorithm makes computational implementation straight-
forward. Juárez discussed the application of this method to several prob-
lems, including the sedimentation of particles in Newtonian flows, free
surface problems, and thermal flow problems, and presented results of numerical experiments.

Steve Cox of the Department of Computational and Applied Mathematics at Rice University gave a talk on the determination of spatial and temporal heterogeneities from multipotential single nerve-cell recordings. There is by now general agreement that individual nerve cells distribute their excitable machinery in a far from uniform fashion, Cox explained. At a coarse level, the cause of the heterogeneity is clear: The basal and apical dendritic trees, cell body, and axon serve quite separate functions. At finer levels, precise identification of the detailed distribution of synapses and ion channels is a difficult, though extremely important, problem. Cox described the underlying physiology, surveyed previous efforts to tackle the problem, and presented new theoretical results.

J. Roberto Mercado of the Instituto Mexicano de Tecnología del Agua closed the SMM-SIAM session with a talk titled "The Group of Möbius in Channels," in which he analyzed the Korteveg-de Vries equation model of one-directional waves in channels. He described a method for obtaining solutions of this equation in the context of the Ablowitz conjecture, which states that a (nonlinear) PDE is integrable whenever the solutions have the Painlevé property---i.e., when their only movable singularities are poles. By using the symmetries of the homographic transformations belonging to the Möbius group, Mercado showed how solutions of the KdV equations are obtained in terms of solitons. These transformations, whose invariants are the Schwarzian derivatives and the celerity, give rise to a certain type of Lax transformations. Mercado used the Bäcklund transformations to obtain the solutions he presented.

On the way back to the Mexico City airport, session participants enjoyed a tour of the pyramids of Teotihuacan. Participants and organizers are grateful to both SMM and SIAM for the opportunity to hold the session and for the support provided to the speakers in the joint session. We look forward to continuing the tradition with a joint session at the next meeting, scheduled for October
10-15, 2004, in Ensenada, Baja California.

Pedro González-Casanova is a faculty member at Universidad Nacional Autónoma de México. Humberto Madrid is a faculty member at Universidad Autónoma de Coahuila. Roger Knobel is an associate professor in the Department of Mathematics at the University of Texas-Pan American.

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