On The Estrada Index Of Cacti, 2014 University of Wyoming

#### On The Estrada Index Of Cacti, Erfang Shan, Hongzhuan Wang, Liying Kang

*Electronic Journal of Linear Algebra*

Let G be a simple connected graph on n vertices and λ_1, λ_2, . . . , λ_n be the eigenvalues of the adjacency matrix of G. The Estrada index of G is defined as EE(G) = \sum_{i=1}^n e^{λi}. A cactus is a connected graph in which any two cycles have at most one common vertex. In this work, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cycles was determined. Also, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cut ...

Exact Results For Perturbation To Total Positivity And To Total Nonsingularity, 2014 University of Wyoming

#### Exact Results For Perturbation To Total Positivity And To Total Nonsingularity, Miriam Farber, Mitchell Faulk, Charles R. Johnson, Evan Marzion

*Electronic Journal of Linear Algebra*

A study of the maximum number of equal entries in totally positive and totally nonsingular m-by-n, matrices for small values of m and n, is presented. Equal entries correspond to entries of the totally nonnegative matrix J that are not changed in producing a TP or TNS matrix. It is shown that the maximum number of equal entries in a 7-by-7 totally positive matrix is strictly smaller than that for a 7-by-7 totally non-singular matrix, but, this is the first pair (m; n) for which these maximum numbers differ. Using point-line geometry in the projective plane, a family of values ...

The Combinatorial Inverse Eigenvalue Problem Ii: All Cases For Small Graphs, 2014 University of Wyoming

#### The Combinatorial Inverse Eigenvalue Problem Ii: All Cases For Small Graphs, Wayne Barrett, Curtis G. Nelson, John Henry Sinkovic, Tianyi Yang

*Electronic Journal of Linear Algebra*

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n by n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n - 1 real numbers \lambda_1\geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots \geq \lambda_{n-1} \geq \mu_{n-1} \geq \lambda_{n-1}, and a vertex v of G, the question is addressed of whether or not there exists A in S(G) with eigenvalues \lambda_1, \ldots, \lambda_ n such that A(v) has eigenvalues \mu_1, \ldots, \mu_{n-1}, where A(v) denotes the matrix with ...

Commuting Maps On Rank-K Matrices, 2014 University of Wyoming

#### Commuting Maps On Rank-K Matrices, Xiao-Wei Xu, Xiaofei Yi

*Electronic Journal of Linear Algebra*

In this short note we give a new proof and a slight improvement of the Franca Theorem. More precisely we prove: Let n \geq 3 be a natural number, and let Mn(K) be the ring of all n × n matrices over an arbitrary field K with center Z. Fix a natural number 2≤s≤n. If G:Mn(K)→Mn(K) is an additive map such that G(x)x=xG(x) for every rank-s matrix x ∈ Mn(K), then there exist an element λ ∈ Z and an additive map μ : Mn(K) → Z such that G(x) = λx ...

Traces Of Matrix Products, 2014 University of Wyoming

#### Traces Of Matrix Products, John Greene

*Electronic Journal of Linear Algebra*

Given two noncommuting matrices, A and B, it is well known that AB and BA have the same trace. This extends to cyclic permutations of products of A’s and B’s. It is shown here that for 2×2 matrices A and B, whose elements are independent random variables with standard normal distributions, the probability that Tr(ABAB) > Tr(AB) is exactly 1/\sqrt{2} .

Note On Rainbow Connection In Oriented Graphs With Diameter 2, 2014 Georgia Southern University

#### Note On Rainbow Connection In Oriented Graphs With Diameter 2, Rebecca Holliday, Colton Magnant, Pouria Salehi Nowbandegani

*Theory and Applications of Graphs*

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter $2$. For a tournament $T$ of diameter $2$, we show $2 \leq \overrightarrow{rc}(T) \leq 3$. Furthermore, we provide a general upper bound on the rainbow $k$-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of $k^{th}$ diameter $2$ has rainbow $k$-connection number at most approximately $k^{2}$.

The Scope Of Gödel’S First Incompleteness Theorem, 2014 Indiana University – Purdue University Fort Wayne

#### The Scope Of Gödel’S First Incompleteness Theorem, Bernd Buldt

*Philosophy Faculty Publications*

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.

Hamiltonicity And $\Sigma$-Hypergraphs, 2014 Georgia Southern University

#### Hamiltonicity And $\Sigma$-Hypergraphs, Christina Zarb

*Theory and Applications of Graphs*

We define and study a special type of hypergraph. A $\sigma$-hypergraph $H= H(n,r,q$ $\mid$ $\sigma$), where $\sigma$ is a partition of $r$, is an $r$-uniform hypergraph having $nq$ vertices partitioned into $ n$ classes of $q$ vertices each. If the classes are denoted by $V_1$, $V_2$,...,$V_n$, then a subset $K$ of $V(H)$ of size $r$ is an edge if the partition of $r$ formed by the non-zero cardinalities $ \mid$ $K$ $\cap$ $V_i \mid$, $ 1 \leq i \leq n$, is $\sigma$. The non-empty intersections $K$ $\cap$ $V_i$ are called the parts of $K$, and $s ...

Online Resource Platform For Mathematics Education, 2014 Dublin Institute of Technology

#### Online Resource Platform For Mathematics Education, Marisa Llorens, Edmund Nevin, Eileen Mageean

*Conference Papers*

Engineering education is facing many challenges: a decline in core mathematical skills; lowering entry requirements; and the diversity of the student cohort. One approach to confronting these challenges is to make subject content appropriate to the communication styles of today’s student. To achieve this, a pedagogical shift from the traditional hierarchical approach to learning to one that embraces the use of technology as a tool to enhance the student learning experience is required. By including the student as co-creator of course content, a greater sense of engagement is achieved and a change to one where students become agents of ...

Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, 2014 Georgia Southern University

#### Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, Shinya Fujita, Colton Magnant, Kenta Ozeki

*Theory and Applications of Graphs*

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Ogden College News, 2014 Western Kentucky University

#### Ogden College News, Cheryl Stevens, Dean

*Ogden College of Science & Engineering Publications*

No abstract provided.

The Scientific Theories Of Michael Faraday And James Clerk Maxwell, 2014 Purdue University

#### The Scientific Theories Of Michael Faraday And James Clerk Maxwell, Shannon P. Israelsen

*The Purdue Historian*

This paper explores the role of mathematical formalism in physics theories through an analysis of the work of physicists Michael Faraday and James Clerk Maxwell in the field of electromagnetism. After a brief description of the educational backgrounds of each scientist, their views on the nature of physics theory and its relationship to mathematics are contrasted. Faraday sought a mathematics-free theory to describe the results of his experiments, producing a family of concepts that became the roots of electromagnetic field theory. Maxwell built upon these concepts, giving them robust mathematical expression and incorporating them into one unified theoretical picture. An ...

Spatiotemporal Crime Analysis, 2014 Purdue University

#### Spatiotemporal Crime Analysis, James Q. Tay, Abish Malik, Sherry Towers, David Ebert

*The Summer Undergraduate Research Fellowship (SURF) Symposium*

There has been a rise in the use of visual analytic techniques to create interactive predictive environments in a range of different applications. These tools help the user sift through massive amounts of data, presenting most useful results in a visual context and enabling the person to rapidly form proactive strategies. In this paper, we present one such visual analytic environment that uses historical crime data to predict future occurrences of crimes, both geographically and temporally. Due to the complexity of this analysis, it is necessary to find an appropriate statistical method for correlative analysis of spatiotemporal data, as well ...

The Bernoulli Family: Their Massive Contributions To Mathematics And Hostility Toward Each Other, 2014 Chapman University

#### The Bernoulli Family: Their Massive Contributions To Mathematics And Hostility Toward Each Other, Dung (Yom) Bui, Mohamed Allali

*e-Research: A Journal of Undergraduate Work*

Throughout the history of mathematics, there are several individuals with significant contributions. However, if we look at the contribution of a single family in this field, the Bernoulli probably outshines others in terms of both the number of mathematicians it produced and their influence on the development of mathematics. The most outstanding three Bernoulli mathematicians are Jacob I Bernoulli (1654-1705), Johann I Bernoulli (1667-1748), and Daniel Bernoulli (1700-1782), all three of whom were the most influential math experts in the academic community yet very hostile to each other. Their family structure and jealousy toward each other might have fueled their ...

Continuous Dependence Of Solutions Of Equations On Parameters, 2014 Rose-Hulman Institute of Technology

#### Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map *T : C^{n} -> C^{n}*.

Network Analysis Of World Trade Using The Baci-Cepii Dataset, 2014 SelectedWorks

#### Network Analysis Of World Trade Using The Baci-Cepii Dataset, Luca De Benedictis, Silvia Nenci, Gianluca Santoni, Lucia Tajoli, Claudio Vicarelli

*Luca De Benedictis*

In this paper we explore the BACI-CEPII database using Network Analysis. Starting from the visualization of the World Trade Network, we then define and describe the topology of the network, both in its binary version and in its weighted version, calculating and discussing some of the commonly used network’s statistics. We finally discuss some specific topic that can be studied using Network Analysis and International Trade data, both at the aggregated and sectoral level. The analysis is done using multiple software (Stata, R, and Pajek). The scripts to replicate part of the analysis are included in the appendix, and ...

Cold Hard Fractals, 2014 Northern Michigan University

#### Cold Hard Fractals, David Buhl, Sam Morey

*Journal Articles*

Although many college mathematics students enjoy studying fractal geometry, more times than not, they question whether the content can be useful outside the classroom. This article describes a mathematical investigation that applied content from a fractal geometry unit to the real-world phenomena of ice formations on Lake Superior created by the recent winter.

Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, 2014 Western University

#### Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, Youlong Yan

*University of Western Ontario - Electronic Thesis and Dissertation Repository*

The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One important device relevant for this study is the notion of tilting sheaf.

This thesis is concerned with the existence of tilting sheaves on some smooth projective varieties. The main technique we use in this thesis is Galois descent theory. We first construct tilting bundles on general Brauer-Severi varieties. Our main result shows the existence of tilting bundles on some Brauer-Severi schemes. As an application, we prove that there are tilting bundles on an arithmetic toric variety whose toric variety ...

Composite Dilation Wavelets With High Degrees, 2014 Illinois Wesleyan University

#### Composite Dilation Wavelets With High Degrees, Tian-Xiao He

*Tian-Xiao He*

No abstract provided.

Choosing Between Parametric And Non-Parametric Tests, 2014 Minnesota State University, Mankato

#### Choosing Between Parametric And Non-Parametric Tests, Russ Johnson

*Journal of Undergraduate Research at Minnesota State University, Mankato*

A common question in comparing two sets of measurements is whether to use a parametric testing procedure or a non-parametric procedure. The question is even more important in dealing with smaller samples. Here, using simulation, several parametric and nonparametric tests, such as, t-test, Normal test, Wilcoxon Rank Sum test, van-der Waerden Score test, and Exponential Score test are compared.