Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

On Time Discretizations for Spectral Methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.     

Depth notions for orthogonal regression

Global depth, tangent depth and simplicial depths for classical and orthogonal regression are compared in examples, and properties that are useful for calculations are derived. The robustness of the maximum simplicial depth estimates is shown in examples. Algorithms for the calculation of depths for orthogonal regression are proposed, and tests for multiple regression are transferred to orthogonal regression. These tests are distribution free in the case of bivariate observations. For a particular test problem, the powers of tests that are based on simplicial depth and tangent depth are compared by simulations.     

Estimating a survival function with incomplete cause-of-death data

We propose a random censorship model which permits uncertainty in the cause of death assessments for a subset of the subjects in a survival experiment. A nonparametric maximum likelihood approach and a “self-consistency” approach are considered. The solution sets corresponding to both approaches are found. They are infinite and identical. Only some of the solutions are consistent; i.e., the MLEs and self-consistent estimators are not consistent in general. Two estimates are thus proposed and their asymptotic properties are studied. It is shown that both estimates are strongly consistent and converge to Gaussian processes. The covariance structures of these Gaussian processes are derived.     

Periodic and solitary-wave solutions of the Degasperis–Procesi equation

Travelling-wave solutions of the Degasperis–Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis–Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.     

State models and asymptotic behaviour of 2D positive systems

Homogeneous 2D positive systems are 2D state-space models whosevariables are alwalys nonnegative and, consequently, are describedby a pair of nonnegative square matrices (A, B). In the paper,the properties of these pairs are discussed both in the generalcase and under particular assumptions like finite memory, separability,and property L. Various aspects of the positive asymptotic dynamic are considered;in particular, sufficient conditions are provided guaranteeingthat the local states are eventually strictly positive. Finally,some results on the convergence of the states towards a constantasymptotic distribution are presented.     

Differential systems with Fuchsian linear part: Correction and linearization, normal forms and multiple orthogonal polynomials

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable.More generally, normal forms are found.The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.     

Enumeration of point-determining graphs

Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

Economic design of screening procedures when the rejected items are reprocessed

In a screening inspection, all of the items are subject to acceptance inspection. If an item fails to meet the predetermined specifications, it is rejected. In this paper, economic screening procedures are considered when the rejected items are reprocessed. It is assumed that the reprocessed and nonreprocessed items are produced by the same manufacturing process and therefore their quality characteristics are identically and independently distributed. Screening procedures based on the performance variable of interest and a variable which is correlated with the performance variable are considered. Cost models are constructed which involve the cost incurred by imperfect quality, reprocessing cost, and inspection cost. Methods of finding the optimal screening procedures are presented and an example is given.