Shape Preserving Spline Interpolation

A rational spline solution to the problem of shape preserving interpolation is discussed. The rational spline is represented in terms of first derivative values at the knots and provides an alternative to the spline-under-tension. The idea of making the shape control parameters dependent on the first derivative unknowns is then explored. The monotonic or convex shape of the interpolation data can then be preserved automatically through the solution of the resulting non-linear consistency equations of the spline

A Philosophy Of Christian Librarianship

While a number of Christian librarians have explored the implications of the Christian world view for particular issues in library practice, few have attempted to develop a thoroughgoing philosophy of Christian librarians/zip. Those who have done so have generally failed to center their proposals around the Christian view of truth. The knowability, objectivity, unity, practicality, and spirituality of truth should impact the way librarians at Christian colleges carry out major library functions, including collection development, reference services, bibliographic instruction, research and publication, and management

The Lord\u27s Prayer

These notes describe the context, structure, and interpretation of the Lord\u27s Prayer. Special attention is given to the version of the prayer that appears in Matthew 6:9-13, with interpretive comments provided for each phrase

Collapsing regions and black hole formation

Up to a conjecture in Riemannian geometry, we significantly strengthen a recent theorem of Eardley by proving that a compact region in an initial data surface that is collapsing sufficiently fast in comparison to its surface-to-volume ratio must contain a future trapped region. In addition to establishing this stronger result, the geometrical argument used does not require any asymptotic or energy conditions on the initial data. It follows that if such a region can be found in an asymptotically flat Cauchy surface of a spacetime satisfying the null-convergence condition, the spacetime must contain a black hole with the future trapped region therein. Further, up to another conjecture, we prove a strengthened version of our theorem by arguing that if a certain function (defined on the collection of compact subsets of the initial data surface that are themselves three-dimensional manifolds with boundary) is not strictly positive, then the initial data surface must contain a future trapped region. As a byproduct of this work, we offer a slightly generalized notion of a future trapped region as well as a new proof that future trapped regions lie within the black hole region.Comment: 11 pages, REVTeX 3.