Positive Definite Functions And Dual Pairs Of Locally Convex Spaces, 2018 Chapman University

#### Positive Definite Functions And Dual Pairs Of Locally Convex Spaces, Daniel Alpay, Saak Gabriyelyan

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

Using pairs of locally convex topological vector spaces in duality and topologies defined by directed families of sets bounded with respect to the duality, we prove general factorization theorems and general dilation theorems for operator-valued positive definite functions.

Distributed Evolution Of Spiking Neuron Models On Apache Mahout For Time Series Analysis, 2017 Cylance, Inc.

#### Distributed Evolution Of Spiking Neuron Models On Apache Mahout For Time Series Analysis, Andrew Palumbo

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Conference Program, 2017 University of Dayton

#### Conference Program, University Of Dayton

*Summer Conference on Topology and Its Applications*

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications.

Disjoint Infinity Borel Functions, 2017 University of Denver

#### Disjoint Infinity Borel Functions, Daniel Hathaway

*Summer Conference on Topology and Its Applications*

Consider the statement that every uncountable set of reals can be surjected onto R by a Borel function. This is implied by the statement that every uncountable set of reals has a perfect subset. It is also implied by a new statement D which we will discuss: for each real a there is a Borel function f_{a} : RtoR and for each function g : RtoR there is a countable set G(g) of reals such that the following is true: for each a in R and for each function g : R to R, if f_{a} is disjoint from g ...

On A Construction Of Some Class Of Metric Spaces, 2017 Adam Mickiewicz University of Poznan

#### On A Construction Of Some Class Of Metric Spaces, Dariusz Bugajewski

*Summer Conference on Topology and Its Applications*

In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations.

On Di-Injective T0-Quasi-Metric Spaces, 2017 North-West University (South Africa)

#### On Di-Injective T0-Quasi-Metric Spaces, Collins Amburo Agyingi

*Summer Conference on Topology and Its Applications*

We prove that every q-hyperconvex T0-quasi-metric space (X, d) is di-injective without appealing to Zorn’s lemma. We also demonstrate that QX as constructed by Kemajou et al. and Q(X) (the space of all Katˇetov function pairs on X) are di-injective. Moreover we prove that di-injective T0-quasi-metric spaces do not contain proper essential extensions. Among other results, we state a number of ways in which the the di-injective envelope of a T0-quasi-metric space can be characterized.

Some New Completeness Properties In Topological Spaces, 2017 Gazi University

#### Some New Completeness Properties In Topological Spaces, Cetin Vural, Süleyman Önal

*Summer Conference on Topology and Its Applications*

One of the most widely known completeness property is the completeness of metric spaces and the other one being of a topological space in the sense of Cech. It is well known that a metrizable space X is completely metrizable if and only if X is Cech-complete. One of the generalisations of completeness of metric spaces is subcompactness. It has been established that, for metrizable spaces, subcompactness is equivalent to Cech-completeness. Also the concept of domain representability can be considered as a completeness property. In [1], Bennett and Lutzer proved that Cech-complete spaces are domain representable. They also proved, in ...

A New Class Of Dendrites Having Unique Second Symmetric Product, 2017 Universidad Autonoma del Estado de Mexico

#### A New Class Of Dendrites Having Unique Second Symmetric Product, David Maya, José G. Anaya, Fernando Orozco Zitli

*Summer Conference on Topology and Its Applications*

The second symmetric product of a continuum X, F_{2}(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F_{2}(X) provided that each continuum Y satisfying that F_{2}(X) and F_{2}(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F_{2}(X) will be presented.

Compactness Via Adherence Dominators, 2017 Morgan State University

#### Compactness Via Adherence Dominators, Bhamini M. P. Nayar, Terrence A. Edwards, James E. Joseph, Myung H. Kwack

*Summer Conference on Topology and Its Applications*

This talk is based on a joint work by T. A. Edwards, J. E. Joseph, M. H. Kwack and B. M. P. Nayar that apperared in the *Journal of Advanced studies in Topology,* Vol. 5 (4), 2014), 8 - 15. B

An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the family of closed subsets of X satisfying A(Ω) ⊆ π(Ω) where A(Ω) is the adherence of Ω. The notations π(Ω) and A(Ω) are used for the values of the functions π and A and π ...

Cohen Reals And The Sequential Order Of Groups, 2017 Tennessee Technological University

#### Cohen Reals And The Sequential Order Of Groups, Alexander Shibakov

*Summer Conference on Topology and Its Applications*

We show that adding uncountably many Cohen reals to a model of diamond results in a model with no countable sequential group with an intermediate sequential order. The same model has an uncountable group of sequential order 2. We also discuss related questions.

