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Articles 1 - 15 of 15
Full-Text Articles in Mathematics
Pixley-Roy Hyperspaces Of Ω-Graphs, Joe Mashburn
Pixley-Roy Hyperspaces Of Ω-Graphs, Joe Mashburn
Joe D. Mashburn
The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two ω-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of Rn are also shown to be homeomorphic.
A Note On Irreducibility And Weak Covering Properties, Joe Mashburn
A Note On Irreducibility And Weak Covering Properties, Joe Mashburn
Joe D. Mashburn
A space X is irreducible if every open cover of X has a minimal open refinement. Interest in irreducibility began when Arens and Dugendji used this property to show that metacompact countably compact spaces are compact. It was natural, then, to find out what other types of spaces would be irreducible and therefore compact in the presence of countable compactness or Lindelof in the presence of N1-compactness. … It is shown in this paper that T1 δθ -refinable spaces and T1 weakly δθ-refinable spaces are irreducible. Since examples of Lindelof spaces that are neither T1 nor irreducible can be easily …
Countable Covers Of Spaces By Migrant Sets, Zoltan Balogh, Joe Mashburn, Peter Nyikos
Countable Covers Of Spaces By Migrant Sets, Zoltan Balogh, Joe Mashburn, Peter Nyikos
Joe D. Mashburn
The motivation for this note is a paper by Hidenori Tanaka in which he shows that the Pixley-Roy hyperspace of a metric space X is normal if and only if X is an almost strong q-set.
Sobriety In Delta Not Sober, Joe Mashburn
Sobriety In Delta Not Sober, Joe Mashburn
Joe D. Mashburn
We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology.
Dissertation: The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn
Dissertation: The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn
Joe D. Mashburn
The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises. In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions. Retracts are discussed in section 3, where it is seen that they are not sufficient …
On The Decomposition Of Order-Separable Posets Of Countable Width Into Chains, Gary Gruenhage, Joe Mashburn
On The Decomposition Of Order-Separable Posets Of Countable Width Into Chains, Gary Gruenhage, Joe Mashburn
Joe D. Mashburn
partially ordered set X has countable width if and only if every collection of pairwise incomparable elements of X is countable. It is order-separable if and only if there is a countable subset D of X such that whenever p, q ∈ X and p < q, there is r ∈ D such that p ≤ r ≤ q. Can every order-separable poset of countable width be written as the union of a countable number of chains? We show that the answer to this question is "no" if there is a 2-entangled subset of IR, and "yes" under the Open Coloring …
The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn
The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn
Joe D. Mashburn
A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point. It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete …
Oif Spaces, Zoltan Balogh, Harold Bennett, Dennis Burke, Gary Gruenhage, David Lutzer, Joe D. Mashburn
Oif Spaces, Zoltan Balogh, Harold Bennett, Dennis Burke, Gary Gruenhage, David Lutzer, Joe D. Mashburn
Joe D. Mashburn
A base β of a space X is called an OIF base when every element of B is a subset of only a finite number of other elements of β. We will explore the fundamental properties of spaces having such bases. In particular, we will show that in T2 spaces, strong OIF bases are the same as uniform bases, and that in T3 spaces where all subspaces have OIF bases, compactness, countable compactness, or local compactness will give metrizability.
Linearly Ordered Topological Spaces And Weak Domain Representability, Joe Mashburn
Linearly Ordered Topological Spaces And Weak Domain Representability, Joe Mashburn
Joe D. Mashburn
It is well known that domain representable spaces, that is topological spaces that are homeomorphic to the space of maximal elements of some domain, must be Baire. In this paper it is shown that every linearly ordered topological space (LOTS) is homeomorphic to an open dense subset of a weak domain representable space. This means that weak domain representable spaces need not be Baire.
A Spectral Order For Infinite Dimensional Quantum Spaces: A Preliminary Report, Joe Mashburn
A Spectral Order For Infinite Dimensional Quantum Spaces: A Preliminary Report, Joe Mashburn
Joe D. Mashburn
In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approximations of which pure states. In a previous paper the author extended the Bayesian order to infinite dimensional spaces of classical states. The order on infinite dimensional spaces retains many of the characteristics important to physics, …
Three Counterexamples Concerning Ω-Chain Continuous Functions And Fixed-Point Properties, Joe Mashburn
Three Counterexamples Concerning Ω-Chain Continuous Functions And Fixed-Point Properties, Joe Mashburn
Joe D. Mashburn
A partially ordered set is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.
A Spectral Order For Infinite Dimensional Quantum Spaces, Joe Mashburn
A Spectral Order For Infinite Dimensional Quantum Spaces, Joe Mashburn
Joe D. Mashburn
In this paper we extend the spectral order of Coecke and Martin to infinite-dimensional quantum states. Many properties present in the finite-dimensional case are preserved, but some of the most important are lost. The order is constructed and its properties analysed. Most of the useful measurements of information content are lost. Shannon entropy is defined on only a part of the model, and that part is not a closed subset of the model. The finite parts of the lattices used by Birkhoff and von Neumann as models for classical and quantum logic appear as subsets of the models for infinite …
An Order Model For Infinite Classical States, Joe Mashburn
An Order Model For Infinite Classical States, Joe Mashburn
Joe D. Mashburn
In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory,2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and …
A Note On Reordering Ordered Topological Spaces And The Existence Of Continuous, Strictly Increasing Functions, Joe Mashburn
A Note On Reordering Ordered Topological Spaces And The Existence Of Continuous, Strictly Increasing Functions, Joe Mashburn
Joe D. Mashburn
The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with Rn and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from Rn into R that was strictly increasing with respect to the new order imposed on Rn. It is the purpose of this paper to explore the structural characteristics of ordered topological spaces which have this …
A Comparison Of Three Topologies On Ordered Sets, Joe Mashburn
A Comparison Of Three Topologies On Ordered Sets, Joe Mashburn
Joe D. Mashburn
We introduce two new topologies on ordered sets: the way below topology and weakly way below topology. These are similar in definition to the Scott topology, but are very different if the set is not continuous. The basic properties of these three topologies are compared. We will show that while domain representable spaces must be Baire, this is not the case with the new topologies.