Frontiers of Applied Mathematics

PREFACE.

CONTENTS.

Networks of neurons in the nervous system can produce a variety of temporal patterns of different frequency; the same network can produce different rhythms at different times, or several rhythms at one time. We focus here on the gamma (30–90 Hz) and theta (4–12 Hz) rhythms produced in the hippocampus, a part of the nervous system critical for learning and recall. We discuss experiments and models that suggest that there are separate subnetworks that produce the different rhythms; the sharing of components of the networks induces competition between the rhythms, which can lead to suppression of one of the rhythms, or nesting of the rhythms. We show how low-dimensional maps can help to understand the properties of the cells and networks to allow this to happen.

It is proposed that the action potentials propagates as ion-acoustic waves in the axons rather than the diffusion process according to the telegraph equation. A non-linear oscillator model is also proposed to account for the spike threshold and the signal coding of the action potentials.

In this paper, the effects of a conventional form of negative feedback on receptor synthesis rate as a mechanism to induce robustness of biological development of the Drosophila wing imaginal disc is analyzed through the initial-boundary value problem in a basic partial differential equation model for the extracellular biological development activities. The existence, uniqueness, linear stability and monotonicity of the steady state signaling gradient are established rigorously. Solutions are then obtained for a special case of the steady state problem to show explicitly the effects of the chosen form of negative feedback. It is evident from the results that the principal effect of such a feedback mechanism is to render the signaling bound morphogen gradient more uniformly distributed except for a narrow layer adjacent to the edge of the posterior compartment of the wing disc. While the change in the magnitude of the maximum signaling morphogen concentration near the ligand source may be kept at an acceptable level, the leveling and flattening of the gradient render it less differential in space (except in the boundary layer) and hence more deviating from the desired gradient for the target biological pattern. The conclusion is then shown to apply to the general case with the help of the theoretical results on monotonicity. These results suggest that negative feedback on the receptor synthesis rate of the chosen form is not effective for promoting robustness. In fact, it drives the system somewhat in the opposite direction, a conclusion supported by the simulation results. The findings in turn suggest more useful forms of the negative feedback for mediating receptor synthesis to offset a higher ligand synthesis rate for our types of systems.

CSAW (conditioned self-avoiding walk) is a model of protein folding that combines the features of SAW (self-avoiding walk) and the Monte-Carlo method. It simulates the Brownian motion of a chain-molecule in the presence of interactions. The model is designed so that one can add desired features step by step. It can be used as guide to strucuture prediction, as well as theoretical laboratory.

The polypeptide collapse is an essential dynamics in protein folding. To understand the mechanism of the collapse, the changes in protein compactness in the process of protein folding was observed by using time-resolved small-angle x-ray scattering. The characterized dynamics for several proteins with different topologies suggested a common folding mechanism termed “collapse and search” dynamics, in which the polypeptide collapse precedes the formation of the native contact formation. The collapsed intermediates demonstrated the scaling relationship between radius of gyration (R_g) and chain length with a scaling exponent of 0.35 ± 0.11, which is close to the theoretical value (1/3) for the collapsed globules predicted by a statistical theory of polymers. Thus, it was suggested that the initial collapse is caused by the coil-globule transition of polymers. Since the collapse is essential to the folding of larger proteins, further investigations on the collapse likely lead to an important insight into the protein folding phenomena.

Biopolymers are ubiquitous in a biological living system. They are involved in almost all cellular biochemical reactions, playing versatile biological roles. The mechanical characteristic of a biopolymer is directly related to its functions. In this paper, we briefly review theoretical efforts of our group on the structural transitions of three major types of biopolymers (DNA, RNA, proteins) and on the structural organization in the spider capture silk. These theoretical studies demonstrated that, weak non-covalent interactions, such as wan der Waals attraction, hydrogen bonding, hydrophobic interaction, and screened electrostatic interactions, enable a biopolymer to achieve an optimal combination of structural stability and deformability in accordance with its specific biological functions.

Spiral structure is the most distinguished feature common in all astrophysical disks. One of the ways the spirals can be generated is through an external periodic force in a mechanism known as the resonance excitation. We will use numerical simulations to demonstrate this process for galactic disks. A rotating bar potential or a potential due to spiral waves, both of stellar origin, acts as a periodic forcing to be imposed on a gaseous disk in a disk galaxy. We will show how the spiral density waves are generated, the mass of the disk is re-distributed, and the instability results. In other words, we show in simulations the structure, evolution and instability of the disk subject to such a periodic forcing. The instability which leads to turbulence and chaos of the disk can be identified to be of the types of Rayleigh’s shear instability and Toomre’s gravitational instability, or a combination of both. The results are generally shown in movies and some analyses are given. The work is in parts supported by a grant from National Science Council, Taiwan, NSC94-2752-M-001-002-PAE.

