Abstract. Let E be an elliptic curve over Q with L-function L E (s). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions L E (1, χ), as χ runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than X for which L E (1, χ) vanishes is asymptotic to b E X 1/2 log e E X for some constants b E , e E depending only on E. We also compute explicitely the conjecture of Keating and Snaith about the moments of the special values L E (1, χ) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of L E (s).

Abstract. Let E be an elliptic curve over Q, with L-function L E (s). For any primitive Dirichlet character χ, let L E (s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions L E (s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that L E (1, χ) = 0, but that for any fixed prime k ≥ 7, there are only finitely many character of order k such that L E (1, χ) vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions K/Q of prime degree such that E acquires new rank over K.

Abstract. Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L (E, 1, χ) of the twisted L-function as χ ranges over Dirichlet characters of given order.

Aim: Stakeholders have an important role in both prevention and management of work related musculoskeletal disorders (WRMSDs). However, the scope of practice is unclear regarding the methods and levels of stakeholder engagement (SE) in management of WRMSDs. The aim of this review is to map the literature regarding various methods and levels of SE and how it is best implemented when managing WRMSDs. Methods: A systematic literature search was conducted in several major databases using key search phrases. Various methods and levels of SE in management

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