Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.Comment: 23 page

Learning Terrain Dynamics: A Gaussian Process Modeling and Optimal Control Adaptation Framework Applied to Robotic Jumping

The complex dynamics characterizing deformable terrain presents significant impediments toward the real-world viability of locomotive robotics, particularly for legged machines. We explore vertical, robotic jumping as a model task for legged locomotion on presumed-uncharacterized, nonrigid terrain. By integrating Gaussian process (GP)-based regression and evaluation to estimate ground reaction forces as a function of the state, a 1-D jumper acquires the capability to learn forcing profiles exerted by its environment in tandem with achieving its control objective. The GP-based dynamical model initially assumes a baseline rigid, noncompliant surface. As part of an iterative procedure, the optimizer employing this model generates an optimal control strategy to achieve a target jump height. Experiential data recovered from execution on the true surface model are applied to train the GP, in turn, providing the optimizer a more richly informed dynamical model of the environment. The iterative control-learning procedure was rigorously evaluated in experiment, over different surface types, whereby a robotic hopper was challenged to jump to several different target heights. Each task was achieved within ten attempts, over which the terrain's dynamics were learned. With each iteration, GP predictions of ground forcing became incrementally refined, rapidly matching experimental force measurements. The few-iteration convergence demonstrates a fundamental capacity to both estimate and adapt to unknown terrain dynamics in application-realistic time scales, all with control tools amenable to robotic legged locomotion

Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse

A semianalytic theory for the pulse dynamics in similariton amplifiers and lasers is presented, based on a model pulse with adaptive shape. By changing a single parameter, this test function can be continuously tweaked between a pure Gaussian and a pure parabolic profile and can even represent sech-like pulses, the shape of a soliton. This approach allows us to describe the pulse evolution in the self-similar and other regimes of optical propagation. Employing the method of moments, the evolution equations for the characteristic pulse parameters are derived from the governing nonlinear Schr\"odinger or Ginzburg-Landau equation. Due to its greatly reduced complexity, this description allows for extensive parameter optimization, and can aid intuitive understanding of the dynamics. As an application of this approach, we model a soliton-similariton laser and validate the results against numerical simulations. This constitutes a semianalytic model of the soliton-similariton laser. Due to the versatility of the model pulse, it can also prove useful in other application areas

Appearance of the central singularity in spherical collapse

We analyze here the structure of non-radial nonspacelike geodesics terminating in the past at a naked singularity formed as the end state of inhomogeneous dust collapse. The spectrum of outgoing nonspacelike null geodesics is examined analytically. The local and global visibility of the singularity is also examined by integrating numerically the null geodesics equations. The possible implications of existence of such families towards the appearance of the star in late stages of gravitational collapse are considered. It is seen that the outgoing non-radial geodesics give an appearance to the naked central singularity as that of an expanding ball whose radius reaches a maximum before the star goes within its apparent horizon. The radiated energy (along the null geodesics) is shown to decay very sharply in the neighbourhood of the singularity. Thus the total energy escaping via non-radial null geodesics from the naked central singularity vanishes in the scenario considered here.Comment: 25 pages, 6 figure

Optimization of an Alkylpolyglucoside-Based Dishwashing Detergent Formulation.

The aim of this work was to formulate and optimize the washing performance of an alkylpolyglucoside-based dishwashing detergent. The liquid detergent was formulated with five ingredients of commercial origin: anionic (linear sodium alkylbenzenesulfonate and sodium laurylethersulfate), nonionic (C12–C14 alkylpolyglucoside) and zwitterionic (a fatty acid amide derivative with a betaine structure) surfactants, and NaCl for viscosity control. In addition to the plate test, other properties were investigated including ‘‘cloud point’’, viscosity, and emulsion stability. Statistical analysis software was used to generate a central composite experimental design. Then, a second order design and analysis of experiments approach, known as the Response Surface Methodology, was set up to investigate the effects of the five components of the formulation on the studied properties in the region covering plausible component ranges. The method proved to be efficient for locating the domains of concentrations where the desired properties were met

Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

phi^4 Kinks - Gradient Flow and Dynamics

The symmetric dynamics of two kinks and one antikink in classical (1+1)-dimensional $\phi^4$ theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink-antikink pair production is explored.Comment: 23 pages, LaTex, 11 eps figures. We have added some clarification of our metho

Polariton propagation in weak confinement quantum wells

Exciton-polariton propagation in a quantum well, under centre-of-mass quantization, is computed by a variational self-consistent microscopic theory. The Wannier exciton envelope functions basis set is given by the simple analytical model of ref. [1], based on pure states of the centre-of-mass wave vector, free from fitting parameters and "ad hoc" (the so called additional boundary conditions-ABCs) assumptions. In the present paper, the former analytical model is implemented in order to reproduce the centre-of-mass quantization in a large range of quantum well thicknesses (5a_B < L < inf.). The role of the dynamical transition layer at the well/barrier interfaces is discussed at variance of the classical Pekar's dead-layer and ABCs. The Wannier exciton eigenstates are computed, and compared with various theoretical models with different degrees of accuracy. Exciton-polariton transmission spectra in large quantum wells (L>> a_B) are computed and compared with experimental results of Schneider et al.\cite{Schneider} in high quality GaAs samples. The sound agreement between theory and experiment allows to unambiguously assign the exciton-polariton dips of the transmission spectrum to the pure states of the Wannier exciton center-of-mass quantization.Comment: 15 pages, 15 figures; will appear in Phys.Rev.

Enterprise Education Competitions: A Theoretically Flawed Intervention?

The demand for including enterprise in the education system, at all levels and for all pupils is now a global phenomenon. Within this context, the use of competitions and competitive learning activities is presented as a popular and effective vehicle for learning. The purpose of this chapter is to illustrate how a realist method of enquiry – which utilises theory as the unit of analysis – can shed new light on the assumed and unintended outcomes of enterprise education competitions. The case developed here is that there are inherent flaws in assuming that competitions will ‘work’ in the ways set out in policy and guidance. Some of the most prevalent stated outcomes – that competitions will motivate and reward young people, that they will enable the development of entrepreneurial skills, and that learners will be inspired by their peers – are challenged by theory from psychology and education. The issue at stake is that the expansion of enterprise education policy into primary and secondary education increases the likelihood that more learners will be sheep dipped in competitions, and competitive activities, without a clear recognition of the potential unintended effects. In this chapter, we employ a realist-informed approach to critically evaluate the theoretical basis that underpins the use of competitions and competitive learning activities in school-based enterprise education. We believe that our findings and subsequent recommendations will provide those who promote and practice the use of competitions with a richer, more sophisticated picture of the potential flaws within such activities.Peer reviewedFinal Published versio

Antibodies Against β2-Glycoprotein I Complexed With an Oxidised Lipoprotein Relate to Intima Thickening of Carotid Arteries in Primary Antiphospholipid Syndrome

To explore whether antibodies against β2-glycoprotein I (β2GPI) complexed to 7-ketocholesteryl-9-carboxynonanoate (oxLig-1) and to oxidised low-density lipoproteins (oxLDL) relate to paraoxonase activity (PONa) and/or intima media thickness (IMT) of carotid arteries in primary antiphospholipid syndrome (PAPS). As many as 29 thrombotic patients with PAPS, 10 subjects with idiopathic antiphospholipid antibodies (aPL) without thrombosis, 17 thrombotic patients with inherited thrombophilia and 23 healthy controls were investigated. The following were measured in all participants: β2GPI−oxLDL complexes, IgG anti-β2GPI−oxLig-1, IgG anti-β2GPI−oxLDL antibodies (ELISA), PONa, (para-nitrophenol method), IMT of common carotid (CC) artery, carotid bifurcation (B), internal carotid (IC) by high resolution sonography. β2GPI−oxLDL complex was highest in the control group (p < 0.01), whereas, IgG anti-β2GPI−oxLig1 and IgG anti-β2GPI−oxLDL were highest in PAPS (p < 0.0001). In healthy controls, β2GPI−oxLDL complexes positively correlated to IMT of the IC (p = 0.007) and negatively to PONa after correction for age (p < 0.03). PONa inversely correlated with age (p = 0.008). In PAPS, IgG anti-2GPI−oxLig-1 independently predicted PONa (p = 0.02) and IMT of B (p = 0.003), CC, (p = 0.03) and of IC (p = 0.04). In PAPS, PONa inversely correlated to the IMT of B, CC and IC (p = 0.01, 0.02 and 0.003, respectively). IgG anti-2GPI−oxLig-1 may be involved in PAPS related atherogenesis via decreased PON activity