Computing Admissble Transformation Volume

The ability to quantify part dimensional quality with respect to design specifications is of fundamental importance in product design and manufacturing. Our earlier work has proposed the use of admissible transformation volume as a part dimensional quality metric. That is, part quality is quantified based on how much an as-manufactured part shape can move while still remaining within a tolerance zone. A transformation is admissible if upon such a transformation a manufactured part shape falls within the design tolerance zone. A collection of such transformations in the transformation space forms an admissible transformation volume (ATV). In this paper, we present two properties of ATV: transformation invariant and decomposability. We then describe algorithms for computing ATV and how ATV properties facilitate complex tolerance check and reveal new insight on part producibility.

Linear Maps on Hermitian Matrices: The Stabilizer of an Inertia Class

Let T be a linear transformation acting on the space of n x n complex matrices. Let G(k) be the set of all hermitian matrices with k positive and n — k negative eigenvalues. Let T map some indefinite inertia class G(k) onto itself. We classify all such T. The possibilities are congruence, congruence followed by transposition, and, if n = 2k, it is possible that — T can be a congruence or a congruence followed by transposing. In other words, negation is an admissible transformation when n = 2k.