Applied Mathematics Commons^™

Order, Distance, Closure And Convergence: Reconciling Competing Fundamental Topological Concepts, Walter Tholen

Summer Conference on Topology and Its Applications

Already in Hausdorff’s 1914 book, often considered the cradle of general topology, one finds traces of a discussion on the relative strengths of the concepts mentioned in the title of this talk. In fact, one may argue that Hausdorff anticipated the basic ideas of how to unify these concepts, which were developed only later on by many mathematicians over the course of a century, as propagated in Hofmann, Seal & Tholen. Indeed, Hausdorff thought of ordering points by assigning to every pair of them a (truth) value, just as a metric assigns to them a number. More importantly, he also …

Virtual Seifert Surfaces And Slice Obstructions For Knots In Thickened Surfaces, Micah Chrisman, Hans U. Boden, Robin Gaudreau

Summer Conference on Topology and Its Applications

Here we introduce the notion of virtual Seifert surfaces. Virtual Seifert surfaces may be thought of as a generalization of Gauss diagrams of virtual knots to spanning surfaces of a knot. This device is then employed to extend the Tristram-Levine signature function to AC knots. Using the AC signature functions and Tuarev’s graded genus invariant, we determine the slice status of all 76 almost classical knots having at most six crossings. The slice obstructions for AC knots are then extended to all virtual knots via the parity projection map. This map, which is computable from a Gauss diagram, sends a …

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.

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. On the …

The Isbell-Hull Of An Asymmetrically Normed Space, Olivier Olela Otafudu, Jurie Conradie, Hans-Peter Künzi

Summer Conference on Topology and Its Applications

In this talk, we discuss an explicit method to define the linear structure of the Isbell-hull of an asymmetrically normed space.

On Continua With Regular Non-Abelian Self Covers, Mathew Timm

Summer Conference on Topology and Its Applications

We look at a planar 2-dimensional continuum X which satisfy the following:

Given any finite group G there is an |G|-fold regular self cover f:X → X with G as its group of deck transformations.

A Trace Formula For Foliated Flows (Working Paper), Jesús A. Álvarez López, Yuri A. Kordyukov, Eric Leichtnam

Summer Conference on Topology and Its Applications

The talk, based on work in progress, will be about our progress to show a trace formula for foliated flows on foliated spaces, which has been conjectured by V. Guillemin, and later by C. Deninger with more generality. It describes certain Leftchetz distribution of the foliated flow, acting on some version of the leafwise cohomology, in terms of local data at the closed orbits and fixed points.

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.

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.

Rigidity And Nonrigidity Of Corona Algebras, Paul Mckenney, Alessandro Vignati

Summer Conference on Topology and Its Applications

Shelah proved in the 1970s that there is a model of ZFC in which every homeomorphism of the Cech-Stone remainder of the natural numbers is induced by a function on the natural numbers. More recently, Farah proved that in essentially the same model, every automorphism of the Calkin algebra on a separable Hilbert space must be induced by a linear operator on the Hilbert space. I will discuss a common generalization of these rigidity results to a certain class of C*-algebras called corona algebras. No prerequisites in C*-algebra will be assumed.

Balanced And Functionally Balanced P-Groups, Menachem Shlossberg

Summer Conference on Topology and Its Applications

In relation to Itzkowitz's problem, we show that a c-bounded P-group is balanced if and only if it is functionally balanced. We prove that for an arbitrary P-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform P-space. In particular, we show that this group is (functionally) balanced precisely when its subsets B_n, consisting of words of length at most n, are all (resp., functionally) balanced.

Compactly Supported Homeomorphisms As Long Direct Limits, Rafael Dahmen, Gábor Lukács

Summer Conference on Topology and Its Applications

Let λ be a limit ordinal and consider a directed system of topological groups (G_α)_{α < λ} with topological embeddings as bonding maps and its directed union G=∪_{α < λ}G_α. There are two natural topologies on G: one that makes G the direct limit (colimit) in the category of topological spaces and one which makes G the direct limit (colimit) in the category of topological groups.

For λ = ω it is known that these topologies almost never coincide (Yamasaki's Theorem).

In my talk last year, I introduced the Long Direct Limit Conjecture, stating that …

Uncountable Discrete Sets And Forcing, Akira Iwasa

Summer Conference on Topology and Its Applications

Suppose that a space X has no uncountable discrete subspace. We will discuss if forcing can create an uncountable discrete subspace of X.

