Boundary problem

Abstract: The fixed end point problem in non-parametric form General end conditions The index form Self-adjoint systems The functional on a Riemannian space The critical sets of functions The boundary problem in the large Closed extremals Solution of the Poincare continuation problem.

Abstract: If a probability density function has bounded support, kernel density estimates often overspill the boundaries and are consequently especially biased at and near these edges. In this paper, we consider the alleviation of this boundary problem. A simple unified framework is provided which covers a number of straightforward methods and allows for their comparison: ‘generalized jackknifing’ generates a variety of simple boundary kernel formulae. A well-known method of Rice (1984) is a special case. A popular linear correction method is another: it has close connections with the boundary properties of local linear fitting (Fan and Gijbels, 1992). Links with the ‘optimal’ boundary kernels of Muller (1991) are investigated. Novel boundary kernels involving kernel derivatives and generalized reflection arise too. In comparisons, various generalized jackknifing methods perform rather similarly, so this, together with its existing popularity, make linear correction as good a method as any. In an as yet unsuccessful attempt to improve on generalized jackknifing, a variety of alternative approaches is considered. A further contribution is to consider generalized jackknife boundary correction for density derivative estimation. En route to all this, a natural analogue of local polynomial regression for density estimation is defined and discussed.

Abstract: We present an investigation on the spring analogy. The spring analogy serves for deformation in a moving boundary problem. First, two different kinds of springs are discussed: the vertex springs and the segment springs. The vertex spring analogy is originally used for smoothing a mesh after mesh generation or refinement. The segment spring analogy is used for deformation of the mesh in a moving boundary problem. The difference between the two methods lies in the equilibrium length of the springs. By means of an analogy to molecular theory, the two theories are generalized into a single theory that covers both. The usual choice of the stiffness of the spring is clarified by the mathematical analysis of a representative one-dimensional configuration. The analysis shows that node collision is prevented when the stiffness is chosen as the inverse of the segment length. The observed similarity between elliptic grid generation and the spring analogy is also investigated. This investigation shows that both methods update the grid point position by a weighted average of the surrounding points in an iterative manner. The weighting functions enforce regularity of the mesh

Abstract: In this second part of the paper, the implementation of the model presented in part I into the computer software ‘MatCalc’, is described. The new model is compared to the classical treatment of a moving boundary problem in phase transformations based on discrete composition profiles and good agreement is obtained. Selected examples demonstrate the model's potential for application to the evolution of the precipitate microstructure in multi-component systems where classical models frequently encounter difficulties due to the system's complexity.