Uniaxial symmetry in nematic liquid crystals

Within the Landau-de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. We show that the uniaxial symmetry constraint is very restrictive and can in general not be satisfied, except in very symmetric situations. For one- and two-dimensional configurations, we characterize completely the uniaxial equilibria: they must have constant director. In the three dimensional case we focus on the model problem of a spherical droplet with radial anchoring, and show that any uniaxial equilibrium must be spherically symmetric. It was known before that uniaxiality can sometimes be broken by energy minimizers. Our results shed a new light on this phenomenon: we prove here that in one or two dimensions uniaxial symmetry is always broken, unless the director is constant. Moreover, our results concern all equilibrium configurations, and not merely energy minimizers.Comment: contains a new presentation of results in arXiv:1307.0295, and new result

Bifurcation analysis in a frustrated nematic cell

Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behaviour of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture agrees with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and J.R. Kelly. Some of the methods that we use in the present paper apply to other situations, and we present the proofs in a general setting. More precisely, the paper contains the proof of a general uniqueness result for a class of perturbed quasilinear elliptic systems, and general considerations about symmetric solutions and their stability, in the spirit of Palais' Principle of Symmetric Criticality

Interactive task design: Metachat and the whole learner

In this chapter the focus is on conversations about language between adult learners online, in synchronous and asynchronous postings. Socio-affective and social-semiotic perspectives are used, thus distancing the work somewhat from cognitive ways of looking at tasks. Because adults come to the task with diverse knowledge of both L2 and L1, the expectation is that metalinguistic interaction will enable them to swap expert and novice roles with each other within the constantly changing dynamics of the classroom. This if shown to be the case would advance an educational agenda favouring learner-directedness. Secondly, as metalinguistic conversations develop in directions that the learners feel like following, a greater degree of contingency can arise. This is considered in this paper as motivational for adults, and also as progressive, following van Lier (1996: 180) for whom in a contingent conversation "the agenda is shared by all participants and educational reality may be transformed". However, in seeking to satisfy his condition of contingency, the problem of designing tasks for greater spontaneity proves difficult. Therefore this study provide an ethnographic account of metalinguistic conversations by learners engaged in an online task, Simuligne, designed to address this difficulty. After studying data from the project forums, chat rooms and emails, we introduce a new perspective on the function of these conversations, which holds pointers for task design

Is there language teaching after global English?

This study documents a case of language education decline, and the role that distance-teaching expertise, allied with Information and Communication Technology (ICT) experience, can play in alleviating the problem. In the United Kingdom a number of factors have led to a crisis in the teaching and learning of European Languages Other Than English (ELOTE). One of the main determiners is the dominance of English as a lingua franca for Continental Western European countries, and another the political reluctance of the part of British governments to engage fully with the European Union. In the country where English is the mother tongue, the position of ELOTE is particularly critical. After quantifying the decline in demand for these languages, I will look at different ways in which language-teaching professionals have attempted to fight back, and I will focus on the benefits that may be derived from a strategy that combines ICT capacity with distance-learning methodologies, using the UK Open University (UKOU) as an example. The lessons drawn by that institution in different discipline areas over two decades will be applied to languages

Mechanism design with partially-specified participation games

This paper considers the implementation of an economic outcome under complete information when the strategic and informational details of the participation game are partially-specified. This means that full participation is required to be a subgame-perfect equilibrium for a large variety of extensive modifications of the simultaneous-move participation game in the same vein as Kalai [Large Robust Games, Econometrica 72 (2004) 1631-1665].mechanism design ; robust implementation ; strong Nash equilibrium ; partial subgame perfection ; collusion on participation

The future of the multilateral trading system

This 12-page transcript of the Australian lecture discussing the future of the multilateral trading system in the present era of global economic uncertainty

Identification and estimation of sequential English auctions

Brendstrup (2007) and Brendstrup and Paarsch (2006) claim that sequential English auction models with multi-unit demand can be identified from the distribution of the last stage winning price and without any assumption on bidding behavior in the earliest stages. We show that their identification strategy is not correct and that non-identification occurs even if equilibrium behavior is assumed in the earliest stages. For two-stage sequential auctions, an estimation procedure that has an equilibrium foundation and that uses the winning price at both stages is developed and supported by Monte Carlo experiments. Identification under general affiliated multi-unit demand schemes is also investigated.sequential auctions ; nonparametric identification ; nonparametric estimation

Singular perturbation of manifold-valued maps with anisotropic energy

We establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of $$E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),$$ where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are known in the isotropic case $W(\nabla u)=\vert \nabla u\vert^2$ and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider, all of these ingredients are absent. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than $\varepsilon$: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of $W$-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.Comment: The initial proof of the energy improvement lemma 2.2 contained a gap and has been corrected in this new versio