Stable Relations and their Loci in Operator Algebra Variables

由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，近似为多项式的零点的点接近于精确零点。同一组多项式在应用于矩阵时可能变得“不稳定”(当应用于算子代数变量时更是如此)。也就是说，可以有压缩矩阵，当代入多项式时，得到的结果接近于零，但没有接近近似解的实际矩阵根。我们称这种近似解为虚解。在计算机模型的背景下，这些虚幻的近似解可能具有误导性。对于不要求关系为多项式的关系，存在稳定性的概念。这个项目的大部分内容将研究当应用于算子代数变量时保持稳定的关系。这些都是罕见的，但往往与算子代数的有趣不变量有关。稳定性结果也将局限于矩阵领域，希望在数值分析中找到一些应用。多项式关系具有我们每天依赖的稳定性性质。以激光切割一些扁平材料为例，其中的切割点被确定为一组方程的零点。如果激光的x和y坐标接近于满足这些方程，我们就继续发射激光，因为我们确信，在这个近似解附近有一个精确的解，因此我们正在接近我们应该切割的地方。在计算机建模中，一个人的模型通常不仅仅是基于数字变量。特别值得一提的是，许多数值分析都涉及矩阵运算。结果表明，对于普通值是稳定的方程，对于矩阵值是不稳定的。也就是说，方程可以有不接近任何真解的近似解。我们把这样的事情称为幻影近似解。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。

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.