The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version called parity prove following. Let $E_1$ $E_2$ two curves defined over number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness Shafarevich-Tate $E_2$, show Swinnerton-Dyer correctl...

Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...

Michael Dyer Artificial Intelligence Lab Computer Science Department UCLA Abst rac t This paper diacusses a method for learning thematic level structures, i.e. abstract plan/goal combinations, by observing the bad planning behavior of narrative characters. The learning method discussed is a one-trial, schema acquisition method, which is similar to DeJong's [DeJong, 1983]. The method uses constr...

Douglas J. Kelly*†||, Sean C. Warren*†, Sunil Kumar*, João L. Lagarto*, Benjamin T. Dyer*‡, Anca Margineanu*, Eric W.-F. Lam, Chris Dunsby*¶ and Paul M. W. French* *Photonics Group, Department of Physics, Imperial College London South Kensington Campus, London, SW7 2AZ, UK Institute of Chemical Biology, Department of Chemistry Imperial College London, South Kensington Campus London, SW7 2AZ, UK...

13. Dyer AR, Stamler J, Shekelle RB, Schoenberger J: The relationship of education to blood pressure. Circulation 54: 987, 1976 14. Becker M (ed): The health belief model and personal health behavior. Health Educ Monogr 2: 236, 1974 15. Baric L: Recognition of the "at risk" role: a means to influence health behavior. Int J Health Educ 12: 24, 1969 16. American Heart Association Long Range Plan,...

Currently microalgae are considered as an efficient biological producer of oil.Present study was conducted to explore the lipid yield percentage three microalgal strains Chlorellasp., Oedogonium sp. and Spirogyra Bligh Dyer, Soxhlet, Folch Microwave assistedextraction methods were employed for oil extraction. It observed that Dyer method withChloroform-methanol (1:2, v/v) gave maximum extractio...

To Takayuki Oda on his 60th birthday.

Let (W,S) be a Coxeter system and let w → w∗ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w−1 = w∗) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Pσ y,w w...