Positivity of Lyapunov exponents for Anderson-type models on two coupled strings

We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol'dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices) to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies in the first model and for sufficiently large energies in the second model). The methods used allow for singularly distributed random parameters, including Bernoulli distributions.Comment: 19 page

The band spectrum of the periodic airy-schrodinger operator on the real line

We introduce the periodic Airy-Schr\"odinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a semiclassical regime in which the results are stated for a fixed value of the semiclassical parameter and are thus estimates instead of asymptotic results. We prove that there exists a sequence of explicit constants, which are zeroes of classical functions, giving upper bounds of the semiclassical parameter for which the spectral bands are in the semiclassical regime. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, in the semiclassical regime, we get estimates of the edges of every spectral bands and thus of the widths of every spectral bands and spectral gaps

Exploring reflection in pre-service teacher education: a social perspective on the application, value and factors influencing reflection

This study explores the implementation of reflection from the perspective of teacher trainees in two ‘four-week’ teacher training courses in the southeast of the UK and provides a social perspective into reflection. The study reveals that participants conceptualise the different aspects of reflection, written, oral, and reflections made about others, differently in terms of their application and value, and shows a number of factors impacting the understanding and practice of such aspects. Accordingly, the application of written reflection is viewed as an idiosyncratic process and is understood as a systematic activity that is underexplored, while oral reflection is described as an interactional process, characterised by discussion and participation, and portrayed to be a practice of an important value. Participants’ views on the reflections made on others, both peers and experienced teachers, define such reflections in terms of a social process. Three factors impacting reflection are psychological involving trainees’ emotions, contextual, identifying aspects such as the context of reflection, trainees’ status as novice teachers, experience in reflection, and the guidance required, and social, revealing the impact of community dynamics on participants’ reflections. The study followed a qualitative approach and was framed within an ethnographic methodology with an autoethnographic aspect, recounting my insider experience during fieldwork. Data were collected in the two teacher training courses, using interviews, observations of the field, and an analysis of participants’ reflections, then analysed thematically. The findings of the study have methodological implications, focusing on the importance of insider experiences in research, pedagogical implications, suggesting a multi-dimensional framework for reflection, useful in teacher education contexts and beyond, and theoreticalimplications, proposing the integration of a social perspective to the study of reflection

A matrix-valued point interactions model

We study a matrix-valued Schr\"odinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the H\"older continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its H\"older continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents

Localization for a matrix-valued Anderson model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr\"odinger operators, acting on L^2(\R)\otimes \C^N, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the F\"urstenberg group of these operators, valid on an interval $I\subset \R$, they exhibit localization properties on $I$, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the F\"urstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters

Comparison of Selection Methods in On-line Distributed Evolutionary Robotics

In this paper, we study the impact of selection methods in the context of on-line on-board distributed evolutionary algorithms. We propose a variant of the mEDEA algorithm in which we add a selection operator, and we apply it in a taskdriven scenario. We evaluate four selection methods that induce different intensity of selection pressure in a multi-robot navigation with obstacle avoidance task and a collective foraging task. Experiments show that a small intensity of selection pressure is sufficient to rapidly obtain good performances on the tasks at hand. We introduce different measures to compare the selection methods, and show that the higher the selection pressure, the better the performances obtained, especially for the more challenging food foraging task

Effects of urban effluents on spatial structure, morphology and total phenols of Posidonia oceanica: Comparison with a reference site

In the Mediterranean, the discharge of urban effluents in coastal areas adversely affects Posidonia oceanica, since it induces nutrient enrichment and a decrease of water clarity, but data from the southern Mediterranean are very sparse. In this paper we examine the differences in P. oceanica parameters from a disturbed site (Bou Ismaïl) and a reference site (Kouali), at meadow level (shoot density, meadow cover), at individual level (leaf biometry, A coefficient, epiphytic index) and at biochemical level (total phenols). The differences were examined at the upper limit (− 5 m) and the lower limit (− 20 m), during an annual cycle. Results indicated a significantly lower mean meadow cover at the upper limits of the disturbed site compared to the reference site. The leaf length and leaf area were also lower at the disturbed site, and this difference was recorded all year round at the upper limit. The epiphytic index and the A coefficient of adult leaves were higher only in July and October at the upper limits of the disturbed site. In contrast, total phenol concentration of the leaves did not show any response to disturbances related to urban effluents. However, the biosynthesis of phenolic substances at the two sites was significantly higher in the intermediate leaves, in winter, when growth slows down. The unexpected higher value of total phenol concentration recorded in the intermediate leaves and sheaths in spring at the reference site is discussed