Some results concerning Abel's equation in the theory of differential equations and consequences for the affine geometry of closed plane curves

Aequationes mathematicae -     

Gaussian Closure of Transient Unsaturated Flow in Random Soils

The Gaussian closure approximation, previously used by the authors to solve steady state stochastic unsaturated flow problems in randomly heterogeneous soils, is extended here to transient flow. The method avoids linearizing the governing flow equations or the soil constitutive relations. It places no theoretical limit on the variance of constitutive parameters and applies to a broad class of soils with flow properties that scale according to a linearly separable model. Closure is obtained by treating the dimensionless pressure head as a multivariate Gaussian function. It yields a system of coupled nonlinear differential equations for the first and second moments of . We apply the Gaussian closure technique to the problem of transient infiltration into a randomly stratified soil. In each layer, hydraulic conductivity and water content vary exponentially with . Elsewhere we show that application of the technique to other constitutive relations is straightforward. Our solution for the mean and variance of in a one-dimensional layer with random conductivity compares well with Monte Carlo results over a wide range of parameters, provided that the spatial variability of the constitutive exponent is small. The solution provides considerable insight into the behavior of the transient unsaturated stochastic flow problem.     

Nanostructure of open water-channel reversed micelles. I. 1H NMR spectroscopy and molecular modeling

A recently proposed model for the rodlike reversed micelles of nickel(II) bis(2-ethylhexyl)phosphate is examined in greater detail using 1H NMR spectroscopy and molecular modeling. 1H NMR spectra show that the solubilized water molecules are situated in a different environment compared with the water molecules in classical (AOT) reversed micelles. Geometry optimization and molecular dynamics simulation clearly indicate that the water molecules are not located in the interior core of the reversed micelles, but instead the water molecules exist in compartments or channels in the surface of these rodlike reversed micelles, thereby confirming the open water-channel model of reversed micelles. Molecular modeling was also employed to examine the effects of surfactant molecular structure, cosurfactant, solvent aromaticity, and temperature on the nanostructure of the reversed micellar aggregates. It is significant that molecular modeling provides an interpretation of the nanostructure of reversed micellar aggregates that is consistent with a variety of known experimental observations reported in the liquid/liquid extraction literature. These findings show that the structure of reversed micelles is much richer at the nanoscale level than previously recognized.     

Wrap-around partitioning for block bidiagonal linear systems

A stable vector algorithm for the solution of block diagonallinear systems is obtained by a permutation of the unknownscalled wrap-around partitioning combined with standard QR factorization.Wrap-around partitioning uses blocking and selects the unknownsin the blocks in turns. After a suitable orthogonal eliminationstep one ends up with a reduced system which is again blockbidiagonal and so wrap-around partitioning can be applied again.Using a simple model for vectorization overhead it is shownthat small block sizes give best performance. The minimal blocksize 2, which corresponds to cyclic reduction, is suboptimaldue to memory bank conflicts.     

Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient

Summary In this paper the differential equation (1) y″=q(t)y is considered where q(t) is a real continuous function with period π. There is proved a necessary and sufficient condition for the stability of the trivial solution of Equation (1) when the zeros of the characteristic equation λ² - Aλ+1=0, coincide. Moreover, there is shown the construction of all Equations (1) admitting only periodic or half-periodic solutions with period π.     

Global optimality of extremals: An example

The question of the existence and the location of Darboux points (beyond which global optimality is lost) is crucial for minimal sufficient conditions for global optimality and for computation of optimal trajectories. Here, we investigate numerically the Darboux points and their relationship with conjugate points for a problem of minimum fuel, constant velocity, horizontal aircraft turns to capture a line. This simple second-order optimal control problem shows that ignoring the possible existence of Darboux points may play havoc with the computation of optimal trajectories.The authors are indebted to G. Moyer for his constructive comments. This research was supported, for the first author, by a National Research Council Associateship at NASA Ames Research Center.on leave from the Technion, Israel Institute of Technology, Haifa, Israel.