Some examples of special Lagrangian tori

A short paper giving some examples of smooth hypersurfaces M of degree n+1 in complex projective n-space that are defined by real polynomial equations and whose real slice contains a component diffeomorphic to an n-1 torus, which is then special Lagrangian with respect to the Calabi-Yau metric on M.

Calibrated embeddings in the special Lagrangian and coassociative cases

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.

I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Numerical study about natural escape and capture routes by the Moon via Lagrangian points L1 and L2

The lunar sphere of influence, whose radius is some 66,300 km, has regions of stable orbits around the Moon and also regions that contain trajectories which, after spending some time around the Moon, escape and are later recaptured by lunar gravity. Both the escape and the capture occur along the Lagrangian equilibrium points L1 and L2. In this study, we mapped out the region of lunar influence considering the restricted three-body Earth-Moon-particle problem and the four-body Sun-Earth-Moon-particle (probe) problem. We identified the stable trajectories, and the escape and capture trajectories through the L I and L2 in plots of the eccentricity versus the semi-major axis as a function of the time that the energy of the osculating lunar trajectory in the two-body Moon-particle problem remains negative. We also investigated the properties of these routes, giving special attention to the fact that they supply a natural mechanism for performing low-energy transfers between the Earth and the Moon, and can thus be useful on a great number of future missions. (C) 2007 Published by Elsevier Ltd on behalf of COSPAR.

The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current

This article reviews several recently developed Lagrangian tools and shows how their com- bined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time depen- dent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.

Public sector evaluation of festivals and special events

This research project set out to explore Unitary Authority (UA) involvement in festivals and special events across Wales. It considers the level and nature of UA involvement and investigates activity by event purpose; reasons for, and characteristics of, UA engagement; and, crucially, the extent and nature of event evaluation. The study’s aim was to begin the development of a baseline of information for further research into the growing use of festivals and special events as a strategy for local economic development in Wales. A quantitative survey approach facilitated a comprehensive snapshot of UA responses whilst also incorporating discursive elements. A telephone survey was designed and undertaken with representatives of all 22 UA departments responsible for festivals and events in Wales. The research reveals a significant level of festival and special event activity across Wales, supported primarily for its perceived socio-cultural value. However, evaluation would appear to be focused on improving processes and measuring economic outputs rather than assessing whether socio-cultural objectives are being achieved. Whilst overwhelmingly positive about efforts to improve approaches to evaluation, respondents held clear views about the complications most likely to hamper any such efforts. These responses focused upon the need for flexibility, cost effectiveness and comparability across festival and special event typologies.

Faster pairings on special Weierstrass curves

This paper presents efficient formulas for computing cryptographic pairings on the curve y 2 = c x 3 + 1 over fields of large characteristic. We provide examples of pairing-friendly elliptic curves of this form which are of interest for efficient pairing implementations.

The enactment of the New Basics Project in a special school

This study investigates the impact of the New Basics Project on teachers at a special school for students with intellectual impairments. The study is aimed at exploring the complex nature of the work of special educators as they enact the New Basics curriculum with a particular focus on the teachers’ opinions about challenges that arose for their curriculum, pedagogy and assessment practices. Attention is also paid to how the principal’s leadership supported the enactment of the New Basics in respect to what he did and why he used particular strategies. The nine teachers and their principal were involved in a series of in-depth, semi-structured interviews from one of only three special schools in phase one of the New Basics trial in Queensland, Australia. These interviews produced data from the special educators as they were confronted with a new curriculum that challenged their previous teaching practices. The enactment of the New Basics curriculum occurred within the context of a state-sanctioned mandate to provide alternative programs to those offered in mainstream schools, for students with special needs. This thesis explores these teachers' experiences using critical theory as a basis for analyzing their opinions on issues such as the role of the special educator, tensions between old and new curricula, pedagogical and assessment practices, and connections between the at-school learning experiences for intellectually impaired students and the realities of post-school life. The investigation also examines the leadership conduct of the principal in changing times at the school. The findings suggest that the New Basics has played a significant role in providing structures for developing communities of practice amongst teachers; in supporting special educators to focus more on the educational needs of the students (e.g., literacy, numeracy, financial planning) and less on their medical needs (e.g., toileting, feeding, personal hygiene); and supporting school leadership that empowers and listens critically to teachers as essential components of the successful enactment of curriculum reforms like the New Basics.

The Special Broadcasting Service after 30 years : public service media and new ways of thinking about media and citizenship

This article considers the distinctive ways in which the Special Broadcasting Service (SBS) has evolved over its history since 1980, and how it has managed competing claims to being a multicultural yet broad-appeal broadcaster, and a comprehensive yet low-cost media service. It draws attention to the challenges presented by a global rethinking of the nature of citizenship and its relationship to media, for which SBS is well placed as a leader, and the challenges of online media for traditional public service media models, where SBS has arguably been a laggard, particularly when compared with the Australian Broadcasting Corporation (ABC). It notes recent work that has been undertaken by the author with others into user-created content strategies at SBS and how its online news and current affairs services have been evolving in recent years.