Steiner 3-Designs

斯坦纳2-设计类似于普通平面（欧几里德平面或射影平面），除了点的数量和线的数量是有限的。定义属性是（i）任何两个点包含在一条直线中，以及（ii）所有直线（视为点的子集）包含相同数量的点。在统计学术语中，点与处理相同，线称为区组，2-设计称为不完全区组设计。不完全区组设计（或2-设计）需要构建最佳试验方案以确定最佳处理。因此，重要的是确定对于什么参数v1（点的总数）和k（点的数目），可以构造Steiner 2-设计。经过几十年的研究，R. M. Wilson，Steiner 2-设计的存在性和构造问题至少对大量的点得到了解决。Steiner 3-设计类似于圆几何，其中任何3-点包含在唯一的圆中，其中点和圆（块）的数量是有限的。斯坦纳3-设计是基本的数学对象，在统计实验计划，数学通信理论（编码理论），密码学，计算机网络和许多数学分支中有应用。确定参数v（点的总数）和k、Steiner 3-设计存在的圆（块）大小以及相应的构造问题的问题是非常具有挑战性的问题。PI和他的合作者已经在这个问题上取得了重大进展。PI现在提出解决大v的存在性问题。 从历史上看，连续数学，即微积分，微分方程等，在物理现象及其理想化解决方案的制定中发挥了重要作用。这些数学方法解决了许多工程实际问题，如桥梁，飞机和宇宙飞船，电气网络的建设，也提供了理论物理的基础。随着计算机的出现，方法的“离散数学”，其中一个研究空间的100多个点正在发挥越来越重要的作用，在不同的领域，如规划最佳临床试验（实验设计），调度，网络，密码学等斯坦纳2-设计就像欧几里德几何除了数量的点和线是有限的。 斯坦纳3-设计研究有限几何中的三维构型。斯坦纳设计在试验的最佳规划、计算机网络、存在噪声的通信和密码学中有重要的应用。这个项目将集中在构造施泰纳设计的数学方法。

Steiner 2-designs are analogous to ordinary planes, (Euclidean or projective plane), except that the number of points and the number of lines is finite. The defining properties are (i) any two points is contained in exactly one line and (ii) all lines (viewed as subsets of points) contain the same number of points. In statistical terminology, points are same as treatments and lines are called blocks and the 2-design is called an incomplete block design. Incomplete block designs (or 2-designs) are needed to construct optimum experimental plans to determine the best treatment. Therefore it is important to determine for what parameters v, (the total number of points) and k (the number of points), a Steiner 2-design can be constructed. After decades of research culminating in the work of R. M. Wilson, the existence and construction problem for Steiner 2-designs is solved for at least large number of points. Steiner 3-designs are analogous to circle geometry in which any 3-points is contained in a unique circle, where the numbers of points and circles (blocks) are finite. Steiner 3-designs are fundamental mathematical objects which have application in statistical experimental plans, mathematical communication theory (coding theory), cryptography, computer networking and many branches of mathematics. The problem of determining parameters, v (the total number of points) and k, the circle (block) size for which a Steiner 3-design exists and the corresponding construction problem are very challenging problems. The PI and his collaborators already made significant progress on this problem. The PI now proposes to solve the existence problem for large v. Historically, continuous mathematics, i.e. calculus, differential equations, etc. played a fundamental role in the formulation of physical phenomena and their idealized solutions. These kinds of mathematical approaches solved many practical problems of engineering like building of bridges, aircraft and space ships, electrical networks and also provided the foundation of theoretical physics. With the advent of computers, methods of 'discrete mathematics' where one studies a space of finitely many points are playing increasingly important roles in diverse areas like planning optimum clinical trials (experimental designs), scheduling, networks, cryptography, etc. A Steiner 2- design is like Euclidean geometry except that the number of points and lines are finite. Steiner 3-designs studies 3-dimensional configurations in finite geometry. Steiner designs have important applications in optimum planning of trials, computer networks, communication in the presence of noise, and cryptography. This project will concentrate on mathematical methods for construction of Steiner designs.