Mathematics Commons^™

2012 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2012 Fall Colloquium.

Cycles, The Degree Distance, And The Wiener Index, Daniel Gray, Hua Wang

Mathematical Sciences: Faculty Publications

The degree distance of a graph G is D'(G)=(1/2)∑ⁿ_i=1∑ⁿ_j=1(d_i+d_j)L_{i ,j}, where d_i and d_j are the degrees of vertices v_i, v_j ∈ V (G), and L_i,j is the distance between them. The Wiener index is defined as W(G)=(1/2)∑ⁿi₌₁ ∑ⁿ_j-1L_{i, j}. An elegant result (Gutman; Klein, Mihalic, Plavsic and Trinajstic) is known regarding their correlation, that D'(T)=4W(T)-n(n-1)for a tree T with n vertices. In …

Integer Compositions, Gray Code, And The Fibonacci Sequence, Linus Lindroos

College of Graduate Studies: Theses & Dissertations

In this thesis I show the relation of binary and Gray Code to integer compositions and the Fibonacci sequence through the use of analytic combinatorics, Zeckendorf's Theorem, and generating functions.

Integer Solutions To Optimization Problems And Modular Sequences Of Nexus Numbers, Jeremy T. Davis

College of Graduate Studies: Theses & Dissertations

In this thesis, we examine the use of integers through two ideas. As mathematics teachers, we prefer students not use calculators on assessments. In order to require this, students compute the problems by hand. We take a look at the classic Calculus I optimization box problem while restricting values to integers. In addition, sticking with the integer theme, we take a new look at the nexus numbers. Nexus numbers are extensions of the hex and rhombic dodecahedral numbers. We put these numbers into a sequence, and through a few computations of modular arithmetic, we analyze the sequences and their patterns …

A Non-Parametric Approach To Change-Point Detection In Cross-Asset Correlations, L. Kaili Diamond

College of Graduate Studies: Theses & Dissertations

In this thesis we explore the problem of detecting change-points in cross-asset correlations using a non-parametric approach. We began by comparing and contrasting several common methods for change-point detection as well as methods for measuring correlation. We finally settle on a statistic introduced in early 2012 by Herold Dehling et.al. and test this statistic against real world financial data. We provide the estimated change-point for this data as well as the asymptotic p-value associated with this statistic. Once this process was complete we went on to use simulated data to measure the accuracy, power, and type 1 error associated with …

Compositions, Bijections, And Enumerations, Charles R. Dedrickson Iii

College of Graduate Studies: Theses & Dissertations

In this thesis we give an introduction to colored-compositions of an integer. This is a generalization of traditional integer compositions, and we show a few results for n-color compositions which are analogous to regular compositions with both combinatorial and analytic proofs. We also show several bijections between various types of compositions to certain types of numeric strings, and provide a generalization of a classic bijection between compositions and binary strings.

An Analysis Of Differences In Approaches To Systems Of Linear Equations Problems Given Multiple Choice Answers, Amber Lagasse

Honors Theses and Capstones

This descriptive study focuses on the approaches college students (ages 20 -24) use when solving systems of linear equations problems that have multiple choice answers. Participants were from a midsize public university in the northeast. Four approaches were considered – three forwards approaches: 1) substitution, 2) elimination, and 3) graphing, and one backwards approach: plugging in the x and y values from each multiple choice option. Participants solved systems of linear equations problems and answered questions based on their methods in a structured clinical interview. Each participant also filled out a questionnaire. It was shown from the results of this …

A Unified Approach To Generalized Stirling Functions, Tian-Xiao He

Scholarship

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

The Secant Conjecture In The Real Schubert Calculus, Luis D. García-Puente, Nickolas Hein, Christopher Hillar, Abraham Martín Del Campo, James Ruffo, Frank Sottile, Zach Teitler

Mathematics Faculty Publications and Presentations

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

Packing Tight Hamilton Cycles In Uniform Hypergraphs, Deepak Bal, Alan Frieze

Department of Mathematics Faculty Scholarship and Creative Works

We say that a k-uniform hypergraph C is a Hamilton cycle of type l, for some 1 ≤ l ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, and for every pair of consecutive edges E i-1\E i in C (in the natural ordering of the edges) we have |E i-1 \ E i| = l. We define a class of (ε, p)-regular hypergraphs, that includes random hypergraphs, for which we can prove the existence of a decomposition of almost all edges into type l Hamilton cycles, …