Locally Compact Groups: Traditions And Trends, Karl Heinrich Hofmann, Wolfgang Herfort, Francesco G. Russo

Summer Conference on Topology and Its Applications

For a lecture in the Topology+Algebra and Analysis section, the subject of locally compact groups appears particularly fitting: Historically and currently as well, the structure and representation theory of locally compact groups draws its methods from each of theses three fields of mathematics. Nowadays one might justifiably add combinatorics and number theory as sources. The example of a study of a class of locally compact groups called “near abelian,” undertaken by W. Herfort, K. H. Hofmann, and F. G. Russo, may be used to illustrate the liaison of topological group theory with this different areas of interest. Concepts like the …

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 …

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.

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(Ω) =⋂_Ω π F= …

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, …

On Cardinality Bounds Involving The Weak Lindelöf Degree And H-Closed Spaces, Nathan Carlson, Angelo Bella, Jack Porter

Summer Conference on Topology and Its Applications

1. Bella and Carlson give several classes of spaces X for which |X| ≤ 2^wL(X)χ(X). This includes locally compact spaces and, more recently, extremally disconnected spaces. Three proofs of the former lead to more general results. One such result is that any regular space X with a π-base consisting of elements with compact closure satisfies |X| ≤ 2^wL(X)χ(X). It is also shown that if X is locally compact and power homogeneous that |X| ≤ 2^wL(X)t(X), an extension of De la Vega's Theorem.

2. Porter and Carlson give a new cardinality bound for any Hausdorff …

Integration Over Curves And Surfaces Defined By The Closest Point Mapping, Catherine Kublik, Richard Tsai

Mathematics Faculty Publications

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of …

Effects Of Cell Cycle Noise On Excitable Gene Circuits, Alan Veliz-Cuba, Chinmaya Gupta, Matthew R. Bennett, Krešimir Josić, William Ott

Mathematics Faculty Publications

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Convolutions And Green’S Functions For Two Families Of Boundary Value Problems For Fractional Differential Equations, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question …

Identification Of Control Targets In Boolean Molecular Network Models Via Computational Algebra, David Murrugarra, Alan Veliz-Cuba, Boris Aguilar, Reinhard Laubenbacher

Mathematics Faculty Publications

Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered …

Theorems On Boundedness Of Solutions To Stochastic Delay Differential Equations, Youssef Raffoul, Dan Ren

Mathematics Faculty Publications

In this report, we provide general theorems about boundedness or bounded in probability of solutions to nonlinear delay stochastic differential systems. Our analysis is based on the successful construction of suitable Lyapunov functionals. We offer several examples as application of our theorems.

Optimal Control Analysis Of Ebola Disease With Control Strategies Of Quarantine And Vaccination, Muhammad Dure Ahmad, Muhammad Usman, Adnan Khan, Mudassar Imran

Mathematics Faculty Publications

The 2014 Ebola epidemic is the largest in history, affecting multiple countries in West Africa. Some isolated cases were also observed in other regions of the world.

Upper And Lower Solution Method For Boundary Value Problems At Resonance, Samerah Al Mosa, Paul W. Eloe

Mathematics Faculty Publications

We consider two simple boundary value problems at resonance for an ordinary differential equation. Employing a shift argument, a regular fixed point operator is constructed. We employ the monotone method coupled with a method of upper and lower solutions and obtain sufficient conditions for the existence of solutions of boundary value problems at resonance for nonlinear boundary value problems. Three applications are presented in which explicit upper solutions and lower solutions are exhibited for the first boundary value problem. Two applications are presented for the second boundary value problem. Of interest, the upper and lower solutions are easily and explicitly …

Math Department Newsletter, 2016, University Of Dayton. Department Of Mathematics

Department of Mathematics Newsletters

No abstract provided.

Uniform Stability In Nonlinear Infinite Delay Volterra Integro-Differential Equations Using Lyapunov Functionals, Youssef Raffoul, Habib Rai

Mathematics Faculty Publications

In [10] the first author used Lyapunov functionals and studied the exponential stability of the zero solution of nite delay Volterra Integro-dierential equation. In this paper, we use modified version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-dierential equation

x′(t) = Px(t) + ^t∫ _−∞ C(t, s)g(x(s))ds.

Coding Strategies, The Choquet Game And Domain Representability, Lynne Yengulalp

Mathematics Faculty Publications

We prove that if the NONEMPTY player has a winning strategy in the strong Choquet game on a regular space X then NONEMPTY has a winning coding strategy in that game (a strategy that only depends on the previous 2 moves). We also prove that any regular domain representable space is generalized subcompact.

Asymptotically Periodic Solutions Of Volterra Integral Equations, Muhammad Islam

Mathematics Faculty Publications

We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.