About
64
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111
Citations
Introduction
Additional affiliations
August 1995 - present
Position
- Associate Professor of Mathematics
Description
- Algebra, Logic, Nonstandard Analysis, Metric Fixed Point Theory, Engineering, Education, Advising and Mentoring Junior Researchers, Advising and Mentoring Student Research Assistants, Research Policy, Conference Organization, Refereeing
August 1995 - present
Position
- Associate Professor of Mathematics
Description
- Curriculum Development, Teaching, Mentoring, Advising, Assessment, Policy
August 1989 - August 1995
Position
- Research Assistant
Description
- Algebra, Logic, Nonstandard Analysis, Metric Fixed Point Theory, Engineering, Education, Advising and Mentoring Student Research Assistants, Conference Organization
Education
August 1987 - August 1989
August 1985 - August 1987
August 1979 - December 1985
Publications
Publications (64)
We show that various representation spaces for continua contain a homeomorphic copy of the harmonic sequence, so even though it is not a T0 space, it contains infinite metrizable subspaces. It remains an open question whether this representation space (topologized using continuous surjections) contains a closed subspace homeomorphic to the harmonic...
We generalize the concept of a norm on a vector space to one of a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light f...
Previously, the authors used the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of some compactifications of regular spaces. They now show that any Hausdorff compactification is obtainable with their method.
Previously, the authors used the insights of Robinson's non-standard analysis as a powerful tool to extend and simplify the construction of some compactifications of regular spaces. They now show that any Hausdorff compactification is obtainable with their method.
When relatively cold fluid is injected to hot host rock featured with fractures, the fluid-rock temperature difference induces cooling of the self-propping fractures and could cause the failure of the asperity. Although fracture sustainability has been extensively studied, little research has focused on thermal effects on asperity failure. A finite...
Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$ , acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$ .
We define a binary metric as a symmetric, distributive lattice ordered magma-valued function of two variables, satisfying a “triangle inequality”. Using the notion of a Kuratowski topology, in which topologies are specified by closed sets rather than open sets, we prove that every topology is induced by a binary metric. We conclude with a discussio...
The notion of an ε mapping is crucial to the study of continua, for it provides one avenue for naturally generalizing the notion of chainability to likeness between spaces when neither is chainable. Steven Leth has previously spoken in this conference about nonstandard methods for continua. We here continue this kind of investigation, starting with...
We will expand on a presentation that introduces "binary metric spaces" by considering how nonstandard methods applied to Boolean algebra and distributive lattices, as studied by Gonshor, and by Gehrke, Insall, and Kaiser, can be used to introduce various topologies on a given binary metric space. One of our goals is to find how to recover the orig...
Ercan’s work aimed to find a generalized notion metric that can map any topology. We build on his work by defining a generalized metric whose axioms and balls are given on {0, 1}T rather than on its projections. We accomplish that by switching our gaze from the open sets to the closed sets. That is, rather than specifying topologies in terms of the...
Given any equivalence relation on the "remote" points of an enlargement of a regular space, the equivalence classes form the boundary points of a standard compactification (Insall, Loeb, and Marciniak, 2014). Now we can show that every Hausdorff compactification is produced in precisely this way! Examples show the necessity of the Hausdorff assumpt...
We show how to retrieve a pressure field that satisfies the Clay Institute restricted Navier-Stokes equations from a smooth and bounded velocity vector field. Then we introduce a strategy for generalizing the nonlinear term of the Navier-Stokes equations, which can be repeated for many different conditions to search for a counterexample of the Navi...
We use steps to a solution to an educational problem as formal objects and descriptions of the steps as formal attributes. From there, we build a formal concept lattice which can be traced through to figure out where a student has made a mistake. Based on how far the student has gone through the lattice, we award a partial credit grade.
From a measure space, ( X , μ X ) (X, \mu ^{\mathbb {X}}) we define a measure μ P ( X ) \mu ^{\mathbb {P}(\mathbb {X})} on the power set of X X . If ( X , τ ) (X, \tau ) is a compactum, whose topology τ \tau is compatible with the measure μ X \mu ^{\mathbb {X}} on X X , then the measure μ P ( X ) \mu ^{\mathbb {P}(\mathbb {X})} restricts to a natur...
When polluting firms acquire firms in the same industry, do they seek out greener firms in order to make use of their cleaner technology, especially under threat of penalty for pollution? How does penalty for pollution affect the incentive for acquisition? The result of this study indicates that firms who are penalized for their pollution do not ne...
