Mathematical Sciences: Quantum Self-Trapping

研究者继续他的量子化研究， 几个微分方程尤其是阿布洛维茨-拉迪克 方程和离散非线性薛定谔方程的一种形式 称为离散自陷方程。 方程出现了 在描述各种非线性系统的行为时， 许多来自生物学背景。 他利用这些结果， 加强对几种应用的理解： 量子理论的两个方程，特别是集中在 孤子结合能问题，光谱数据分析 为分子晶体，并模拟生长动力学， 微管 更好地理解细胞的行为是一个核心问题， 生物学的主题 该项目使用数学工具， 考虑细胞生物学的几个相关方面， 关注工具本身。 因此 研究人员研究微管的结构（一种 "蛋白质晶体"），进行了更一般的分析， 分子晶体，并发展了某些类型的理论， 描述非线性现象的微分方程。 一 微管是微管蛋白以管状形状的排列。 微管蛋白是一种蛋白质，它在细胞的增殖能力中起着关键作用。 细胞保持其形状并移动。 所以更好的理解 微管的生长可以使我们更清楚地了解 细胞发育;这在不同的领域都很重要， 胎儿发育和肿瘤生长。

The investigator continues his studies of quantization of several differential equations. particularly the Ablowitz-Ladik equation and a form of discrete nonlinear Schrodinger equation called the discrete self-trapping equation. The equations arise in describing the behavior of a variety of nonlinear systems, many coming from a biological context. He uses these results to enhance understanding of several applications: comparisons of quantum theories for the two equations, especially focussing on the issue of soliton binding energies, analysis of spectral data for a molecular crystal, and modelling the growth dynamics of microtubule. A better understanding of how cells behave is a central theme in biology. This project uses mathematical tools to consider several related aspects of cell biology and also concerns itself with the tools themselves. Hence the investigator studies the structure of microtubule (a kind of "protein crystal"), undertakes a more general analysis of molecular crystals, and develops the theory of certain kinds of differential equations that describe nonlinear phenomena. A microtubule is an arrangement of tubulin in a tube-like shape. Tubulin is a protein that plays a key role in the ability of cells to hold their shape and to move. So a better understanding of the growth of microtubule could lead to a clearer picture of cell development; this is important in areas as diverse as the development of fetuses and the growth of tumors.