‎It is shown that the knowledge of a surjective morphism $Xto Y$ of complex‎ ‎curves can be effectively used‎ ‎to make explicit calculations‎. ‎The method is demonstrated‎ ‎by the calculation of $j(ntau)$ (for some small $n$) in terms of $j(tau)$ for the elliptic curve ‎with period lattice $(1,tau)$‎, ‎the period matrix for the Jacobian of a family of genus-$2$ curves‎ ‎complementing the classi...

We introduce Integral Curve Coordinates, which identify each point in a bounded domain with a parameter along an integral curve of the gradient of a function f on that domain; suitable functions have exactly one critical point, a maximum, in the domain, and the gradient of the function on the boundary points inward. Because every integral curve intersects the boundary exactly once, Integral Cur...

An algorithm is presented that in context of public key use of Elliptic Curve Cryptography allows discovery of the private key in worst case O(n). Charles Sauerbier December 2009 Humans have for as long as recorded history sought means to keep secrets and to communicate in secret. In modern times we have turned to electronic computations as means to facilitate both. The later problem of communi...

Assume we are given a hyperbolic, reducible morphism W . Every student is aware that exp (E2) ∼= ` ( −1, σC,B · C ) P̄−1 (−2) ∧ · · · × c ( S−5,−O′′ )

It is a classic idea in algebraic geometry to use degeneration method. In particular, it achieved successes recently in the studying of moduli spaces of vector bundles (See [Gi], [GL1], [GL2], [NR] and [S1]). In connection of string theory, it needs also to study the degeneration of moduli spaces of G-bundles for any reductive algebraic group G (See [F1],[F2]). Let X → B be a proper flat family...

The systematic data set on isotopic effects in spectator fragmentation collected recently at the GSI laboratory [1] permits investigating the N/Z dependence of the nuclear caloric curve which is of interest for several reasons. It is, for example, a basic assumption of isotopic reaction studies that a variation of the isotopic composition of the system will not change the reaction mechanism as ...

We describe an isomorphism of categories conjectured by Kontsevich. If M and M̃ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on M̃. We prove this equivalence when M is an elliptic curve and M̃ is its dual curve, exhibiting the dictionary in detail.