We discuss molecular orbital basis sets that contain both Gaussian and polynomial (ramp) functions. We show that, by modeling ramp-Gaussian products as sums of ramps, all of the required one- and two-electron integrals can be computed quickly and accurately. To illustrate our approach, we construct R-31+G, a mixed ramp-Gaussian basis in which the core basis functions of the 6-31+G basis are rep...

We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtai...

The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of inte...

Abstract. In this paper we analyse a hybrid approximation of functions on the sphere S ⊂ R by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear sy...

Support Vector Machine (SVM) is one of the most robust and accurate method amongst all the supervised machine learning techniques. Still, the performance of SVM is greatly influenced by the selection of kernel function. This research analyses the characteristics of the two well known existing kernel functions, local Gaussian Radial Basis Function and global Polynomial kernel function. Based on ...

GB-spline curves can be considered as the generalization of B-spline curve incorporating a shape parameter into the polynomial basis functions. The geometric effect of the alteration of the shape parameter is discussed in this paper, including constrained shape control of the curve.

We introduce a basis of rational polynomial-like functions P 0 ,. .. , P n−1 for the free module of functions Z/nZ → Z/mZ. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions lcm(k) P k where lcm(k) is the least common multiple of 2,. .. , k (viewed in Z/mZ). As a consequence , when n ≥ m, the number of such functions is indep...

H–bases are bases for polynomial ideals, characterized by the fact that their homogeneous leading terms are a basis for the associated homogeneous ideal. In the computation of H–bases without term orders, an important task is to determine the orthogonal projection of a homogeneous polynomial to certain subspaces of homogeneous polynomials with respect to a given inner product. One way of doing ...