Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

Invariant Solutions of Richards' Equation for Water Movement in Dissimilar Soils

Scaling methods allow a single solution to Richards' equation (RE) to suffice for numerous specific cases of water flow in unsaturated soils. During the past half-century, many such methods were developed for similar soils. In this paper, a new method is proposed for scaling RE for a wide range of dissimilar soils. Exponential-power (EP) functions are used to reduce the dependence of the scaled RE on the soil hydraulic properties. To evaluate the proposed method, the scaled RE was solved numerically considering two test cases: infiltration into relatively dry soils having initially uniform water content distributions, and gravity-dominant drainage occurring from initially wet soil profiles. Although the results for four texturally different soils ranging from sand to heavy clay (adopted from the UNSODA database) showed that the scaled solution were invariant for a wide range of flow conditions, slight deviations were observed when the soil profile was initially wet in the infiltration case or deeply wet in the drainage case. The invariance of the scaled RE makes it possible to generalize a single solution of RE to many dissimilar soils and conditions. Such a procedure reduces the numerical calculations and provides additional opportunities for solving the highly nonlinear RE for unsaturated water flow in soils.

Controlling the transport of an ion: classical and quantum mechanical solutions

The accurate transport of an ion over macroscopic distances represents a challenging control problem due to the different length and time scales that enter and the experimental limitations on the controls that need to be accounted for. Here, we investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. The applicability of each of the control methods depends on the length and time scales of the transport. Our comprehensive set of tools allows us make a number of observations. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepacket propagation, wavepacket squeezing is the only quantum effect that comes into play for a large range of trapping parameters. We show that this can be corrected by a compensating force derived from invariant based inverse engineering, without a significant increase in the operation time.

On bifurcation and symmetry of solutions of symmetric nonlinear equations with odd-harmonic forcings

In this work we study existence, bifurcation, and symmetries of small solutions of the nonlinear equation Lx = N(x, p, epsilon) + mu f, which is supposed to be equivariant under the action of a group OHm, and where f is supposed to be OHm-invariant. We assume that L is a linear operator and N(., p, epsilon) is a nonlinear operator, both defined in a Banach space X, with values in a Banach space Z, and p, mu, and epsilon are small real parameters. Under certain conditions we show the existence of symmetric solutions and under additional conditions we prove that these are the only feasible solutions. Some examples of nonlinear ordinary and partial differential equations are analyzed. (C) 1995 Academic Press, Inc.

Unpacking ‘lessons learned’ : investigating failures and considering alternative solutions

One of the most common ways to share project knowledge is to capture the positive and negative aspects of projects in the form of lessons learned (LL). If effectively used, this process can assist project managers in reusing project knowledge and preventing future projects from repeating mistakes. Nevertheless, the process of capturing, storing, reviewing and reusing LL often remains suboptimal. Despite the potential for rich knowledge capture, lessons are often documented as simple, line-item statements devoid of context. Findings from an empirical investigation across four cases revealed a range of reasons related to the perceived quality, process and visibility of LL that lead to their limited use and application. Drawn from the cross-case analysis, this paper investigates an integrated approach to LL involving the use of a collaborative Web-based tool, which is easily accessible, intelligible and user-friendly, allowing more effective sharing of project knowledge and overcoming existing problems with LL.

Two dynamic discrete choice estimation problems and simulation method solutions

This paper considers two problems that frequently arise in dynamic discrete choice problems but have not received much attention with regard to simulation methods. The first problem is how to simulate unbiased simulators of probabilities conditional on past history. The second is simulating a discrete transition probability model when the underlying dependent variable is really continuous. Both methods work well relative to reasonable alternatives in the application discussed. However, in both cases, for this application, simpler methods also provide reasonably good results.

A framework for understanding and generating integrated solutions for residential peak energy demand

Supplying peak energy demand in a cost effective, reliable manner is a critical focus for utilities internationally. Successfully addressing peak energy concerns requires understanding of all the factors that affect electricity demand especially at peak times. This paper is based on past attempts of proposing models designed to aid our understanding of the influences on residential peak energy demand in a systematic and comprehensive way. Our model has been developed through a group model building process as a systems framework of the problem situation to model the complexity within and between systems and indicate how changes in one element might flow on to others. It is comprised of themes (social, technical and change management options) networked together in a way that captures their influence and association with each other and also their influence, association and impact on appliance usage and residential peak energy demand. The real value of the model is in creating awareness, understanding and insight into the complexity of residential peak energy demand and in working with this complexity to identify and integrate the social, technical and change management option themes and their impact on appliance usage and residential energy demand at peak times.

