Stable Relations and their Loci in Operator Algebra Variables

算子代数变量中的稳定关系及其轨迹

• 批准号: 9970799
• 负责人: Terry Loring
• 金额: $ 6.77万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970799&HistoricalAwards=false
• 关键词: Stable; Relations; their; Loci; Operator

AbstractLoringThe abstract properties of curves in a unit cube defined by polynomials are well understood. In particular, points that approximately are zeros of the polynomials are close to exact zeros. The same set of polynomials can become "unstable" when applied to matrices (and even more so when applied to operator algebra variables). That is, there can be contractive matrices that, when substituted into the polynomials give results near zero, and yet there is no actual matrix-root close to the approximate solution. We call such approximate solutions phantom solutions. These phantom approximate solutions can be misleading in the context of computer models. There is a notion of stability for a relation that does not require the relation to be polynomial. Much of this project will study relations that remain stable when applied to operator algebra variables. These are rare, but tend to be related to interesting invariants of operator algebras. Stability results limited to the realm of matrices will also be explored, with the hope of finding some applications to numerical analysis.Polynomial relations have stability properties that we rely on every day. Take the example of a laser cutting some flat material, where the cut-points are determined as the zero-set of a set of equations. If the x and y coordinates of the laser come close to satisfying these equations, we go ahead and fire the laser, for we are sure that close to this approximate solution there is an exact solution, and thus we are cutting close to where we should. In computer modeling, one's model is often based on more that just numerical variables. In particular, much of numerical analysis involves matrix-valued computations. It turns out that equations that are stable for ordinary values become unstable for matrix values. That is, there can be approximate solutions to equations that are not close to any true solutions. We call such things phantom approximate solutions. This project will study these phantom approximate solutions and ways to detect and avoid them.

摘要Loring多项式定义的单位立方体中曲线的抽象性质是很好理解的。 特别地，近似为多项式的零的点接近于精确的零。 当应用于矩阵时，同一组多项式可能变得“不稳定”（当应用于算子代数变量时甚至更不稳定）。 也就是说，可以有压缩矩阵，当代入多项式时给出接近零的结果，但没有接近近似解的实际矩阵根。 我们称这种近似解为幻影解。 这些虚幻的近似解在计算机模型的上下文中可能会产生误导。 有一个稳定的关系，不需要关系是多项式的概念。 本专题的大部分内容将研究应用于算子代数变量时保持稳定的关系。 这些是罕见的，但往往与有趣的算子代数不变量。 稳定性结果限于领域的矩阵也将探讨，希望找到一些应用程序的数值分析。多项式关系的稳定性，我们每天都依赖。 以激光切割一些平面材料为例，其中切割点被确定为一组方程的零集。 如果激光的x和y坐标接近满足这些方程，我们就继续发射激光，因为我们确信在接近这个近似解的地方有一个精确解，因此我们正在接近我们应该切割的地方。 在计算机建模中，一个人的模型往往是基于更多的只是数字变量。 特别是，许多数值分析涉及矩阵值计算。 结果表明，对于普通值稳定的方程对于矩阵值变得不稳定。 也就是说，方程可能有近似解，但不接近任何真解。 我们称之为幻影近似解。 本项目将研究这些幻影近似解以及检测和避免它们的方法。