Fractal solutions of linear and nonlinear dispersive partial differential equations

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.Comment: 28 page

Near-linear dynamics in KdV with periodic boundary conditions

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction

Stable directions for small nonlinear Dirac standing waves

We prove that for a Dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues close enough, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schr\"{o}dinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation.Comment: 62 page

On non-local variational problems with lack of compactness related to non-linear optics

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.Comment: 30 page

Effect of mixture proportions on the drying shrinkage and permeation properties of high strength concrete containing class F fly ash

Sustainability of concrete can be improved by using large volume of fly ash as a replacement of cement and by ensuring improved durability of concrete. Durability of concrete containing large volume of class F fly ash is dependent on the design of mixture proportions. This paper presents an experimental study on the effect of mixture proportions on the drying shrinkage and permeation properties of high strength concrete containing large volume local class F fly ash. Concrete mixtures were designed with and without adjustments in the water to binder ratio (w/b) and the total binder content to take into account the incorporation of fly ash up to 40% of total binder. Concretes, in which the mixture proportions were adjusted for fly ash inclusion achieved equivalent strength of the control concrete and showed enhanced properties of drying shrinkage, sorptivity, water permeability and chloride penetration as compared to the control concrete. The improvement of durability properties was less significant when no adjustments were made to the w/b ratio and total binder content. The results show the necessity of the adjustments in mixture proportions of concrete to achieve improved durability properties when using class F fly ash as a cement replacement

An evaluation of three DoE-guided meta-heuristic-based solution methods for a three-echelon sustainable distribution network

This article evaluates the efficiency of three meta-heuristic optimiser (viz. MOGA-II, MOPSO and NSGA-II)-based solution methods for designing a sustainable three-echelon distribution network. The distribution network employs a bi-objective location-routing model. Due to the mathematically NP-hard nature of the model a multi-disciplinary optimisation commercial platform, modeFRONTIER®, is adopted to utilise the solution methods. The proposed Design of Experiment (DoE)-guided solution methods are of two phased that solve the NP-hard model to attain minimal total costs and total CO2 emission from transportation. Convergence of the optimisers are tested and compared. Ranking of the realistic results are examined using Pareto frontiers and the Technique for Order Preference by Similarity to Ideal Solution approach, followed by determination of the optimal transportation routes. A case of an Irish dairy processing industry’s three-echelon logistics network is considered to validate the solution methods. The results obtained through the proposed methods provide information on open/closed distribution centres (DCs), vehicle routing patterns connecting plants to DCs, open DCs to retailers and retailers to retailers, and number of trucks required in each route to transport the products. It is found that the DoE-guided NSGA-II optimiser based solution is more efficient when compared with the DoE-guided MOGA-II and MOPSO optimiser based solution methods in solving the bi-objective NP-hard three-echelon sustainable model. This efficient solution method enable managers to structure the physical distribution network on the demand side of a logistics network, minimising total cost and total CO2 emission from transportation while satisfying all operational constraints

Loneliness and Social Internet Use: Pathways to Reconnection in a Digital World?

With the rise of online social networking, social relationships are increasingly developed and maintained in a digital domain. Drawing conclusions about the impact of the digital world on loneliness is difficult because there are contradictory findings, and cross-sectional studies dominate the literature, making causation difficult to establish. In this review, we present our theoretical model and propose that there is a bidirectional and dynamic relationship between loneliness and social Internet use. When the Internet is used as a way station on the route to enhancing existing relationships and forging new social connections, it is a useful tool for reducing loneliness. But when social technologies are used to escape the social world and withdraw from the “social pain” of interaction, feelings of loneliness are increased. We propose that loneliness is also a determinant of how people interact with the digital world. Lonely people express a preference for using the Internet for social interaction and are more likely to use the Internet in a way that displaces time spent in offline social activities. This suggests that lonely people may need support with their social Internet use so that they employ it in a way that enhances existing friendships and/or to forge new ones