Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in G-metric spaces

In this paper, we present some coincidence point theorems in the setting of quasi-metric spaces that can be applied to operators which not necessarily have the mixed monotone property. As a consequence, we particularize our results to the field of metric spaces, partially ordered metric spaces and G-metric spaces, obtaining some very recent results. Finally, we show how to use our main theorems to obtain coupled, tripled, quadrupled and multidimensional coincidence point results.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Damping models in the truncated derivative nonlinear Schrödinger equation

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schrödinger equation, has been analyzed; wave 1 is linearly unstable with growth rate , and waves 2 and 3 are stable with damping 2 and 3, respectively. The dependence of gross dynamical features on the damping model as characterized by the relation between damping and wave-vector ratios, 2 /3, k2 /k3, and the polarization of the waves, is discussed; two damping models, Landau k and resistive k2, are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist as against flow contraction just requiring.In the case of right-hand RH polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if 2+3/2.

Measuring and predicting the dynamics of linear monodisperse entangled polymers in rapid flow through an abrupt contraction. A small angle neutron scattering study

Small-angle neutron scattering measurements on a series of monodisperse linear entangled polystyrene melts in nonlinear flow through an abrupt 4:1 contraction have been made. Clear signatures of melt deformation and subsequent relaxation can be observed in the scattering patterns, which were taken along the centerline. These data are compared with the predictions of a recently derived molecular theory. Two levels of molecular theory are used: a detailed equation describing the evolution of molecular structure over all length scales relevant to the scattering data and a simplified version of the model, which is suitable for finite element computations. The velocity field for the complex melt flow is computed using the simplified model and scattering predictions are made by feeding these flow histories into the detailed model. The modeling quantitatively captures the full scattering intensity patterns over a broad range of data with independent variation of position within the contraction geometry, bulk flow rate and melt molecular weight. The study provides a strong, quantitative validation of current theoretical ideas concerning the microscopic dynamics of entangled polymers which builds upon existing comparisons with nonlinear mechanical stress data. Furthermore, we are able to confirm the appreciable length scale dependence of relaxation in polymer melts and highlight some wider implications of this phenomenon.

Robust Controller Design of Networked Control Systems with Nonlinear Uncertainties

We address robust stabilization problem for networked control systems with nonlinear uncertainties and packet losses by modelling such systems as a class of uncertain switched systems. Based on theories on switched Lyapunov functions, we derive the robustly stabilizing conditions for state feedback stabilization and design packet-loss dependent controllers by solving some matrix inequalities. A numerical example and some simulations are worked out to demonstrate the effectiveness of the proposed design method.

Nonlinear pedagogy : implications for teaching games for understanding (TGfU)

Nonlinear Dynamics, provides a framework for understanding how teaching and learning processes function in Teaching Games for Understanding (TGfU). In Nonlinear Pedagogy, emergent movement behaviors in learners arise as a consequence of intrinsic self-adjusted processes shaped by interacting constraints in the learning environment. In a TGfU setting, representative, conditioned games provide ideal opportunities for pedagogists to manipulate key constraints so that self-adjusted processes by players lead to emergent behaviors as they explore functional movement solutions. The implication is that, during skill learning, functional movement variability is necessary as players explore different motor patterns for effective skill execution in the context of the game. Learning progressions in TGfU take into account learners’ development through learning stages and have important implications for organisation of practices, instructions and feedback. A practical application of Nonlinear Pedagogy in a national sports institute is shared to exemplify its relevance for TGfU practitioners.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis