We present a method for the automatic assembly of apictorial jigsaw puzzles. This method relies on integral area invariants for shape matching and an optimization process to aggregate shape matches into a final puzzle assembly. Assumptions about indi
vidual piece shape or arrangement are not necessary. We illustrate our method by solving example puzzles of various shapes and sizes.
Calculation of the vacuum polarization, $<phi^2(x)>$, and expectation value of the stress tensor, $<T_{mu u}(x)>$, has seen a recent resurgence, notably for black hole spacetimes. To date, most calculations of this type have been done only in four di
mensions. Extending these calculations to $d$ dimensions includes $d$-dimensional renormalization. Typically, the renormalizing terms are found from Christensens covariant point splitting method for the DeWitt-Schwinger expansion. However, some manipulation is required to put the correct terms into a form that is compatible with problems of the vacuum polarization type. Here, after a review of the current state of affairs for $<phi^2(x)>$ and $<T_{mu u}(x)>$ calculations and a thorough introduction to the method of calculating $<phi^2(x)>$, a compact expression for the DeWitt-Schwinger renormalization terms suitable for use in even-dimensional spacetimes is derived. This formula should be useful for calculations of $<phi^2(x)>$ and $<T_{mu u}(x)>$ in even dimensions, and the renormalization terms are shown explicitly for four and six dimensions. Furthermore, use of the finite terms of the DeWitt-Schwinger expansion as an approximation to $<phi^2(x)>$ for certain spacetimes is discussed, with application to four and five dimensions.
