is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies tha...

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Séveri group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over th...

where v ranges over all places of Q and Qv is the completion of Q at v, denote its TateShafarevich group. As usual, L(E/Q, s) is the complex L-function of E over Q. Since E is now known to be modular, Kolyvagin’s work [11] shows that X(E/Q) is finite if L(E/Q, s) has a zero at s = 1 of order ≤ 1, and that gE/Q is equal to the order of the zero of L(E/Q, s) at s = 1. His proof relies heavily on ...

First we recall some standard definitions. By a d-dimensional convex body we mean a compact convex subset of Rd with non-empty interior. Two subsets of Rd with non-empty interiors are non-overlapping if they have no common interior point, and we say that they touch each other if they are non-overlapping and their intersection is non-empty. Denote by H(K) the translative kissing number of a d-di...

Dyer et al. (1) report a rapid decline in maize varieties grown by Mexican farmers, arguing that their results are in contrast with previously reported work. We believe that the support for this rapid decline is weak, and that the contrast with previously reported work is false. Dyer et al. (1) set up a straw man: they show that average household maize diversity reported in case studies has inc...

Selçuklular, Ça?r? Bey ile ba?layan Anadolu’ya ak?nlar?n? Sultan Tu?rul, Alparslan ve Melik?ah dönemlerinde sürdürerek burada kal?c? fetihler yapm??lard?r. Anadolu’da özellikle Malazgirt Sava?? sonras?nda h?zlanan Selçuklu ak?nlar? Sinop’un al?nmas?na vesile olmu? ilk olarak buray? komutanlar?ndan Karatekin fethetmi?tir (1084/1085). Karatekin’in Sinop’ta hâkimiyeti uzun sürmemi? tahminen 1086 y...

We propose an alternative characterisation of object-based distributed systems in terms of algebraic structures and topological spaces. Some examples are given in order to attest the adequacy of this approach to the subject. We also illustrate a method of transference of results from these mathematical theories that can further contribute to the advancement of distributed systems theory.