Minkowski Geometric Algebra of Complex Sets: Theory, Algorithms, and Applications

Calif BerkeleyMinkowski几何代数的Rida FaroukiU研究了由给定的复数CSET算子得到的所有复数对的和所填充的复数集。由于Rn中的向量加和Minkowski和已经得到了深入的研究，从理论和计算的角度来看，由复数乘法规则诱导的R2中的Minkowski积是一个相对未被探索的概念。从概念上讲，Minkowski几何代数是实数区间算术到复数集的自然推广。然而，随着实数到复数的过渡，实区间的平凡几何及其加法和乘法闭包被放弃。Minkowski几何代数的二维特征赋予了它丰富的几何内容，其中简单的操作数(如圆盘)产生用解析曲线描述的精细结果，如Cassini的笛卡尔椭圆及其高阶推广，而对于一般集合，需要复杂的算法来逼近Minkowski组合。该代数不仅是监视复杂C变量计算中不确定性传播的基本工具，还为两个二维形状算子和(正或逆)波前折射问题的光学描述提供了一种通用的语言。它也是具有不确定参数的系统的稳定性分析的基础在劳斯-丘尔维茨判据和哈里托诺夫稳健性定理的背景下。

ABSTRACT0202179Rida FaroukiU of Calif BerkeleyMinkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers drawn from given complex Cset operands.Whereas the Minkowski sum under vector addition in R n has been thoroughly studied,from both the theoretical and computationalperspective,the Minkowski product in R 2 induced by the multiplication rule for complex numbers is a relatively unexplored oncept.Conceptually, Minkowski geometric algebra is the natural generalization of real interval arithmetic to complex Cnumber sets.With the transition from real to complex,however,the trivial geometry of real intervals and their consequent closure under addition and multiplication is relinquished. Thetwo Cdimensional haracter of Minkowski geometric algebra endows it with a rich geometrical ontent,in which simple operands (e.g.,circular disks) yield subtle results described by analytic curves such as the Cartesian oval ovals of Cassini and their higher Corder generalizations,while sophisticated algorithms are required to approximate Minkowski combinations for general sets.Apart from being a basic tool to monitor the propagation of uncertainty in complex Cvariable computations,the algebra o .ers a versatile language fortwo Cdimensional shape operators and the description of (direct or inverse) wavefront re .ection/refraction problems in optics.It is also fundamental to the stability analysis of systems with uncertain parameters,in the context of the Routh CHurwitz criterion and the Kharitonov robustness theorem.