Randall L. Dougherty

Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic hermite interpolation

The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper de- scribes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data. 1. Introduction. Piecewise polynomial interpolants, especially those based on Hermite polynomials (polynomials determined by their values and values of one or more derivatives at both ends of an interval), have a number of desirable properties. They are easy to compute once the derivative values are chosen. If the derivative values are chosen locally (e.g., by finite difference methods), then the interpolant at a given point will depend only on the given data at nearby mesh points. If the derivatives are computed by spline methods, then the interpolant will have an extra degree of continuity at the mesh points. In either case, the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases. These methods, however, do not necessarily preserve the shape of the given data. When the data arise from a physical experiment, it may be vital that the interpolant preserve nonnegativity (f(x) > 0), nonpositivity (f(x) 0 or f(x) 0), or concavity (f(x) < 0). In this and other cases, geometric considerations, such as preventing spurious behavior near rapid changes in the data, may be more important than the asymptotic accuracy of the interpolation method. One can construct a shape-preserving interpolant by constraining the derivatives for the Hermite polynomials to meet conditions which imply the desired properties ((4), (5), (8), (11)—(15), (20)), by adding new mesh points

The Degree-Diameter Problem for Several Varieties of Cayley Graphs I: The Abelian Case

We consider the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs) for both directed graphs and undirected graphs. The problem is closely related to that of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. For two generators (dimensions), these methods yield optimal Abelian graphs with a given diameter k. (The results in two dimensions are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with three generators and a given diameter k, which we conjecture to be as large as possible; for the directed case, we obtain partial results. These results are connected to efficient lattice coverings of

Unflippable Tetrahedral Complexes

{bf R}^3

A divide-and-conquer algorithm for grid generation

by octahedra or by tetrahedra; computations on Cayley graphs lead us to such lattice coverings, which we conjecture to be optimal. (The problem of finding such optimal coverings can be reduced to a finite number of nonlinear optimization problems.) We discuss the asymptotic behavior of the Abelian degree-diameter problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.

A (55,16,19) binary Goppa code and related codes having large minimum distance

Abstract nWe present a 16-vertexntetrahedralization of S3 (the 3-sphere) for which no topologicalnbistellar flip other than a 1-to-4 flip (i.e., a vertexninsertion) is possible. This answers a question of Altshulernet al. which asked if any two n-vertex tetrahedralizations ofnS3 are connected by a sequence of 2-to-3 and 3-to-2nflips. The corresponding geometric question is whether twontetrahedralizations of a finite point set S in ℝ3 in “generalnposition” are always related via a sequence of geometric 2-to-3nand 3-to-2 flips. Unfortunately, we show that thisntopologically unflippable complex and others with its propertiesncannot be geometrically realized in ℝ3. nn

Method for lossless encoding of image data by approximating linear transforms and preserving selected properties for image processing

We present a new algorithm for generating a logically rectangular curvilinear mesh for a region in the plane bounded by four smooth curves. The curvilinear grid is generated using a divide-and-conquer algorithm where at each step a single grid curve or line is generated using position and derivative information derived from previously generated curves. The grid curves are produced by shape-preserving cubic Hermite interpolation that allow for corners or kinks along the boundary. After all grid curves have been generated, the mesh is modified by a rezoning algorithm to improve smoothness, orthogonality, resolution of a particular function defined in the region, or other desirable qualities.

Covering radius computations for binary cyclic codes

A (55,16,19) binary Goppa code is used to construct (57,17,17), (58,17,18), (59,17,19), and (60,17,20) codes. The first two codes have smaller redundancy than previously known codes (linear or nonlinear) of the same length and minimum distance. The last two codes have parameters previously attained only by nonlinear codes. >