# Mathematics Commons™

## All Articles in Mathematics

14,428 full-text articles. Page 1 of 420.

Brun's 1920 Theorem On Goldbach's Conjecture, 2018 Utah State University

#### Brun's 1920 Theorem On Goldbach's Conjecture, James A. Farrugia

##### All Graduate Theses and Dissertations

One form of Goldbach’s Conjecture asserts that every even integer greater than 4is the sum of two odd primes. In 1920 Viggo Brun proved that every sufficiently large even number can be written as the sum of two numbers, each having at most nine prime factors. This thesis explains the overarching principles governing the intricate arguments Brun used to prove his result.

Though there do exist accounts of Brun’s methods, those accounts seem to miss the forest for the trees. In contrast, this thesis explains the relatively simple structure underlying Brun’s arguments, deliberately avoiding most of his ...

2018 Georgia Southern University

#### An Efficient Algorithm To Test Forcibly-Connectedness Of Graphical Degree Sequences, Kai Wang

##### Theory and Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al's classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage's classic algorithm to enumerate graphical partitions of ...

On The Maximal Numerical Range Of Some Matrices, 2018 New York University Abu Dhabi

#### On The Maximal Numerical Range Of Some Matrices, Ali N. Hamed, Ilya M. Spitkovsky

##### Electronic Journal of Linear Algebra

The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in particular, when $W_0(A)=W(A)$. The set $W_0(A)$ is also described explicitly for matrices unitarily similar to direct sums of $2$-by-$2$ blocks, and some insight into the behavior of $W_0(A)$ is provided when $\mathcal L$ has ...

Finite Asymptotic Clusters Of Metric Spaces, 2018 Institute of Applied Mathematics and Mechanics of NAS of Ukraine

#### Finite Asymptotic Clusters Of Metric Spaces, Viktoriia Bilet, Oleksiy Dovgoshey

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $\Omega_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, \rho_{X})$ whose maximal cliques coincide with $\Omega_{\infty, \tilde r}^{X ... Second-Order Know-How Strategies, 2018 Lafayette College #### Second-Order Know-How Strategies, Pavel Naumov, Jia Tao ##### Faculty Research and Reports The fact that a coalition has a strategy does not mean that the coalition knows what the strategy is. If the coalition knows the strategy, then such a strategy is called a know-how strategy of the coalition. The paper proposes the notion of a second-order know-how strategy for the case when one coalition knows what the strategy of another coalition is. The main technical result is a sound and complete logical system describing the interplay between the distributed knowledge modality and the second-order coalition know-how modality. Quantitative Literacy And Civic Virtue, 2018 University of Colorado, Denver #### Quantitative Literacy And Civic Virtue, William Briggs ##### Numeracy Mathematics educators are occasionally called upon to justify the existence or the offering of quantitative literacy courses. This paper argues that effective quantitative literacy courses have different goals than algebra courses and are legitimate alternatives to algebra courses for non-STEM students. Furthermore, quantitative literacy courses affirm the historic relationship between citizenship and education. In today’s world of proliferating news sources, social media, and fake news, quantitative literacy has become an essential component of the long-held ideal of civic virtue. 2018 University of Central Florida #### A Note On The Matrix Arithmetic-Geometric Mean Inequality, Teng Zhang ##### Electronic Journal of Linear Algebra This note proves the following inequality: If$n=3k$for some positive integer$k$, then for any$n$positive definite matrices$\bA_1,\bA_2,\dots,\bA_n$, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$,$j_2$,$j_3$all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where$\|\cdot\|\$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R ...

From Sets To Metric Spaces To Topological Spaces, 2018 Ursinus College

#### From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville

##### Topology

No abstract provided.

Nearness Without Distance, 2018 Ursinus College

#### Nearness Without Distance, Nicholas A. Scoville

##### Topology

No abstract provided.

Determining The Determinant, 2018 Xavier University

#### Determining The Determinant, Danny Otero

##### Linear Algebra

No abstract provided.

Mathematical Models, 2018 University of North Georgia

#### Mathematical Models, Patty Wagner, Marnie Phipps

##### Mathematics Grants Collections

This Grants Collection for Mathematical Models was created under a Round Nine ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Initial Proposal
• Final Report

2018 Illinois Mathematics and Science Academy

#### A Computer-Aided Decomposition Of The Complete Digraph Into Orientations Of K4 − E With A Double Edge, Hanson Hao

##### The International Student Science Fair 2018

The abstract is available as an Additional File.

Teaching Apportionment, 2018 Humboldt State University

#### Teaching Apportionment, Charles M. Biles

##### IdeaFest: Interdisciplinary Journal of Creative Works and Research from Humboldt State University

No abstract provided.

Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, 2018 Louisiana State University and Agricultural and Mechanical College

#### Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni

##### LSU Doctoral Dissertations

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.

Rank Function And Outer Inverses, 2018 Manipal University, Manipal

#### Rank Function And Outer Inverses, Manjunatha Prasad Karantha, K. Nayan Bhat, Nupur Nandini Mishra

##### Electronic Journal of Linear Algebra

For the class of matrices over a field, the notion of rank of a matrix' as defined by the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the rank' in the class of matrices over a commutative ring. Even this determinantal rank and ...

Facial Unique-Maximum Colorings Of Plane Graphs With Restriction On Big Vertices, 2018 Iowa State University

#### Facial Unique-Maximum Colorings Of Plane Graphs With Restriction On Big Vertices, Bernard Lidicky, Kacy Messerschmidt, Riste Škrekovski

##### Mathematics Publications

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This ...

Templates For Representable Matroids, 2018 Louisiana State University and Agricultural and Mechanical College

#### Templates For Representable Matroids, Kevin Manuel Grace

##### LSU Doctoral Dissertations

The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors.

Chapter 1 gives a brief introduction to ...

2018 Golden Gate University School of Law

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2018 Golden Gate University School of Law

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Average Mixing Matrix Of Trees, 2018 University of Waterloo

#### Average Mixing Matrix Of Trees, Chris Godsil, Krystal Guo, John Sinkovic

##### Electronic Journal of Linear Algebra

The rank of the average mixing matrix of trees with all eigenvalues distinct, is investigated. The rank of the average mixing matrix of a tree on n vertices with n distinct eigenvalues is bounded above by ⌈n/2⌉. Computations on trees up to 20 vertices suggest that the rank attains this upper bound most of the time. An infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound, is given. A lower bound on the rank of the average mixing matrix of a tree, is also given.