An approach is presented for addressing the challenge of model rebuilding after molecular replacement in cases where the placed template is very different from the structure to be determined. The approach takes advantage of the observation that a template and target structure may have local structures that can be superimposed much more closely than can their complete structures. A density-guide...

A Hurwitz correspondence is a multi-valued self-map of the moduli space of genus zero curves with n marked points, M0,n. Considering two n-marked curves (C, a1, ..., an) and (D, b1, ..., bn), given a map ρ from {1, ..., n} to itself and some ramification data, one obtains a Hurwitz correspondence on M0,n which sends a marked curve (D, b1, ..., bn) to any of the marked curves (C, a1, ..., an) fo...

Kodaira and Néron classified and described the geometry of the special fibers of the Néron model of an elliptic curve defined over a discrete valuation ring with a perfect residue field. Tate described an algorithm to determine the special fiber type by manipulating the Weierstrass equation. In the case of non-perfect residue fields, we discover new fiber types which are not on the Kodaira-Néro...

In 2007, Stange proposed a novel method for computing the Tate pairing on an elliptic curve over a finite field. This method is based on elliptic nets, which are maps from Z to a ring that satisfies a certain recurrence relation. In the present paper, we explicitly give formulae based on elliptic nets for computing the following variants of the Tate pairing: the Ate, Atei, R-Ate, and optimal pa...

In this paper, we present two efficient algorithms computing scalar multiplications of a point in an elliptic curve defined over a small finite field, the Frobenius map of which has small trace. Both methods use the identity which expresses multiplication-by-m maps by polynomials of Frobenius maps. Both are applicable for a large family of elliptic curves and more efficient than any other metho...

Given a divisor D on a tropical curve Γ, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system |D|. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the lin...