Statistical Studies in Biomolecular Sequences and Topics in Total Positivity

本研究将集中在三个方面：（i）包含干扰效应的重组随机模型的理论研究。这些包括多标记交叉分布的特征和高阶交叉干扰形式的分析。除了它们作为一类随机过程的内在兴趣之外，这些应该有助于构建多个遗传标记的遗传图谱。(ii)分子序列的统计学研究，强调基于分数的方法。这些在识别DNA和蛋白质序列中的独特序列特征以及在识别蛋白质序列比较中的重要共同片段中发挥作用。 (iii)多变量全正性理论的新方向和发展，其动机是与图上的随机游动相关的应用，特别是在n维超立方体上。决定超立方体上的概率分布何时为多元全正的一般问题是表征重组随机结构的正干扰和负干扰的基础。 对重组过程的研究在开发遗传图谱（例如，与其他容易观察的标记相关的疾病基因图谱）方面具有重要意义。优化和随机比较的研究可以提供见解，在开发计算机算法相关的组合学和感兴趣的排队系统，分支现象，出生和死亡的随机过程。生物分子序列中模式的统计学研究可以用作区分DNA和蛋白质序列中的偶然发生与适当的功能和结构属性的基准。

This research will concentrate on three areas: (i) Theoretical studies of recombination stochastic models incorporating interference effects. These include characterizations of multimarker crossover distributions and analysis of higher order crossover interference forms. Apart from their intrinsic interest as a class of stochastic processes, these should be helpful in constructing genetic maps of multiple genetic markers. (ii) Statistical studies of molecular sequences emphasizing score based methods. These play a role in identifying distinctive sequence features in DNA and protein sequences and in identifying significant common segments in protein sequence comparisons. (iii) New directions and developments in the theory of multivariate total positivity motivated by applications related to random walks on graphs, especially on the n-dimensional hypercube. The general problem of deciding when a probability distribution on the hypercube is multivariate totally positive is fundamental in characterizing positive and negative interference for recombination stochastic structures. The research on recombination processes is of importance in developing genetic maps (for example, mapping of disease genes in relation to other easily observable markers). The studies on majorization and stochastic comparisons can provide insights in developing computer algorithms relevant in combinatorics and of interest in queuing systems, branching phenomena, and birth and death stochastic processes. Statistical studies of patterns in biomolecular sequences can be used as benchmarks for distinguishing chance occurrences from proper functional and structural attributes in DNA and protein sequences.