A Weighted Möbius Function, 2017 Portland State University

#### A Weighted Möbius Function, Derek Garton

*Mathematics and Statistics Faculty Publications and Presentations*

Fix an odd prime ℓ and let **G** be the poset of isomorphism classes of finite abelian** ℓ-**groups, ordered by inclusion. If** ξ:G→R ^{≥0} **is a discrete probability distribution on

**G**and

**A ∈ G**, define the

**A**th moment of

**ξ**to be . The question of determining conditions that ensure

**ξ**is completely determined by its moments has been of recent interest in many problems of Cohen–Lenstra type. Furthermore, recovering

**ξ**from its moments requires a new Möbius-type inversion formula on

**G**. In this paper, we define this function, relate it to the classical Möbius function ...

Vertex Weighted Spectral Clustering, 2017 East Tennessee State University

#### Vertex Weighted Spectral Clustering, Mohammad Masum

*Electronic Theses and Dissertations*

Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to ...

Primes, Divisibility, And Factoring, 2017 Central Washington University

#### Primes, Divisibility, And Factoring, Dominic Klyve

*Number Theory*

No abstract provided.

Babylonian Numeration, 2017 Central Washington University

Gaussian Integers And Dedekind's Creation Of An Ideal: A Number Theory Project, 2017 Colorado State University-Pueblo

#### Gaussian Integers And Dedekind's Creation Of An Ideal: A Number Theory Project, Janet Heine Barnett

*Number Theory*

No abstract provided.

Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, 2017 Colorado State University-Pueblo

#### Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, Janet Heine Barnett

*Analysis*

No abstract provided.

The Definite Integrals Of Cauchy And Riemann, 2017 Ursinus College

#### The Definite Integrals Of Cauchy And Riemann, Dave Ruch

*Analysis*

Rigorous attempts to define the definite integral began in earnest in the early 1800's. One of the pioneers in this development was A. L. Cauchy (1789-1857). In this project, students will read from his 1823 study of the definite integral for continuous functions . Then students will read from Bernard Riemann's 1854 paper, in which he developed a more general concept of the definite integral that could be applied to functions with infinite discontinuities.

Properties Enjoyed By The Highest Digit In A Base Other Than The Base 10, 2017 Valdosta State University

#### Properties Enjoyed By The Highest Digit In A Base Other Than The Base 10, Sudhir Goel, Kathy Simons

*Georgia Journal of Science*

The number nine in base ten enjoys some nice arithmetic properties. In this paper, we show that these properties are not intrinsic to the number nine; in fact, they are true for the largest digit in any base* b*. Four properties involving the final sums of all the digits of a number in a non-decimal base are explored and proofs of these properties are given in the appendix.

Principle Components For Diagnosing Dispersion In Multivariate Statistical Process Control, 2017 Georgia Academy of Science

#### Principle Components For Diagnosing Dispersion In Multivariate Statistical Process Control, Terrance E. Murphy, Mary Mcshane-Vaughn, Kwok Leung-Tsui

*Georgia Journal of Science*

We provide an easily implemented procedure to help data analysts systematically diagnose which quality characteristics may be driving the dispersion of a multivariate process out of control. Multivariate statistical process control commonly uses Hotelling's *T*^{2} statistic to indicate when a multivariate observation goes out-of-control. Several techniques currently exist that accurately diagnose which specific variables are driving the *T*^{2} statistic out-of-control. For subgroups of independently and identically distributed multivariate normal observations, we advocate decomposing the overall *T*^{2} into independent *T*^{2} statistics for separate monitoring of location and dispersion. We propose a procedure based on principle components ...

The Algebraic Connectivity Of Graphs With Given Stability Number, 2017 Central China Normal University

#### The Algebraic Connectivity Of Graphs With Given Stability Number, Zhang Shunzhe, Zhao Qin, Wen Qin

*Electronic Journal of Linear Algebra*

In this paper, we investigate the algebraic connectivity of connected graphs, and determine the graph which has the minimum algebraic connectivity among all connected graphs of order $n$ with given stability number $\alpha\geq\lceil\frac{n}{2}\rceil$, or covering number, respectively.

Block Stanley Deompositions Ii. Greedy Algorithms, Applications And Open Problems, 2017 Iowa State University

#### Block Stanley Deompositions Ii. Greedy Algorithms, Applications And Open Problems, James Murdock, Theodore Murdock

*Mathematics Publications*

Stanley decompositions are used in applied mathematics (dynamical systems) and *sl2* invariant theory as finite descriptsions of the set of standard monomials of a monomial ideal. The block notation for Stanley decompositions has proved itself in this context as a shorter notation and one that is useful in formulating algorithms such as the "box product." Since the box product appears only in dynamical systems literature, we sketch its purpose and the role of block notation in this application. Then we present a greedy algorithm that produces incompressible block decompositions (called "organized") from the monomial ideal; these are desirable for their ...

Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, 2017 Ferdowsi University of Mashhad

#### Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, Mehdi Vosough, Mohammad Sal Moslehian

*Electronic Journal of Linear Algebra*

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained ...

The Importance Of Mathematical Wording By Math Instructors, 2017 Valdosta State University

#### The Importance Of Mathematical Wording By Math Instructors, Sudhir Goel, Denise Reid

*Georgia Journal of Science*

At times we the math instructors in a classroom are not careful enough to use proper Mathematical wording, e.g. *sin*^{2} + *cos*^{2} = 1 without any reference to the angle. Improper Mathematical wording can be confusing to a student at any level, especially at high school and/or at a college freshman or sophomore level. It may hinder the students’ progress in the present and the subsequent math courses.

Connecting Algebra And Geometry For Extraneous Roots, 2017 Valdosta State University

#### Connecting Algebra And Geometry For Extraneous Roots, Sudhir Goel, Iwan Elstak

*Georgia Journal of Science*

Extraneous roots are an unsolved mystery for our freshmen level college math students. The students are often confused and wonder what extraneous roots are and how do they suddenly appear while solving an algebraic equation. In this paper we clarify the idea of extraneous roots and connect the algebra and geometry behind extraneous roots.

Been There Done That Can We Connect?, 2017 Valdosta State University

#### Been There Done That Can We Connect?, Sudhir Goel, Patricia Bezona

*Georgia Journal of Science*

A standard topic in a College Algebra course is to find the coordinates of the midpoint of a line segment connecting two points. In the early twentieth century [1] and [2], in a few texts, this idea was extended to find the coordinates of a point that divides a line segment in any given ratio. In this paper, we present a neat application of this extended idea, and further use it to find the co-ordinates of the centroid of a triangle with vertices *A*(*x*_{1}, y_{1}), *B*(*x*_{2}, y_{2}) and *C*(x_{3}, y_{3}).

Leslie Matrices And Women Population In The United States Of America, 2017 Valdosta State University

#### Leslie Matrices And Women Population In The United States Of America, Brittney Nelson, Denise T. Reid, Antonija Tangar, Jose A. Velez-Marulanda

*Georgia Journal of Science*

This research tests the accuracy of the Leslie matrix, which is a discrete age-structured method that uses fertility and survival rates, as a tool for predicting women population. Based on available data for the year 2000, we have constructed a Leslie matrix that predicts female population in the United States for every five years from the years 2000 to 2020. To test the accuracy of this method, we compare the aforementioned obtained projected data for the year 2010 with the actual data for women population in the United States obtained by the 2010 U.S. Census.

College Algebra - Large Section Versus Traditional Size, 2017 Georgia Academy of Science

#### College Algebra - Large Section Versus Traditional Size, Andreas Lazari, Denise Reid

*Georgia Journal of Science*

The economic crisis facing our nation forced many companies and universities to downsize and learn to operate with smaller budgets. Valdosta State University (VSU) was not immune to this economic crisis. To deal with this crisis VSU started offering large sections of core area courses, including College Algebra (MATH 1111). It is clear from a financial point of view that large sections will benefit the university during this financial crisis. What was not clear was the impact to student learning and success in College Algebra. In the fall 2010 and fall 2011 terms, VSU offered the first large sections of ...

Connecting Algebra And Geometry To Find Square And Higher Order Roots, 2017 Georgia Academy of Science

#### Connecting Algebra And Geometry To Find Square And Higher Order Roots, Iwan Elstak, Sudhir Goel

*Georgia Journal of Science*

Unfortunately, the repeat rate for all core curriculum courses including calculus-I and II, and also math education courses is too high. K-12 students are barely able to do estimation, an important part of Mathematics, whether it is adding fractions or finding roots, or for that matter, simple percent problems. In this paper we present geometric ways to reason about approximation of square roots and cube roots that are not accessible with simple routine techniques. We combine graphical methods, the use of a geometry software (sketch pad) and convergence of sequences to find higher order roots of positive real numbers and ...

The History Of Congressional Apportionment: 1790-2000, 2017 Humboldt State University

#### The History Of Congressional Apportionment: 1790-2000, Charles M. Biles

*Congressional Apportionment*

No abstract provided.

Shrinkage Function And Its Applications In Matrix Approximation, 2017 University of Florida, Gainesville

#### Shrinkage Function And Its Applications In Matrix Approximation, Toby Boas, Aritra Dutta, Xin Li, Katie Mercier, Eric Niderman

*Electronic Journal of Linear Algebra*

The shrinkage function is widely used in matrix low-rank approximation, compressive sensing, and statistical estimation. In this article, an elementary derivation of the shrinkage function is given. In addition, applications of the shrinkage function are demonstrated in solving several well-known problems, together with a new result in matrix approximation.