Sustainable Technology For Person-Centered Accessible Integrated Multimodal Information Systems, Lawrence J. Harman, Uma Shama, Heather Standring, Sabitha Gopalsamy, Anil Sadhu, Mateusz Pacha-Sucharzewski

Mathematics Faculty Publications

This paper reports on a mobility management technology project conducted by the GeoGraphics Laboratory at Bridgewater State University in Bridgewater, Massachusetts, in the Northeastern United States (U.S.). This study is a part of a much larger mobility management technology deployment by the Cape Cod Regional Transit Authority (CCRTA) that deployed integrated intermodal intelligent transportation system (ITS) to support the mobility of a metropolitan region that has a high proportion of elderly residents and persons with disabilities and is a significant tourist destination for national and international travelers. This paper reports on a research project that is developing smartphone applications to …

An Impulsive Delay Differential Inequality And Applications, Xiaodi Li, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

An impulsive delay differential inequality is formulated in this paper. an estimate of the rate of decay of solutions to this inequality is obtained. It can be applied to the study of dynamical behavior of delay differential equations from the impulsive control point of view. as an application, we consider a class of impulsive control systems with time-varying delays and establish a sufficient condition to guarantee the global exponential stability. It is shown that, via proper impulsive control law, a linear delay differential system can be exponentially stabilized even if it is initially unstable. a numerical example is given to …

If Energy Is Not Preserved, Then Planck's Constant Is No Longer A Constant: A Theorem, Vladik Kreinovich, Andres Ortiz

Departmental Technical Reports (CS)

For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory …

Towards Unique Physically Meaningful Definitions Of Random And Typical Objects, Luc Longpre, Olga Kosheleva

Departmental Technical Reports (CS)

To distinguish between random and non-random sequence, Kolmogorov and Martin-Lof proposed a new definition of randomness, according to which an object (e.g., a sequence of 0s and 1s) if random if it satisfies all probability laws, i.e., in more precise terms, if it does not belong to any definable set of probability measure 0. This definition reflect the usual physicists' idea that events with probability 0 cannot happen. Physicists -- especially in statistical physics -- often claim a stronger statement: that events with a very small probability cannot happen either. A modification of Kolmogorov-Martin-Lof's (KLM) definition has been proposed to …

270: How To Win The Presidency With Just 17.56% Of The Popular Vote, Charles D. Wessell

Math Faculty Publications

With the U.S. presidential election fast approaching we will often be reminded that the candidate who receives the most votes is not necessarily elected president. Instead, the winning candidate must receive a majority of the 538 electoral votes awarded by the 50 states and the District of Columbia. Someone with a curious mathematical mind might then wonder: What is the small fraction of the popular vote a candidate can receive and still be elected president? [excerpt]

Why Clayton And Gumbel Copulas: A Symmetry-Based Explanation, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

In econometrics, many distributions are non-Gaussian. To describe dependence between non-Gaussian variables, it is usually not sufficient to provide their correlation: it is desirable to also know the corresponding copula. There are many different families of copulas; which family shall we use? In many econometric applications, two families of copulas have been most efficient: the Clayton and the Gumbel copulas. In this paper, we provide a theoretical explanation for this empirical efficiency, by showing that these copulas naturally follow from reasonable symmetry assumptions. This symmetry justification also allows us to provide recommendations about which families of copulas we should use …

Reverse Mathematics And Properties Of Finite Character, Damir D. Dzhafarov, Carl Mummert

Mathematics Faculty Research

We study the reverse mathematics of the principle stating that,for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey’s lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these …

Sample Path Properties Of Volterra Processes, Leonid Mytnik, Eyal Neuman

Communications on Stochastic Analysis

No abstract provided.

Feynman-Kac Formula For The Solution Of Cauchy's Problem With Time Dependent Lévy Generator, Aroldo Pérez

Communications on Stochastic Analysis

No abstract provided.