We characterize local connectedness of a given metrized continuum in
terms of the connectedness properties of the space of (3 x 3) distance
matrices realized by the given continuum.
Abstract:
By translating teaching and learning styles into propositional logic, we have established the foundational knowledge needed for the development of a "smart" computerized tutoring program. Here, we use set theory to explore the act of learning and, with statements in terms of propositional logic and with
diagrams, we construct many differe...
In this thesis is presented a protracted discussion of Robinson's article ``Germs'', published in 1967 in the Proceedings of the International Symposium on Applications of Model Theory to Algebra, Analysis and Probability. Then some original results about algebraic extensions of perfect fields sporting a compatible topology were presented, introduc...
The article titled ``Nonstandard Methods and Finiteness Conditions in Algebra'', published in the ZML, was based upon this longer document, but the ZML article includes also some results obtained after award of the PhD.
We show that Fermat's Last Theorem implies or is equivalent to certain results, in mathematics and its applications, which may be surprising to some readers. The proofs, however, are not at all difficult, in most cases.
With the rising public and political concern about greener production, there is unrelenting pressure on individual companies to mitigate and abate pollution and adopt cleaner technologies. Governments adopt several types of environmental policies and regulations that are aimed at protecting the environment and encouraging efficient use of natural r...
We use the insights of Robinson’s nonstandard analysis as a powerful tool to extend and simplify the construction of compactifications of regular spaces. In particular, we deal with the Stone-Cech compactification and compactifications formed from topological ends. For the nonstandard extension of a metric space, the monad of a standard point x is...
Even though most acquisitions take place among manufacturing firms regulated by environmental policies, there are not many studies which model acquisition decisions when such policies are present. In this book chapter, we model acquisition incentives when polluting firms compete in a Cournot oligopoly market. Our result suggests that emission tax c...
Consider a dilute gas composed of identical, isotropic molecules in which the potential energy of interaction between pairs of particles is an analytic function of the distance of separation with either a monomial or a normal binomial Laurent expansion. It is shown that the complete iterative inversion method inverts second virial coefficient data...
In this article, we introduce a new notion of (strong) absolute derivative, for functions defined between metric spaces, and we investigate various properties and uses of this concept, especially regarding the geometry of abstract metric spaces carrying no other structure.
This article uses nets to define ends of topological spaces, modifying the sequence-based idea introduced in 1931 by Hans Freudenthal. We connect the lack of ends with the com-pactness of a topological space more firmly than did Freudenthal's original presentation. Specifically, we prove the following generalization of Freudenthal's related result:...
Abstract. In On representation spaces, a forthcoming article,
José G. Anaya, Félix Capulín, Włodzimierz J. Charatonik, and
Fernando Orozco-Zitli have introduced the representation space C
of all continua (up to homeomorphism). Here, we reproduce the
argument that it is a topological space, and then we investigate its
connectedness properties. Speci...
A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends
only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state
for a gas. The complete iterative inversion method (CIIM) is an algorithm employed in an attempt to...
A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas.
The complete iterative inversion method is a technique employed in an attempt to recover...
We describe a theory of ``multi-structures'', and explore logics and languages that are natural for the study of these mathematical objects. The text is written for upper level undergraduate students and beginning graduate students in Computer Science, Computer Engineering, Mathematics, and Philosophy, although it is expected that students of other...
We introduce the notion of a discrepancy function, as an extended real-valued function that assigns to a pair (A,U) of sets a nonnegative extended real number ω(A,U), satisfying specific properties. The pairs (A,U) are certain pairs of sets such that A⊆U, and for fixed A, the function ω takes on arbitrarily small nonnegative values as U varies. We...
We present several new distance functions on hyperspaces and investigate their properties, relative to some natural application considerations.
Discrete event system simulation models provide an alternative methodology for analyzing behavior of certain types of systems where we wish to emphasize the reaction of components within the system to events that occur either within the system or in the system environment. The implementation of discrete event system simulations has been primarily a...
Martin Davis and Matt Insall discuss a quote by Richard W. Hamming about the physical effect of Lebesgue and Riemann integrals
and whether it made a difference whether one or the other was used, for example, in the design of an airplane. The gist of
Hamming’s quote was that the fine points of mathematical analysis are not relevant to engineering co...
