Alkaloids retamine, anagyrine, lupanine, 17-oxoretamine, 12-alpha-hydroxylupanine were detected, along with four others unidentified compounds in the aerial parts of Genista ephedroides D.C.

Quick biodiversity studies on poorly studied taxa and areas are increasingly popular for setting conservation priorities over a wide range of spatial scales. However, the implementation of such studies is complicated by the variable extent to which the different criteria used in prioritisation are correlated to each other. Using methods of constrained ordination, we examined the specieshabitat ...

We discuss a famous problem about right triangles with rational side lengths. This elementarysounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis). 6.

The need for antioxidant agents, which can prevent oxidative stress, has become a major priority. Saharan plants could constitute a reservoir of effective biomolecules useful as antioxidants. Genista saharae, wich is anendemic plant of Algeria,is a leafless spontaneous fabaceae called "Tellegit" by local people. As it is known to be a source of chemical compounds which are characterized by anti...

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.

We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Ω2n+1(p) and PΩ4n(p) both occur as the Galois group of a Galois extension of the rationals for all integers n ≥ 2 and all primes p ≥ 5. We obtain our representations by studying families of twists of elliptic curves a...

4 The Birch and Swinnerton-Dyer conjecture 18 4.1 Analytic rank 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1.1 Kolyvagin’s proof . . . . . . . . . . . . . . . . . . . . . 18 4.1.2 A variant . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.3 Kato’s proof . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Analytic rank 1 . . . . . . . . . . . . . . . . . . . . . . . . ...

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of MordellW...

In this paper we present a method for explicitly computing rational points on elliptic curves using Heegner points. This method was crucial to the proof of Gross-Zagier, which proves the rank one case of the Birch and Swinnerton-Dyer Conjecture. Although this use of Heegner points can be found in many books and articles, we strive here to present it in a more concrete and complete form, and usi...