Finding better weight functions for generalized Shepard's operator on infinite intervals

In this paper, we investigate the difference of Shepard's generalized operators S(sigma) from the approximated set of data for various weight functions sigma. Bounds are given for the sizes of the 'bumps' shown on the graph of S(sigma) for sigma(d) = 1/d in dimension N = 1, and the best weight function sigma for practical use is proposed

On the equivalence of direct mechanisms and structurally minimal pathways

A reaction‐pathway identification procedure has two distinct phases. The first phase enumerates exhaustively the feasible candidate pathways, and the second phase identifies the ultimate feasible pathway or pathways among them. Probably the most efficient way to execute the first phase is to algorithmically generate the networks of feasible candidate pathways from a predefined set of plausible elementary reactions. The available algorithmic methods for this purpose can be roughly grouped into two major classes, one based on graph theory and the other on linear algebra. Both classes of methods consider any chemical reaction system as a network of elementary reactions, thereby implying that the two classes are interrelated. This paper studies the linear algebraic concept termed direct mechanism introduced in the mid‐eighties and the graph‐theoretical concept termed structurally minimal pathway introduced two decades later. Herein, it has been formally proven that the two concepts are equivalent