Topics in Neural Networks, Stochastic and Dynamical Systems

9626575 Burton摘要原创性研究是由研究人员和他的同事在以下方面进行的：(I)神经网络的主题，例如前馈网络和无监督网络，(Ii)估计给定过去的一些记录的时间序列的预期极值的算法，(Iii)各种连分式展开的动力学和度量性质，以及(Iv)作为随机替换的极限而出现的平稳过程和随机场，以期向随机平铺系统迈进。所使用的方法来自遍历理论、概率和统计物理。这项工作包括四个主题：(I)神经网络的数学性质，(Ii)极值估计，(Iii)连分式算法，和(Iv)随机替换过程。(I)神经网络是一种自我编程的计算机，它通过榜样学习，基本上调整自己以适应其环境。神经网络--尽管在数学上还没有被很好地理解--已经在手写阅读器、DNA分类器和金融预测等领域得到了商业应用。有了更多的理解，就会有更开明的使用。(Ii)调查员和他的同事正在开发极值估计算法，以帮助解决近海石油平台等结构设计中需要的估计问题，这些结构需要很高的抵御风暴的概率。通常情况下，可以用来做出这些估计的历史数据有限。这些算法被用来估计未来20年可能在该结构附近发生的最强大的风暴。(Iii)连分式的性质是用分数逼近实数最经济的方法。它们由含有分数的分数组成，等等。研究人员和他的同事将这些作为动力系统来研究它们的性质。这项工作将概率论、群论和曲面流动的思想联系在一起。(4)调查员正在研究随机替换方案，因为它们是使用简单配方创建具有随机属性的符号串的自然方式。这个想法也可以用来生成具有随机性和确定性属性的平面或空间的平铺。构建从一组瓷砖(或简单形状)开始，并对这些瓷砖进行分解，以便每个分解的每一块都是其中一个原始瓷砖的比例复制品。此过程将继续，分解为越来越精细的比例，缩放比例。这可能会给出本质上的材料模型，就像该程序的非随机版本的情况一样。

9626575 Burton ABSTRACT Original research is pursued by the investigator and his colleagues on: (i) topics of neural networks, such as feed-forward networks and unsupervised networks, (ii) algorithms to estimate the expected extreme value of a time series given some record of the past, (iii) dynamical and metric properties of various continued fraction type expansions, and (iv) stationary processes and random fields that arise as limits of random substitutions with a view toward random tiling systems. The methods employed are taken from ergodic theory, probability, and statistical physics. This work consists of four topics: (i) mathematical properties of neural networks, (ii) extreme value estimation, (iii) continued fraction algorithms, and (iv) random substitution processes. (i) A neural network is a type of self-programming computer that learns by example, essentially adjusting itself to adapt to its environment. Neural networks--though not yet well understood, mathematically--are already in commercial use in such areas as handwriting readers, DNA classifiers, and financial forecasters. With greater understanding will come more enlightened use. (ii) The investigator and his colleagues are developing extreme value estimation algorithms to help solve an estimation problem needed in the design of structures such as off-shore oil platforms which require a high probability of withstanding storms. Typically, there is limited historical data from which to make these estimates. The algorithms are used to estimate the most powerful storm that is likely to occur in the vicinity of the structure in the next 20 years. (iii) Continued fractions have the property of being the most economical way to approximate a real number by a fraction. They consist of fractions containing fractions containing fractions, etc., etc. The investigator and his colleagues look at these as dynamical systems to study their properties. This work ties together ideas from probability theory, group theory, and flows on surfaces. (iv) The investigator is studying random substitution schemes as they are natural ways of creating strings of symbols with random properties, using a simple recipe. This idea may also be used to generate tilings of the plane or space that have properties of randomness and determinism. The construction begins with a set of tiles (or simple shapes) and decompositions of these tiles so that each piece of each decomposition is a scale replica of one of the original tiles. This procedure is continued, decomposing to finer and finer scales, zooming the scale. This may give models of materials in nature, as has been the case for non-random versions of this procedure.