Applied and Computational Mathematics Seminar, Fall 2002

Scope

Mission

Schedule

Winter 2003 schedule

Tuesday 12 November, 3:10-4pm in 320 Morton Hall:
Stochastic Modeling of Intracellular Calcium Dynamics: Optimal Ion Channel Clustering, Peter Jung (Physics and Astronomy)

Ion channels and receptors in the cell membranes and internal membranes are often distributed in discrete clusters. One particularly well studied example is the distribution of inositol 1,4,5-triphosphate receptors in the plasma membrane that controls the flux of Ca ions from the endoplasmic reticulum into the cytosol. Using computational modeling, we show that channel clustering can enhance the cell's calcium signaling capability. Furthermore, we predict optimal signaling cellular capability at cluster sizes and distances that agree with experimentally found values in Xenopus oocyte.

Tuesday 5 November, 3:10-4pm in 320 Morton Hall:
Complexity of Multiple Sequence Alignment, Winfried Just (Math)

Alignment of multiple sequences is one of the most fundamental procedures for the extraction of biologically meaningful information from large sets of biomolecular sequence data such as the data created by the Human Genome Project. The speaker has shown that this problem is NP-hard (computationally intractable) for scoring schemes used by biologists and remains NP-hard even when one tries to restrict the search space by limiting the number and kind of gaps that can be inserted into the sequences to be aligned.
  In this talk it will be explained why the Multiple Sequence Alignment Problem is of central importance in the analysis of genomic information. A brief introduction to the P=NP problem will be given, and the result on NP-hardness will be put into the context of related notions, such as the question whether a polynomial-time approximation scheme for the problem exists.

Tuesday 29 October, 3:10-4pm in 320 Morton Hall:
Slow Passage Through Bifurcation in Hamiltonian Systems, Todd Young (Math)

Consider a system with Hamiltonian H(q,p,lambda), where lambda is a slowly varying parameter, i.e. (d/dt)lambda = O(epsilon). If the system has one degree of freedom then a classical result says that the action, I, is an adiabatic invariant. That is |I(t) - I(0)| = O(epsilon) for t in [0,1/epsilon]. This result requires the assumption that trajectories are oscillatory and the ``frequency" stays bounded away from 0. In the seventies, Neistadt and others showed that part of this result can be retained if the frequency passes through 0 by way of crossing a seperatrix of a saddle fixed point. In particular, suppose that H(q,p,lambda) = H(-q,p,lambda), then the region of interest in phase space, U can be decomposed into two regions, V and W with the following properties: 1. The measure of W is small to all orders in epsilon, and 2. For all initial conditions in V, we have |2I(T) - I(0)| = O(sqrt(epsilon) log(epsilon)), where T = 1/epsilon.
  Neistadt's proof uses the averaging method and is very difficult. The averaging method cannot be generalized to higher degree of freedom problems. Further Neistadt made non-generic assumptions about the local form of the equations near the saddle point. In the first part of the talk I will outline a new simple and geometric proof which can possibly be generalized at least to 2 degree of freedom problems and which does not make any assumptions about the local form of the equations.
  In the second part of the talk I will discuss numerical work on the problem in both 1 and 2 degrees of freedom. In 1 degree of freedom we discovered phenonmena which are not described by the current theory (including my proof).

Tuesday 22 October, 3:10-4pm in 320 Morton Hall:
Long memory process and parameter estimation by the Discrete Wavelet Transform., Xunming Du (Math)

In this talk, we will discuss the notations of the long memory process and the wavelet filters. In the long memory, we will focus on the fractional difference (FD) process. Wavelet variance is introduced here. Based on the relationship between the wavelet variance and the parameter delta for the FD process, we will study the maximal likelihood estimations (MLEs) for the parameter delta.

Tuesday 15 October, 3:10-4pm in 320 Morton Hall:
A Fast Transform for Spherical Harmonics, Martin J. Mohlenkamp (Math)

Spherical Harmonics arise on the sphere in the same way that the (Fourier) exponential functions {exp(i k theta)} arise on the circle. Spherical Harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast transform, evaluating (or expanding in) Spherical Harmonic series on the computer is slow---for large computations prohibitively slow. Here we present a fast transform.

Tuesday 8 October, 3:10-4pm in 320 Morton Hall:
Approximation algorithms for network design problems, Vardges Melkonian (Math)

Survivable network design problems concern the design of those kind of networks that remain functional even if some of their elements (edges, nodes) are broken. Network design problems arise from many sources, including the design of various transportation systems (such as highways and mass-transit systems) as well as telephone and computer networks. The network design problems that we consider are NP-hard; so our goal is to find efficient approximation algorithms for them. While in the last 10 years there has been significant progress in designing approximation algorithms for undirected network design problems little progress has been made on their directed counterparts. The main goal of our work is to explain the difficulties intrinsic to directed networks and to give new algorithms for them. The specific network design problem we consider here is the following: find a minimum cost subgraph such that for each vertex set S there are at least f(S) arcs leaving S. The main algorithms we design for these problems are the linear programming rounding algorithm and the primal-dual method. We give theoretical guarantees for the performance of our algorithms and also present computational results which support their good performance in practice. (This is joint work with Eva Tardos.)

Tuesday 1 October, 3:10-4pm in 320 Morton Hall:
A wavelet-like multiscale system with high energy concentration and its properties, Annie Shen (Math)

It is well known that a non-trivial function cannot be compactly supported in time and frequency domains simultaneously. However, among all possible band-limited functions with a given bandwidth, one can ask which function maximizes the fraction of energy over the prescribed time interval. Prolate spheroidal wave functions (PSWFs) are special functions that lead to the optimal solutions of this concentration problem. This fact was unraveled by Slepian and his collaborators at Bell Lab in 1960s. Since then PSWFs have shown promise for applications in engineering and other areas. The primary interest of this talk is to construct a wavelet-like system based on PSWFs. The system not only possesses a multiscale structure similar to those of wavelets but also preserves the high energy concentration property inherited from PSWFs. Approximation properties are proved theoretically and illustrated by numerical examples. A sampling formula for numerically constructing the basis is given and demonstrated via numerical examples.