Self-Interacting Discrete Models

自交互离散模型

• 批准号: RGPIN-2020-06124
• 负责人: Madras, Neal
• 金额: $ 1.31万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740705
• 关键词: Self; Interacting; Discrete; Models

This proposal consist of mathematical research in three rather different areas, ranging from the pure to the applied.  (1) I will examine mathematical models of polymer molecules.  Polymers are very large molecules made of many smaller units called monomers, perhaps many thousands of identical monomers in one polymer molecule.  Examples include polyethylene, DNA, and proteins.  Physicists and chemists have developed many mathematical models to help explain and predict the (frequently surprising) physical properties of polymers.  However these models are difficult to analyze, both theoretically and computationally.  My goal is to improve the rigorous mathematical understanding of these models by focusing on aspects of certain discrete models, in which the polymer's flexible shape must follow the lines in a three-dimensional grid.  Mathematical confirmation of physical predictions can lead to improved confidence in the broader inferences from these models. (2)  A permutation is simply a rearrangement of a set of numbers or objects.  Permutations arise wherever symmetry plays a role:  in physics, computer science, bioinformatics, and almost all areas of mathematics.  I will look at permutations with certain restrictions called "pattern avoidance".  It turns out that imposing pattern avoidance greatly constrains the set of valid permutations of a (large) set of objects, and the permutations satisfying such a restriction turn out to have surprising structures that can be observed visually on a simple scatterplot.  My goal is to develop methods for characterizing and rigorously analyzing properties of these structures, particularly those that hold for most (but maybe not all) members of a given collection of pattern-avoiding permutations.  This is mainly a project in theoretical combinatorics with a probability angle, and whose main applications are within pure mathematics itself. (3) Mathematical modelling of disease has become a crucial part of public health.  For example, when faced with a limited amount of flu vaccine, is it better to focus efforts on children or on the elderly?  Can we predict the peak demand on hospital resources for a coming flu season?  Which vaccines will produce enough benefit to merit a government paying for them?   I will investigate mathematical properties of a class of models for the spread of an infectious disease that has randomness explicitly in the model.  Specifically, each infected person will infect a random number of other people, and each person lives for a random length of time.  Most models for large populations are essentially deterministic, or else have unrealistic simplifying assumptions about individual lifetimes.  I am particularly interested in mathematically analyzing models with realistic probabilities for lengths of lifetimes.  Simulations can tell us some things about specific models, but mathematics can find deeper patterns that hold even for models that have not yet been simulated.

该建议包括在三个相当不同的领域的数学研究，范围从纯度到应用。 （1）我将检查聚合物分子的数学模型。聚合物是由许多称为单体的较小单元制成的非常大的分子，也许是一个聚合物分子中的数千个相同的单体。例子包括聚乙烯，DNA和蛋白质。物理学家和化学家已经开发了许多数学模型，以帮助解释和预测聚合物的物理特性（通常令人惊讶）。但是，这些模型在理论和计算上都难以分析。我的目标是通过专注于某些离散模型的各个方面来提高对这些模型的严格数学理解，在这种方面中，聚合物的柔性形状必须在三维网格中遵循线条。物理预测的数学确认可以提高对这些模型的更广泛推论的信心。 （2）置换只是一组数字或对象的重排。对称性发挥作用的地方都会出现排列：在物理，计算机科学，生物信息学和几乎所有数学领域中。我将查看具有称为“模式避免”的某些限制的排列。事实证明，施加模式避免的模式极大地限制了一组（大）对象集的有效排列集，并且满足这种限制的排列事实证明具有令人惊讶的结构，这些结构可以在简单的散点图上在视觉上观察到。我的目标是开发这些结构的表征和严格分析的属性，尤其是那些对给定的避免模式排列的成员（也许不是全部）成员的方法。这主要是一个具有概率角度的理论组合学项目，其主要应用在纯数学本身之内。 （3）疾病的数学建模已成为公共卫生的关键部分。例如，当面对有限的流感疫苗时，将精力集中在儿童还是年龄较大的情况下更好？我们可以预测即将到来的流感季节对医院资源的高峰需求吗？哪些疫苗将产生足够的好处，值得政府为其付款？我将研究一类模型的数学特性，用于在模型中明确具有随机性的传染病传播。具体而言，每个受感染的人都会感染其他人的随机数量，每个人都会随机生活。大多数人口的模型本质上都是确定性的，或者对个人生命的简化假设不切实际。我对数学分析的模型特别感兴趣，这些模型具有寿命的现实可能性。模拟可以告诉我们一些有关特定模型的信息，但是数学可以找到更深的模式，甚至对于尚未模拟的模型而言。