by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

The retinoid X receptor (RXR) is activated by its often elusive cognate ligand, 9-cis-retinoic acid (9-cis-RA). In flies and moths, molting is mediated by a heterodimer ecdysone receptor consisting of the ecdysone monomer (EcR) and an RXR homolog, ultraspiracle (USP); the latter is believed to have diverged from its RXR origin. In the more primitive insect, Locusta migratoria (Lm), RXR is more ...

Let E/Q be an elliptic curve defined by the equation y 2 = x 3 + ax + b. For a prime p, p ∤ ∆ = −16(4a 3 + 27b 2) = 0, define N p = p + 1 − a p = |E(F p)|. As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c, p≤x,p∤∆ N p p ∼ c(log x) r , x → ∞. Let α p and β p be the eigenvalues of the Frobenius at p. Define ˜ c n =    α k ...

We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: For any real γ, δ, lim inf |n|→∞ n〈nα− γ〉〈nβ − δ〉 = 0, (0.1) where 〈·〉 denotes the distance from the nearest integer. The existence of even a single pair that satisfies (0.1), solves an open problem of Cassels [Ca] from the 50’s. We then prove that if 1, α, β spa...

In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson’s rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequa...

PURPOSE The purpose of this study was to profile the endogenous phospholipid species in the retinal tissue of the S334ter-3 rat model of retinal degeneration. Retinal tissue was collected from S334ter-3 rats at postnatal day (P) 20, P30, and P60, while control retinal samples were collected from Sprague-Dawley (SD) rats at the same time points for comparison. METHODS Lipids were extracted usi...