Curry signed integrals into Hodge star

[lukem12345]

Converting between primal and dual differential forms requires two pieces of information: the n-dimensional volumes of the primal and dual regions of space, and their sign. In the continuous theory, the Hodge star operator accommodates both of these pieces of information. The discrete theory also accounts for this. See for example $\S 4.1$ from Hirani’s thesis.

Currently, CombinatorialSpaces treats the multiplication by volume-ratios separately from accommodating the sign. In other words, the Hodge star handles only the volume computation, and the sign operator must be called on-demand. And indeed sign must always be called for other discrete operators to compute correctly, since they expect signed integrals.

This two-stage process has been a source of confusion. For example, one may expect dual_derivative(0,sd) * interior_product(sd, u, v) to compute the exterior derivative of the interior product. But this is not so. Before the changes in this PR, the correct answer was to compute dual_derivative(0,sd) * (sign(2,sd) .* interior_product(sd, u, v)), since the Hodge star inside the interior-product computation does not compute the signed volume, which the dual exterior derivative expects. The original 1-1 Lie derivative operation did not account for this.

This has been partially addressed on case-by-case bases. For example, PR 78 explicitly added a sign computation inside the primal-dual 1-1 wedge product.

This PR explicitly captures the sign computation in the Hodge star computation, so discrete operator composition works correctly in all cases.