![Fig. 1. One suggested scenario [13] to explain inter-linked chemical...](profile/Matt-Insall/publication/227208373/figure/fig1/AS:669994014744592@1536750429399/One-suggested-scenario-13-to-explain-inter-linked-chemical-and-isotopic-heterogeneities_Q320.jpg)
The potential energy of a nuclide is enhanced by about 10 MeV per nucleon from the repulsion between like nucleons, and diminished by about 20 MeV per nucleon from the attraction between unlike nucleons. Nuclear stability results mostly from the interplay of these opposing forces, plus Coulomb repulsion of positive charges. Whereas fusion may be th...
Convex sets of states and the corresponding normal functionals defined on the Hilbert space containing them play a central role in the foun- dations of General Quantum Mechanics (see (4, 1)). The design of quantum logics for reasoning about General Quantum Mechanics leads naturally to the theory of orthomodular lattices (4, 1, 2), for example, as l...
Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.
Key words: nonstandard methods, topological field, hyperalgebraic elements, infini...
There are many applications involving arrays that contain non-zero components in regular geometric partitions. These include triangular, diagonal, tridiagonal, banded, etc. When computing with this type of arrays, they are usually stored in a packed form and computations are performed with only the non-zero components. This packed form requires an...
This paper presents an approach to operationally evaluate a temporal specification in a distributed computing environment. First, an algorithm Compute History is proposed to allow every distributed processor to collect events executed by itself and by other processors, and to order the collected events by causality. This algorithm employs neither m...
This paper presents a modal logic, Interval Temporal Logic, built upon a classical predicate logic L. In the logic system, we consider formulas that can be used to reason about timing properties of systems, in particular, interval formulas and responsiveness assertions. These formulas can be used to describe timing constraints of a system. Hence th...
Using nonstandard methods, we generalize the notion of an algebraic primitive element to that of an hyperalgebraic primitive element, and show that under mild restrictions, such elements can be found infinitesimally close to any given element of a topological field.
A responsive computing system is a hybrid of real-time,
distributed and fault-tolerant systems. In such a system, severe
consequences can occur if the run-time behavior does not conform to the
expected behavior or specifications. In this paper, we present a formal
approach to ensure satisfaction of the specifications in the operational
environment...
Let X be a metric space and let CB(X) denote the closed bounded subsets of X with the
Hausdorff metric. Given a complete subspace Y of CB(X), two fixed point theorems, analogues of results
in [1], are proved, and examples are given to suggest their applicability in practice.
In [1], the notion of a local join (meet) endomorphism of a lattice was defied. This notion, though interesting in the context of infinitary lattice operations, is redundant in the finitary setting, so its removal from all hypotheses does not affect the results presented therein. The author regrets this oversight and wishes to thank those readers w...
An abstract is not available.
We discuss the application of nonstandard methods to local versions of certain lattice notions. In a particular case, we find that imposition of certain local conditions imply a surprising global one, namely boundedness of the given lattice.
ABSTRACT In this paper ,an ew technique for ensuring run-time satisfaction of properties-sp ecifically responsiveness property ,as ubset of liveness property ,i nr esponsive systems, is presented. Since whether the run-time behavior of a system is satisfied depends,on the execution (operational) environment, we develop a translation which takes int...
An important application of logic to mathematics is the development of nonstandard analysis. We study some concepts in this important area and show how to prove various results using nonstandard methods. This progress report is a part of a larger project involving nonstandard meth- ods. The superstructure is a fundamental construct in nonstandard a...
By translating teaching and learning styles into propositional logic, we have established the foundational knowledge needed for the development of a "smart" computerized tutoring program. Here, we use set theory to explore the act of learning and, with statements in terms of propositional logic and with diagrams, we construct many different models...
Thesis (M.S.)--University of Houston, 1987. Degree granted by Dept. of Mathematics. Includes bibliographical references (leaf 60).
Questions
Questions (2)
This question is not as much a research question as it is a poll or survey.
Chavoya-Aceves asked a related question: https://www.researchgate.net/post/There_is_a_natural_number_that_is_triangular_the_sum_of_two_triangular_numbers_and_also_the_sum_of_the_squares_of_two_triangular_numbers [Note that he asked if there is a proof of a certain result, asserting that there is a unique pair of triangular numbers m, and n, with specific properties.]
A negative answer to my question would be a theorem that is a special case of the Chavoya-Aceves ``conjecture'', which he stated in the form of a question. Note that Andrej Dujella answered Chavoya-Aceves by providing an algorithm by which to produce two (arbitrarily large it seems) triangular numbers m and n with the properties specified in the question. Our question is the case in which one takes m=n in the original problem.
