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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen
Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen
SIUE Faculty Research, Scholarship, and Creative Activity
We give a number of explicit matrix-algorithms for analysis/synthesis
in multi-phase filtering; i.e., the operation on discrete-time signals which
allow a separation into frequency-band components, one for each of the
ranges of bands, say N , starting with low-pass, and then corresponding
filtering in the other band-ranges. If there are N bands, the individual
filters will be combined into a single matrix action; so a representation of
the combined operation on all N bands by an N x N matrix, where the
corresponding matrix-entries are periodic functions; or their extensions to
functions of a complex variable. Hence our setting entails …
Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song
Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song
SIUE Faculty Research, Scholarship, and Creative Activity
We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn Hinto a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.
Kravchuk Polynomials And Induced/Reduced Operators On Clifford Algebras, G. Stacey Staples
Kravchuk Polynomials And Induced/Reduced Operators On Clifford Algebras, G. Stacey Staples
SIUE Faculty Research, Scholarship, and Creative Activity
Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space V, the action induced on the Clifford algebra over V is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in …
On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples
On Representations Of Semigroups Having Hypercube-Like Cayley Graphs, Cody Cassiday, G. Stacey Staples
SIUE Faculty Research, Scholarship, and Creative Activity
The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z2n. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context …
Operator Calculus Algorithms For Multi-Constrained Paths, Jamila Ben Slimane, Rene' Schott, Ye Qiong Song, G. Stacey Staples, Evangelia Tsiontsiou
Operator Calculus Algorithms For Multi-Constrained Paths, Jamila Ben Slimane, Rene' Schott, Ye Qiong Song, G. Stacey Staples, Evangelia Tsiontsiou
SIUE Faculty Research, Scholarship, and Creative Activity
Classical approaches to multi-constrained routing problems generally require construction of trees and the use of heuristics to prevent combinatorial explosion. Introduced here is the notion of constrained path algebras and their application to multi-constrained path problems. The inherent combinatorial properties of these algebras make them useful for routing problems by implicitly pruning the underlying tree structures. Operator calculus (OC) methods are generalized to multiple non-additive constraints in order to develop algorithms for the multi constrained path problem and multi constrained optimization problem. Theoretical underpinnings are developed first, then algorithms are presented. These algorithms demonstrate the tremendous simplicity, flexibility and speed …
Clifford Algebra Decompositions Of Conformal Orthogonal Group Elements, G. Stacey Staples, David Wylie
Clifford Algebra Decompositions Of Conformal Orthogonal Group Elements, G. Stacey Staples, David Wylie
SIUE Faculty Research, Scholarship, and Creative Activity
Beginning with a finite-dimensional vector space V equipped with a nondegenerate quadratic form Q, we consider the decompositions of elements of the conformal orthogonal group COQ(V), defined as the direct product of the orthogonal group OQ(V) with dilations. Utilizing the correspondence between conformal orthogonal group elements and ``decomposable'' elements of the associated Clifford algebra, ClQ(V), a decomposition algorithm is developed. Preliminary results on complexity reductions that can be realized passing from additive to multiplicative representations of invertible elements are also presented with examples. The approach here is …