Positivity in Algebraic Geometry (Draft for Parts 1 and 2) by R.K. Lazarsfeld

By R.K. Lazarsfeld

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Draw and describe the reachability √ region √of the endpoint v2 of the second link, under three conditions: (1) r ≤ 2/2, (2) 2/2 < r ≤ 1, and (3) r > 1. 10 (Challenge) 2D Angle-Limited Linkages: Two Constraints. Continuing the previous problem, also constrain the v1 joint to only turn within a 90◦ range. To be specific, the angle “v0 v1 v2 is between 90◦ (perpendicular to v0 v1 ) and 180◦ (aligned with v0 v1 ). Again draw and describe the reachability region of the v2 endpoint. Are there critical values of r at which the structure of the reachability region changes?

10, you know that angle constraints greatly complicate the possible motions of the chain. But in many applications, there are angle constraints, so they must be confronted. We consider two applications in this chapter, which are, surprisingly, related: protein folding and a certain pop-up card. Despite the whimsical chapter title, the real focus will be the “maxspan of 90◦ -chains,” the mathematical structure shared between the two applications. Several techniques and ideas from the previous chapters will resurface here, including induction and the triangle inequality.

Second, we will treat the bond angles between adjacent atoms along the backbone as fixed. This is nearly true. Third, we assume the chain permits free dihedral motion about each of its bonds. This is definitely not true, because one bond per amino acid (the so-called “peptide” bond) only permits two dihedral angles, 0◦ and 180◦ . Fourth, we will ignore all chemical and electrostatic forces, leaving only the geometry of dihedral motions and the restriction that the chain cannot pass through itself.

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