Dense Subsets Of Function Spaces With No Non-Trivial Convergent Sequences, 2017 Universidad Autonoma Metropolitana - Iztapalapa

#### Dense Subsets Of Function Spaces With No Non-Trivial Convergent Sequences, Vladimir V. Tkachuk

*Summer Conference on Topology and Its Applications*

We will show that a monolithic compact space X is not scattered if and only if C_{p}(X) has a dense subset without non-trivial convergent sequences. Besides, for any cardinal κ ≥ *c*, the space **R**^{κ} has a dense subspace without non-trivial convergent sequences. If X is an uncountable σ-compact space of countable weight, then any dense set Y ⊂ C_{p}(X) has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if C_{p}(X) has a dense k-subspace, then X is scattered.

On Cohomological Dimensions Of Remainders Of Stone-Čech Compactifications, 2017 Batumi Shota Rustaveli State University

#### On Cohomological Dimensions Of Remainders Of Stone-Čech Compactifications, Vladimer Baladze

*Summer Conference on Topology and Its Applications*

In the paper the necessary and sufficient conditions are found under which a metrizable space has the Stone-Cech compactification whose remainder has the given cohomological dimensions (cf. [Sm], Problem I, p.332 and Problem II, p.334, and [A-N]).

In the paper [B] an outline of a generalization of Cech homology theory was given by replacing the set of all finite open coverings in the definition of Cech (co)homology group (Ĥ^{n}_{f}(X, A;G)) Ĥ_{n}^{f}(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm_{1}].

Following Y. Kodama …

Sequential Decreasing Strong Size Properties, 2017 Universidad Autonoma del Estado de Mexico

#### Sequential Decreasing Strong Size Properties, Miguel A. Lara, Fernando Orozco, Felix Capulín

*Summer Conference on Topology and Its Applications*

Let X be a continuum. A topological property *P* is said to be a sequential decreasing strong size property provided that if μ is a strong size map for C_{n}(X), {t_{n}} is a sequence in the interval (t, 1) such that limt_{n} = t and each fiber μ^{-1} (t_{n}) has the property *P*, then μ^{-1} (t) has the property *P*. We show that the following properties are sequential decreasing strong size properties: be a Kelley continuum, indecomposability, local connectedness, continuum chainability and unicoherence.

Enriched Topology And Asymmetry, 2017 Youngstown State University

#### Enriched Topology And Asymmetry, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton

*Summer Conference on Topology and Its Applications*

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be "variable-basis". This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces …

Which Topological Groups Arise As Automorphism Groups Of Locally Finite Graphs?, 2017 University of Pittsburgh

#### Which Topological Groups Arise As Automorphism Groups Of Locally Finite Graphs?, Xiao Chang, Paul Gartside

*Summer Conference on Topology and Its Applications*

Let Γ be a graph which is countable and locally finite (every vertex has finite degree). Then the automorphism group of Γ, Aut(Γ), with the pointwise topology has a compact, zero dimensional open normal subgroup. We investigate whether the converse holds.

On Quasi-Uniform Box Products, 2017 North-West University (South Africa)

#### On Quasi-Uniform Box Products, Hope Sabao, Olivier Olela Otafudu

*Summer Conference on Topology and Its Applications*

In this talk, we preset the quasi-uniform box product, a topology that is finer than the Tychonov product topology but coarser than the uniform box product.

We then present various notions of completeness of a quasi-uniform space that are preserved by their quasi-uniform box product using Cauchy filter pairs.

On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, 2017 Georgian Technical University

#### On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, Leonard Mdzinarishvili, Anzor Beridze

*Summer Conference on Topology and Its Applications*

The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In Milnor, the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category Top^{2}_{C} of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in Milnor). Besides, he proved that constructed homology theory satisfies partial continuity property on the subcategory Top^{2}_{CM} (see theorem 4 in Milnor) and the …

A Compact Minimal Space Whose Cartesian Square Is Not Minimal, 2017 AGH University of Science and Technology, Krakow

#### A Compact Minimal Space Whose Cartesian Square Is Not Minimal, Jan P. Boronski, Alex Clark, Piotr Oprocha

*Summer Conference on Topology and Its Applications*

A compact metric space X is called *minimal* if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {h^{n}(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from "A compact minimal space Y such that its square YxY is not minimal" whose existence answer the following long standing problem in the negative.

**Problem.** Is minimality preserved under Cartesian product in the class of compact spaces?

Note that for the fixed point property this question …

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 …

Relationships Between Hereditary Sobriety, Sobriety, Td, T1, And Locally Hausdorff, 2017 Youngstown State University

#### Relationships Between Hereditary Sobriety, Sobriety, Td, T1, And Locally Hausdorff, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton, Jamal K. Tartir

*Summer Conference on Topology and Its Applications*

This work augments the standard relationships between sobriety, T_{1}, and Hausdorff by mixing in locally Hausdorff and the compound axioms sober + T_{1} and sober + T_{D}. We show the latter compound condition characterizes hereditary sobriety, and that locally Hausdorff fits strictly between Hausdorff and sober + T_{1}. Classes of examples are constructed, in part to show the non-reversibility of key implications.

Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, 2017 University of Tennessee, Knoxville

#### Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, Thomas Weighill, Jerzy Dydak

*Summer Conference on Topology and Its Applications*

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which …

Entropy In Topological Groups, Part 1, 2017 University of Udine

#### Entropy In Topological Groups, Part 1, 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, …

Topologically Homogeneous Continua, Isometrically Homogeneous Continua, And The Pseudo-Arc, 2017 California State University, Sacramento

#### Topologically Homogeneous Continua, Isometrically Homogeneous Continua, And The Pseudo-Arc, Janusz Prajs

*Summer Conference on Topology and Its Applications*

We use accumulated knowledge on topologically homogeneous continua, and, in particular, on the pseudo-arc, to investigate the properties of isometrically homogeneous continua.

Aperiodic Colorings And Dynamics, 2017 Universidade de Santiago de Compostela

#### Aperiodic Colorings And Dynamics, Ramon Barral Lijo, Jesús A. Álvarez López

*Summer Conference on Topology and Its Applications*

A graph coloring is strongly aperiodic if every colored graph in its hull has no automorphisms. The talk will describe a method to define strongly aperiodic colorings on graphs with bounded degree. This also provides an optimal bound for the strongly distinguishing number of a graph. Then some applications to the theory of foliated spaces and to tilings will be discussed.

Pseudo-Contractibility, 2017 Universidad Autonoma del Estado de Mexico

#### Pseudo-Contractibility, Felix Capulín, Leonardo Juarez-Villa, Fernando Orozco

*Summer Conference on Topology and Its Applications*

Let X, Y be topological spaces and let f, g:X→ Y be mappings, we say that f is pseudo-homotopic to g if there exist a continuum C, points a, b ∈ C and a mapping H:X ×C → Y such that H(x, a)=f(x) and H(x, b)=g(x) for each x ∈ X. The mapping H is called a pseudo-homotopy between f and g. A topological space X is said to be pseudo-contractible if the identity mapping is pseudo-homotopic to a constant mapping in X. i.e., if there exist a continuum C, points a, b ∈ C, x_{0} ∈ X and …

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) both satisfy P, …

Some Properties Of Certain Mixed Type Special Matrix Functions And Polynomials, 2017 Aden University

#### Some Properties Of Certain Mixed Type Special Matrix Functions And Polynomials, Ahmed A. Al-Gonah, Fatima M. Al-Samadi

*Applications and Applied Mathematics: An International Journal (AAM)*

By using certain operational methods, the authors introduce some new mixed type special matrix functions and polynomials. Some properties of these matrix functions and polynomials are established.

Numerical Study Of Soliton Solutions Of Kdv, Boussinesq, And Kaup-Kuperschmidt Equations Based On Jacobi Polynomials, 2017 Rasht Branch – Islamic Azad University

#### Numerical Study Of Soliton Solutions Of Kdv, Boussinesq, And Kaup-Kuperschmidt Equations Based On Jacobi Polynomials, Khadijeh Sadri, Hamideh Ebrahimi

*Applications and Applied Mathematics: An International Journal (AAM)*

In this paper, a numerical method is developed to approximate the soliton solutions of some nonlinear wave equations in terms of the Jacobi polynomials. Wave are very important phenomena in dispersion, dissipation, diffusion, reaction, and convection. Using the wave variable converts these nonlinear equations to the nonlinear ODE equations. Then, the operational Collocation method based on Jacobi polynomials as bases is applied to approximate the solution of ODE equation resulted. In addition, the intervals of the solution will be extended using an rational exponential approximation (REA). The KdV, Boussinesq, and Kaup–Kuperschmidt equations are studied as the test examples. Finally, numerical …

Some Relations On Generalized Rice's Matrix Polynomials, 2017 Assiut University, Qassim University

#### Some Relations On Generalized Rice's Matrix Polynomials, Ayman Shehata

*Applications and Applied Mathematics: An International Journal (AAM)*

The main aim of this paper is to obtain certain properties of generalized Rice’s matrix polynomials such as their matrix differential equation, generating matrix functions, an expansion for them. We have also deduced the various families of bilinear and bilateral generating matrix functions for them with the help of the generating matrix functions developed in the paper and some of their applications have also been presented here.

Elliptic Curve Cryptology, 2017 Union College - Schenectady, NY

#### Elliptic Curve Cryptology, Francis Rocco

*Honors Theses*

In today's digital age of conducting large portions of daily life over the Internet, privacy in communication is challenged extremely frequently and confidential information has become a valuable commodity. Even with the use of commonly employed encryption practices, private information is often revealed to attackers. This issue motivates the discussion of cryptology, the study of confidential transmissions over insecure channels, which is divided into two branches of cryptography and cryptanalysis. In this paper, we will first develop a foundation to understand cryptography and send confidential transmissions among mutual parties. Next, we will provide an expository analysis of elliptic curves and …