Steiner 3-Designs

Steiner 2-设计类似于普通平面(欧几里得平面或射影平面)，不同之处在于点数和线数是有限的。定义属性是(I)任意两个点恰好包含在一条线中，以及(Ii)所有线(被视为点的子集)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。需要不完全区组设计(或2-设计)来构建最佳试验方案以确定最佳处理。因此，确定可以构造Steiner 2设计的参数v(点数)和k(点数)是很重要的。经过几十年的研究，在R.M.Wilson的工作中，Steiner 2-设计的存在和构造问题至少在大量的点上得到了解决。Steiner 3-设计类似于圆几何，其中任何3点都包含在一个唯一的圆中，其中点和圆(块)的数量是有限的。Steiner 3-设计是基本的数学对象，在统计试验计划、数学通信理论(编码理论)、密码学、计算机网络以及数学的许多分支中都有应用。确定参数v(点的总数)和k的问题、存在Steiner 3设计的圆(块)的大小以及相应的构造问题是非常具有挑战性的问题。PI和他的合作者在这个问题上已经取得了重大进展。PI现在建议解决大V的存在问题。历史上，连续数学，即微积分、微分方程等在物理现象的表述及其理想化解中发挥了基础作用。这些数学方法解决了许多实际的工程问题，如建造桥梁、飞机和宇宙飞船、电力网络等，也为理论物理提供了基础。随着计算机的出现，研究有限多个点空间的离散数学方法在不同的领域发挥着越来越重要的作用，如计划最佳临床试验(实验设计)、调度、网络、密码学等。Steiner 2-设计类似于欧几里德几何，只是点和线的数量是有限的。施泰纳3-设计研究有限几何中的三维构型。Steiner设计在试验的最优计划、计算机网络、存在噪声的通信和密码学中有重要的应用。这个项目将集中在施泰纳设计的数学方法上。

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.