Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

The equivalent conductance of solutions of picric acid

Picric acid possesses the property, which is rare among strong electrolytes, of having a convenient distribution ratio between water and certain organic solvents such as benzene, chloroform, etc. Because of this property, picric acid offers peculiar advantages for studying the well known deviations of strong electrolytes from the law of mass action, for; by means of distribution experiments, the activities of picric acid in various aqueous solutions may be compared.

In order to interpret the results of such distribution experiments, it is necessary to know the degree of ionization of picric acid in aqueous solutions.

At least three series of determinations of the equivalent conductance of picric acid have been published, but the results are not concordant; and therefore, the degree of ionization cannot be calculated with any degree of certainty.

The object of the present investigation was to redetermine the conductance of picric acid solutions in order to obtain satisfactory data from which the degrees of ionization of its solutions might be calculated.

Growth studies of m-GaN layers on LiAlO2 by MOCVD

This paper reports that the m-plane GaN layer is grown on (200)-plane LiAlO2 substrate by metal-organic chemical vapour deposition (MOCVD) method. Tetragonal-shaped crystallites appear at the smooth surface. Raman measurement illuminates the compressive stress in the layer which is released with increasing the layer's thickness. The high transmittance (80%), sharp band edge and excitonic absorption peak show that the GaN layer has good optical quality. The donor acceptor pair emission peak located at similar to 3.41 eV with full-width at half maximum of 120 meV and no yellow peaks in the photoluminescence spectra partially show that no Li incorporated into GaN layer from the LiAlO2 substrate.

Fabrication of beta-BaB2O4 layers on (001) Sr2+-doped alpha-BaB2O4 by vapor transport equilibration (VTE)

Crystalline beta-BBO layers have been successfully prepared on (0 0 1)-oriented Sr2+-doped alpha-BBO substrates using vapor transport equilibration technique. The layers were characterized by X-ray diffraction, X-ray rocking curve and transmission spectra. The present results manifest that the VTE treatment time and powder ratio are important factors on the preparation of beta-BBO layers. beta-BBO layers with a highly (0 0 l) preferred orientation were obtained according to XRD profiles. The full width at half-maximum of the rocking curve for the layer is as low as about 1000 in., which shows the high crystallinity of the layer. These results reveal the possibility of fabricating beta-BBO (0 0 1) layers on (0 0 1)-oriented Sr2+-doped alpha-BBO substrates by VTE. (C) 2006 Elsevier Ltd. All rights reserved.