On The Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, 2017 Batumi Shota Rustaveli State University

#### On The Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, Anzor Beridze, Vladimer Baladze

*Summer Conference on Topology and Its Applications*

In this paper an exact homology functor from the category **Mor**_{C} of continuous maps of compact Hausdorff spaces to the category **LES** of long exact sequences of abelian groups is defined (cf. [2], [3], [5]). This functor is an extension of the Hu homology theory, which is uniquely defined on the category of continuous maps of finite CW complexes and is constructed without the relative homology groups [9]. To define the given homology functor we use the Chogoshvili construction of projective homology theory [7], [8]. For each continuous map f:X → Y of compact spaces, using the notion of ...

Entropy Of Induced Continuum Dendrite Homeomorphisms, 2017 Universidade Federal do Rio de Janeiro

#### Entropy Of Induced Continuum Dendrite Homeomorphisms, Jennyffer Bohorquez, Alexander Arbieto

*Summer Conference on Topology and Its Applications*

Let f: D → D be a dendrite homeomorphism. Let C(D) denote the hyperspace of all nonempty connected compact subsets of D endowed with the Hausdorff metric. Let C(f):C(D) → C(D) be the induced continuum homeomorphism. In this talk we sketch the proof of the following result: If there exists a nonrecurrent branch point then the topological entropy of C(f) is ∞.

Sequential Order Of Compact Scattered Spaces, 2017 University of North Carolina at Charlotte

#### Sequential Order Of Compact Scattered Spaces, Alan Dow

*Summer Conference on Topology and Its Applications*

A space is sequential if the closure of set can be obtained by iteratively adding limits of converging sequences. The sequential order of a space is a measure of how many iterations are required. A space is scattered if every non-empty set has a relative isolated point. It is not known if it is consistent that there is a countable (or finite) upper bound on the sequential order of a compact sequential space. We consider the properties of compact scattered spaces with infinite sequential order.

Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, 2017 University of Oklahoma

#### Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, Ignat Soroko, Robert Kropholler, Ian Leary

*Summer Conference on Topology and Its Applications*

An interplay between algebra and topology goes in many ways. Given a space X, we can study its homology and homotopy groups. In the other direction, given a group G, we can form its Eilenberg-Maclane space K(G, 1). It is natural to wish that it is `small' in some sense. If K(G, 1) space has n-skeleton with finitely many cells, then G is said to have type F_{n}. Such groups act naturally on the cellular chain complex of the universal cover for K(G, 1), which has finitely generated free modules in all dimensions up to n ...

The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, 2017 Christopher Newport University

#### The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, James Kelly, Will Brian, Tim Tennant

*Summer Conference on Topology and Its Applications*

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces.

It is known from the work of Bartoll, Martinínez-Giménez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply ...

On Roitman's Principle For Box Products, 2017 Universidad Nacional Autonoma de Mexico

#### On Roitman's Principle For Box Products, Hector Alonso Barriga-Acosta

*Summer Conference on Topology and Its Applications*

One of the oldest problems in box products is if the countable box product of the convergent sequence is normal. It is known that consistenly (e.g., b=d, d=c) the answer is affirmative. A recent progress is due to Judy Roitman that states a combinatorial principle which also implies the normality and holds in many models.

Although the countable box product of the convergent sequence is normal in some models of b < d < c, Roitman asked what happen with her principle in this models. We answer that Roitman's principle is true in some models of b < d < c.

Spaces With No S Or L Subspaces, 2017 University of Wisconsin-Oshkosh

#### Spaces With No S Or L Subspaces, Joan Hart, Kenneth Kunen

*Summer Conference on Topology and Its Applications*

We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T_{3} space that is HS but not HL.

More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The "C" in "ESLC" refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no ...

Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), 2017 California State University, Long Beach

#### Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), Joshua Sack

*Summer Conference on Topology and Its Applications*

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C^{*}(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an *intermediate ring* if C^{*}(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P *describes* a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y ...

Topology And Experimental Distinguishability, 2017 University of Michigan - Ann Arbor

#### Topology And Experimental Distinguishability, Gabriele Carcassi, Christine A. Aidala, David J. Baker, Mark J. Greenfield

*Summer Conference on Topology and Its Applications*

In this talk we are going to formalize the relationship between topological spaces and the ability to distinguish objects experimentally, providing understanding and justification as to why topological spaces and continuous functions are pervasive tools in the physical sciences. The aim is to use these ideas as a stepping stone to give a more rigorous physical foundation to dynamical systems and, in particular, Hamiltonian dynamics.

We will first define an experimental observation as a statement that can be verified using an experimental procedure. We will show that observations are not closed under negation and countable conjunction, but are closed under ...

Entropy In Topological Groups, Part 2, 2017 University of Udine

#### Entropy In Topological Groups, Part 2, Dikran Dikranjan

*Summer Conference on Topology and Its Applications*

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be ...