Mathematics Commons^™

Braid Group Actions On Rational Maps, Eriko Hironaka, Sarah Koch

Summer Conference on Topology and Its Applications

Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if …

How Accurate Are Expert Estimations Of Correlation?, Michael Beer, Zitong Gong, Francisco Alejandro Diaz De La O, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, it is important to know the correlation between different quantities -- finding correlations helps find the causes of different phenomena, and helps to find way to improve the situation. Often, there is not enough empirical data to experimentally determine all possible correlation. In such cases, a natural idea is to supplement this situation with expert estimates. Expert estimates are rather crude. So, to decide whether to act based on these estimates, it is desirable to know how accurate are expert estimates. In this paper, we propose several techniques for gauging this accuracy.

In Memoriam : Elart Von Collan, Boyan N. Dimitrov

Mathematics Publications

No abstract provided.

In Education, Delayed Feedback Is Often More Efficient Than Immediate Feedback: A Geometric Explanation, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Feedback is important in education. It is commonly believed that immediate feedback is very important. That is why instructors stay often late at night grading students' assignments -- to make sure that the students get their feedback as early as possible. However, surprisingly, experiments show that in many cases, delayed feedback is more efficient that the immediate one. In this paper, we provide a simple geometric explanation of this seemingly counter-intuitive empirical phenomenon.

Quantitative Justification For The Gravity Model In Economics, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

The gravity model in economics describes the trade flow between two countries as a function of their Gross Domestic Products (GDPs) and the distance between them. This model is motivated by the qualitative similarity between the desired dependence and the dependence of the gravity force (or potential energy) between the two bodies on their masses and on the distance between them. In this paper, we provide a quantitative justification for this economic formula.

How To Get Beyond Uniform When Applying Maxent To Interval Uncertainty, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the Maximum Entropy (MaxEnt) approach leads to reasonable distributions. However, in an important case when all we know is that the value of a random variable is somewhere within the interval, this approach leads to a uniform distribution on this interval -- while our intuition says that we should have a distribution whose probability density tends to 0 when we approach the interval's endpoints. In this paper, we show that in most cases of interval uncertainty, we have additional information, and if we account for this additional information when applying MaxEnt, we get distributions which are …

Efficient Algorithms For Synchroning Localization Sensors Under Interval Uncertainty, Raphael Voges, Bernardo Wagner, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that a practical need for synchronization of localization sensors leads to an interval-uncertainty problem. In principle, this problem can be solved by using the general linear programming algorithms, but this would take a long time -- and this time is not easy to decrease, e.g., by parallelization since linear programming is known to be provably hard to parallelize. To solve the corresponding problem, we propose more efficient and easy-to-parallelize algorithms.

Maximum Entropy As A Feasible Way To Describe Joint Distributions In Expert Systems, Thongchai Dumrongpokaphan, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In expert systems, we elicit the probabilities of different statements from the experts. However, to adequately use the expert system, we also need to know the probabilities of different propositional combinations of the experts' statements -- i.e., we need to know the corresponding joint distribution. The problem is that there are exponentially many such combinations, and it is not practically possible to elicit all their probabilities from the experts. So, we need to estimate this joint distribution based on the available information. For this purpose, many practitioners use heuristic approaches -- e.g., the t-norm approach of fuzzy logic. However, this …

Math Department Newsletter, 2016-2017, Mathematics Department

Mathematics Newsletter

No abstract provided.

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, …

Introduction (2017), Association Of Christians In The Mathematical Sciences

ACMS Conference Proceedings 2017

No abstract provided.

Klein Four Actions On Graphs And Sets, Darren B. Glass

Math Faculty Publications

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (ℤ/2ℤ)2-action to the properties of its quotients generalizes to results about sets and graphs that admit (ℤ/2ℤ)2-actions.

Are Permanent Or Temporary Teams More Efficient: A Possible Explanation Of The Empirical Data, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in education, stable (long-term) student teams are more effective than temporary (short-term) ones. It turned out that the same phenomenon is true for workers working on a long-term project. However, somewhat surprisingly, for small-scale projects, the opposite is true: teams without any prior collaboration experience are more successful. Moreover, it turns out that if combine in a team members with prior collaboration experience and members without such experience, the efficiency of the team gets even lower. In this paper, we provide a possible explanation for this strange empirical phenomenon.

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, x₀ ∈ X and …

Normal Images Of A Product And Countably Paracompact Condensation, Jila Niknejad

Summer Conference on Topology and Its Applications

In 1997, Buzjakova proved that for a pseudocompact Tychonoff space X and λ = | βX|⁺, X condenses onto a compact space if and only if X×(λ+1) condenses onto a normal space. This is a condensation form of Tamano's theorem. An interesting problem is to determine how much of Buzjakova's result will hold if "pseudocompact" is removed from the hypothesis.

In this talk, I am going to show for a Tychonoff space X, there is a cardinal λ such that if X×(λ+1) condenses onto a normal space, then X condenses onto a countably paracompact space.

On The Tightness And Long Directed Limits Of Free Topological Algebras, Gábor Lukács, Rafael Dahmen

Summer Conference on Topology and Its Applications

For a limit ordinal λ, let (A_α)_{α < λ} be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪_{α < λ} A_α. Let (A, T) and (A, A) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has T ⊇ A, but the inclusion may be strict; however, if the tightness of A is smaller than the cofinality of λ, then A=T.

In 1988, Tkachenko proved …

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, F₂(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F₂(X) provided that each continuum Y satisfying that F₂(X) and F₂(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F₂(X) will be presented.

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 Top²_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 Top²_CM (see theorem 4 in Milnor) and the …

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.

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 {hⁿ(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 …