Osmolality electrolyte and carbohydrate type and oral rehydration solutions: A controlled study to compare the efficacy of two commercially available solutions (osmolalities 240 mmol/L and 340 mmol/L)

An open-label, inpatient study was undertaken to compare the efficacy of two oral rehydration solutions (ORS) given randomly to children aged 1-10 years who had acute gastroenteritis with mild or moderate dehydration (n = 45). One solution contained 60 mmol/L sodium and 1.8% glucose, total osmolality 240 mosm/l (gastrolyte, Rhone-poulenc, Rorer) and the other contained 26 mmol/l sodium, 2.7% glucose and 3.6% sucrose, total osmolality 340 mOsm/l (Glucolyte, Gilseal). Analysis of data indicated that Gastrolyte therapy resulted in significantly fewer episodes and volume of vomiting over all time periods in comparison to Glucolyte and significantly less stool volume during the first 8 h and in the 0-24 h period. The differences between treatments in degree of dehydration at each follow-up period, duration of diarrhea, and duration of hospital stay were not significant. No adverse drug reactions occurred. Six patients received intravenous rehydration treatment and were considered treatment failures. We conclude that oral rehydration therapy is safe and efficacious in the management of dehydration in acute diarrhoea and that the lower osmolar rehydration solution has clinically marginal advantages.

Nonparametric traversability estimation in partially occluded and deformable terrain

Terrain traversability estimation is a fundamental requirement to ensure the safety of autonomous planetary rovers and their ability to conduct long-term missions. This paper addresses two fundamental challenges for terrain traversability estimation techniques. First, representations of terrain data, which are typically built by the rover’s onboard exteroceptive sensors, are often incomplete due to occlusions and sensor limitations. Second, during terrain traversal, the rover-terrain interaction can cause terrain deformation, which may significantly alter the difficulty of traversal. We propose a novel approach built on Gaussian process (GP) regression to learn, and consequently to predict, the rover’s attitude and chassis configuration on unstructured terrain using terrain geometry information only. First, given incomplete terrain data, we make an initial prediction under the assumption that the terrain is rigid, using a learnt kernel function. Then, we refine this initial estimate to account for the effects of potential terrain deformation, using a near-to-far learning approach based on multitask GP regression. We present an extensive experimental validation of the proposed approach on terrain that is mostly rocky and whose geometry changes as a result of loads from rover traversals. This demonstrates the ability of the proposed approach to accurately predict the rover’s attitude and configuration in partially occluded and deformable terrain.

Ion exchange of sodium chloride and sodium bicarbonate solutions using strong acid cation resins in relation to coal seam water treatment

Coal seam gas production has resulted in the production of large volumes of associated water which contains dissolved salts dominated by sodium chloride and sodium bicarbonate. Ion exchange using synthetic resins has been proposed as a method for desalination of coal seam water to make it suitable for various beneficial reuse options. This study investigated the behaviour of solutions of sodium chloride and sodium bicarbonate with respect to exchange with Lanxess S108H strong acid cation (SAC) resin. Equilibrium isotherms were created for solutions of NaCl and NaHCO3 and an actual sample of coal seam water from the Surat Basin in southern Queensland. The exchange of sodium ions arising from sodium bicarbonate was found to be considerably more favourable than exchange of sodium ions from sodium chloride solutions. This latter behaviour was attributed to the secondary decomposition of bicarbonate species under acidic conditions which resulted in the evolution of carbon dioxide and formation of water. The isotherm profiles could not be satisfactorily fitted by a single isotherm model such as the Langmuir expression. Instead, two Langmuir equations had to be simultaneously applied in order to fit the sections of the isotherm attributable to sodium ion exchange from sodium bicarbonate and sodium chloride. The shape of the isotherm profile was dependent upon the ratio of sodium chloride to sodium bicarbonate in solution and there was a high degree of correlation between simulated and actual coal seam water solutions.

Conservation law with the flux function discontinuous in the space variable- II - Convex-concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex-concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine-Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L-1. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented. (C) 2006 Elsevier B.V. All rights reserved.

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.