A new set of special functions is described which has a wide range of applications, from number theory to integrability of non-linear dynamical systems. We study multiple orthogonal polynomials with respect to p > 1 weights satisfying Pearson’s equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights which is based on properties of t...

A colored domino is a rotatable 2× 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P , a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We provide tight bounds on the number of colors required to tile simple and ...

The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors, matrices, etc) into an orthonormal basis (a set of orthogonal, unit-length vectors, bi or tri dimensional matrices). The process consists of taking each array and then subtracting the projections in common with the previous arrays. This paper introduces an enhanced version of the Gra...

The one-dimensional harmonic oscillator wave functions are solutions to a SturmLiouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms...

Abstract— This paper introduces a new set of orthogonal moments function hypergeometric based on the discrete Legendre polynomials. The Legendre moments can be effectively used as pattern features in the analysis of two-dimensional images. The implementation of moments proposed in this paper does not involve any numerical approximation, since the basis set is orthogonal in the discrete domain o...

We present two-dimensional filter banks with directional vanishing moments. The directional-vanishing-moment condition is crucial for the regularity of directional filter banks. However, it is a challenging task to design orthogonal filter banks with directional vanishing moments. Due to the lack of multidimensional factorization theorems, traditional one-dimensional methods cannot be extended ...

Abstract. Stochastic Collocation (SC) has been studied and used in different disciplines for Uncertainty Quantification (UQ). The method consists of computing a set of appropriate points, called collocation points, and then using Lagrange interpolation to construct the probability density function (pdf) of the quantity of interest (QoI). The collocation points are usually chosen as Gauss quadra...

A well-performing set of radial basis functions (RBFs) can emerge from genetic competition among individual RBFs. Genetic selection of the individual RBFs is based on credit sharing which localizes competition within orthogonal niches. These orthogonal niches are derived using singular value decomposition and are used to apportion credit for the overall performance of the RBF network among indi...

It is shown that regularity of orthogonal wavelet matrices, i.e. vanishing of discrete moments of some of their rows, corresponds to maximizing the discrete moments of one of the rows. This leads to an algorithm for calculation of such wavelets by nding a vector which simultaneosly vanishes certain indeenite quadratic forms.

An orthogonal double cover (ODC) of the complete graph Kn by a graph G is a collection G of n spanning subgraphs of Kn, all isomorphic toG, such that any two members ofG share exactly one edge and every edge ofKn is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of le...