arXiv subjects

Daniel J. Greenhoe

Publications and source records attributed to Daniel J. Greenhoe.

5 recordsLinked to original sources

arXiv2016

Properties of distance spaces with power triangle inequalities

Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the "power triangle function". The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.

math.MG↗

arXiv2015

Boolean and ortho fuzzy subset logics

Constructing a fuzzy subset logic L with Boolean properties is notoriously difficult because under a handful of "reasonable" conditions, we have the following three debilitating constraints: (1) Bellman and Giertz in 1973 showed that if L is distributive, then it must be idempotent. (2) Dubois and Padre in 1980 showed that if L has the excluded middle or the non-contradiction property or both, then it must be non-idempotent. (3) Bellman and Giertz also demonstrated in 1973 that even if L is idempotent, then the only choice available for the (meet,join) logic operator pair is the (min,max) operator pair. Thus it would seem impossible to construct a non-trivial fuzzy subset logic with Boolean properties. However, this paper examines these three results in detail, and shows that "hidden" in the hypotheses of the three is the assumption that the operator pair (meet,join) is pointwise evaluated. It is further demonstrated that removing this constraint yields the following results: (A) It is indeed possible to construct fuzzy subset logics that have all the Boolean properties, including that of idempotency, non-contradiction, excluded middle, and distributivity. (B) Even if idempotency holds, (min,max) is not the only choice for (meet,join).

math.LO↗

arXiv2014

Properties and applications of transversal operators

This paper presents some properties and applications of "transversal operators". Two transversal operators are presented: a "translation" operator T and a "dilation" operator D. Such operators are used in common analysis systems including Fourier series analysis, Fourier analysis, Gabor analysis, multiresolution analysis (MRA), and wavelet analysis. Like the unitary Fourier transform operator F, the transversal operators T and D are unitary. Demonstrations of the usefulness of these three unitary operators are found in the proofs of results found in some common analytic systems including MRA analysis and wavelet analysis.

math.GM↗

arXiv2014

Partition of unity systems and B-splines

This paper presents the basic principles of partition of unity systems and B-splines. Analysis of these systems is performed using Fourier analysis, multi-resolution analysis, and wavelet analysis.

math.GM↗

arXiv2014

MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

math.GM↗