The cubic moment of central values of automorphic L-functions

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:Comment: 42 pages, published versio

Mappings of least Dirichlet energy and their Hopf differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

Language and the development of intercultural competence in an 'internationalised' university: staff and student perspectives

Within the currently diverse UK higher education environment, one important aspect of learning is the development of intercultural competence. The study that informs this paper investigated the ways intercultural competence was perceived as being enhanced or inhibited through current language and educational practices at a university that positions itself as internationally engaged and globally recognised. The project employed a multiple-case study design, examining eight academic programmes drawn from four different broad disciplinary groupings: social sciences, science, engineering, and management. Data were collected through individual, focus group and stimulated recall interviews, the latter using class observation recordings as a stimulus. The study revealed the ways in which language was exploited by both staff and students to convey particular meanings within an intercultural context. It was found that language choices, register and style were perceived as contributing to the pragmatic impact of either reinforcing barriers to or promoting intercultural competence development