The dynamics of spiral galaxies is a gold-mine of challenging problems for the astrophysicist and the applied mathematician. In Astrophysics, we may ask how these island-universes formed, evolved, and reached their current structure and thus address the problems of the dynamics of the interstellar medium, of star formation, of stellar dynamics, and of the presence and role of dark matter halos. These fundamental issues have attracted the interest of the scientific community for almost a century. The models conceived and developed to sharpen and to quantify our understanding of the basic dynamical processes at work in this context require a semi empirical approach and general tools that are characteristic of Applied Mathematics. In particular, the beautiful morphology of spiral galaxies poses a number of interesting questions, most of which have found a coherent answer in the framework of the Density Wave Theory. I will give a synthetic description of the main concepts and of the main achievements of the theory, as developed in the course of three decades in the second part of last century. The theory has had a major impact on Astrophysics and has been the inspiration for a number of important theoretical and observational investigations. The relatively recent advent of near-infrared observations (especially those in the K-band, probing the underlying stellar component in galaxies) has confirmed that indeed large-scale spiral arms are associated with a smooth, sinusoidal density perturbation in the stellar disk, that grand design is very frequent and generally two-armed, and that multiple-armed spiral structure is mostly associated with the gaseous interstellar medium. I will also briefly outline other interesting topics in the dynamics of spiral galaxies, where progress may take place as a result of a fruitful exchange between Astrophysics and Applied Mathematics.

During the first ten million years or so after the big bang, the fluctuations in matter and radiation in the universe remained small enough that they could be treated as tiny ripples imprinted on a smooth background. A major challenge in cosmology is in calculating how these small fluctuations grow under gravitational instability into highly collapsed objects. The traditional tool to study cosmological structure formation in the nonlinear region is numerical simulations. In this lecture I describe a complementary approach based on a kinetic theory for the evolution of the phase space distributions of dark matter in galaxy halos.

Star formation is a process that spans many decades in length and time scales. As a consequence, asymptotic methods of the variety pioneered by C. C. Lin and other applied mathematicians can be used to great effect at various stages of the problem. In this lecture, we give an overview of the complete problem in the case of the formation of single stars of sunlike masses, and we highlight a few of the mathematical solutions made possible by an application of asymptotic techniques and ideas — the formation of molecular cloud cores; the gravitational collapse of self-similar, magnetized, rotating toroids; the dissipation by magnetic reconnection of the trapped interstellar flux brought in by gravitational collapse that would otherwise have prevented the formation of circumstellar disks; and the jets and outflows that result when the accreting circumstellar disk interacts with the magnetosphere of the newly formed star. We end by indicating where additional progress requires a better understanding of the role of turbulence, another subject where astronomers of the future can look to the work of C. C. Lin for guidance.

Localized wave solutions, such as solitary waves, often termed solitons, arise naturally in nonlinear optics and fluid dynamics. In optical communications, asymptotic analysis leads to the “classical” and “dispersion-managed” (DM) nonlocal nonlinear Schrödinger equations which have localized pulses as special solutions. Recent research has shown that similar dispersion-managed equations occur in mode-locked lasers. A numerical method is introduced to find these and other localized waves in nonlinear optics, water waves and multi-fluid systems. In water waves the classical equations are reformulated as a system of two equations, one of which is an explicit nonlocal equation, for the wave height and for the velocity potential evaluated on the free surface. The nonlocal system is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney-Luke and nonlinear Schrödinger equations. Two dimensional lumps with sufficient surface tension, are obtained numerically.

No abstract received.

Some aspects of long surface waves are considered. Special attention is focused on the existence of conservation laws for this physical system and related problems.

New solitary-wave structures in two-dimensional periodic media are obtained in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. These new structures bifurcate from the edges of Bloch bands with two linearly independent Bloch modes. Away from these band edges, superposition of these Bloch modes, modulated by nonlinear effects, give rise to composite solitary waves with distinctive intensity and phase profiles such as vortex arrays. Using perturbation methods, coupled nonlinear envelope equations for the two Bloch waves near the band edges are analytically derived. Numerically, these composite solitons are directly computed both near and far away from the band edges, and the analytical results are fully confirmed.