Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , x, . . . } to get a sequence {Pn}∞n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of p...

A randomized Gram--Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical while being at least as numerically stable modified process. Our approach based on random sketching, which a dimension reduction technique consisting in estimation inner products by their small eff...

Selected Common Threads and Root Concepts Linking Old to New Sandra L. Arlinghaus* and Joseph Kerski** Introduction In the twentieth century, much of mathematics experienced a well-known paradigm shift in moving from a focus on mathematical objects to a focus on transformations of objects. Thus, one sees, for example, linear algebra courses that focus on matrix vector products as linear transfo...

Principal component analysis (PCA) is a key statistical technique for multivariate data analysis. For large data sets, the common approach to PCA computation is based on the standard NIPALS-PCA algorithm, which unfortunately suffers from loss of orthogonality, and therefore its applicability is usually limited to the estimation of the first few components. Here we present an algorithm based on ...

NTRU is a very efficient public-key cryptosystem based on polynomial arithmetic. Its security is related to the hardness of lattice problems in a very special class of lattices. This article is motivated by an interesting peculiar property of NTRU lattices. Namely, we show that NTRU lattices are proportional to the so-called symplectic lattices. This suggests to try to adapt the classical reduc...

This paper presents a method for reducing the number of rules for large rule based fuzzy systems. While retaining the linguistic information of the original system, the reduction is achieved by attempting to select the most significant rules that have important contribution in the error reduction. Using a numerically efficient modified Gram-Schmidt orthogonalization routine, a measure called ru...

Blind adaptive beamforming is often used in communication systems to combat co-channel interference. Among a number of techniques, the constant modulus algorithm (CMA) has proven to be an e ective tool for blind beamforming of uncorrelated signals. Unfortunately, CMA beamformers encounter problems for correlated signals and interferences. In this paper, we consider correlated co-channel input s...

In this paper we study the algebraic and geometric structure of the space of compactly supported biorthogonal wavelets. We prove that any biorthogonal wavelet matrix pair (which consists of the scaling lters and wavelet lters) can be factored as the product of primitive parau-nitary matrices, a pseudo identity matrix pair, an invertible matrix, and the canonical Haar matrix. Compared with the f...

This paper presents a distinctive phonetic features (DPFs) based phoneme recognition method by incorporating syllable language models (LMs). The method comprises three stages. The first stage extracts three DPF vectors of 15 dimensions each from local features (LFs) of an input speech signal using three multilayer neural networks (MLNs). The second stage incorporates an Inhibition/Enhancement (...