Given a finite set P ⊆ Rd, called a pattern, tP (n) denotes the maximum number of translated copies of P determined by n points in Rd. We give the exact value of tP (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that tP (n) = n −mr (n), where r is the rational affine dimension of P , and mr (n) is the r-Kruskal-Macaulay fun...

In the Drosophila wing, Hedgehog is made by cells of the posterior compartment and acts as a morphogen to pattern cells of the anterior compartment. High Hedgehog levels instruct L3/4 intervein fate, whereas lower levels instruct L3 vein fate. Transcriptional responses to Hedgehog are mediated by the balance between repressor and activator forms of Cubitus interruptus, CiR and CiA. Hedgehog reg...