# Mathematics Commons™

## All Articles in Mathematics

13045 full-text articles. Page 1 of 371.

Gaussian Guesswork: Infinite Sequences And The Arithmetic-Geometric Mean, 2017 Colorado State University-Pueblo

#### Gaussian Guesswork: Infinite Sequences And The Arithmetic-Geometric Mean, Janet Heine Barnett

##### Calculus

No abstract provided.

The Pell Equation In India, 2017 Ursinus College

#### The Pell Equation In India, Toke Knudson, Keith Jones

##### Number Theory

No abstract provided.

Generating Pythagorean Triples: A Gnomonic Exploration, 2017 Colorado State University-Pueblo

#### Generating Pythagorean Triples: A Gnomonic Exploration, Janet Heine Barnett

##### Number Theory

No abstract provided.

2017 The Graduate Center, City University of New York

#### Some Results In Combinatorial Number Theory, Karl Levy

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The first chapter establishes results concerning equidistributed sequences of numbers. For a given $d\in\mathbb{N}$, $s(d)$ is the largest $N\in\mathbb{N}$ for which there is an $N$-regular sequence with $d$ irregularities. We compute lower bounds for $s(d)$ for $d\leq 10000$ and then demonstrate lower and upper bounds $\left\lfloor\sqrt{4d+895}+1\right\rfloor\leq s(d)< 24801d^{3} + 942d^{2} + 3$ for all $d\geq 1$. In the second chapter we ask if $Q(x)\in\mathbb{R}[x]$ is a degree $d$ polynomial such that for $x\in[x_k]=\{x_1,\cdots,x_k\}$ we have $|Q(x)|\leq 1$, then how big can its lead coefficient be? We prove that there is a unique polynomial, which we call $L_{d,[x_k]}(x)$, with maximum lead coefficient under these constraints and construct an algorithm that generates $L_{d,[x_k]}(x)$.

2017 The Graduate Center, City University of New York

#### Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff's celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of ...

Interstructure Lattices And Types Of Peano Arithmetic, 2017 The Graduate Center, City University of New York

#### Interstructure Lattices And Types Of Peano Arithmetic, Athar Abdul-Quader

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA.

In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA that is pointwise definable. Enayat later showed that there are 20 models with the ...

The Common Invariant Subspace Problem And Tarski’S Theorem, 2017 Nicolaus Copernicus University of Toruń

#### The Common Invariant Subspace Problem And Tarski’S Theorem, Grzegorz Pastuszak

##### Electronic Journal of Linear Algebra

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is ...

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 Ath 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 ...

2017 The Graduate Center, City University of New York

#### Some Metric Properties Of The Teichmüller Space Of A Closed Set In The Riemann Sphere, Nishan Chatterjee

##### All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

Let E be an infinite closed set in the Riemann sphere, and let T(E) denote its Teichmüller space. In this dissertation we study some metric properties of T(E). We prove Earle's form of Teichmüller contraction for T(E), holomorphic isometries from the open unit disk into T(E), extend Earle's form of Schwarz's lemma for classical Teichmüller spaces to T(E), and finally study complex geodesics and unique extremality for T(E).

A Real Options Approach To Criminal Careers, 2017 FURG and PPGOM/UFPel

#### A Real Options Approach To Criminal Careers, Cristiano Aguiar De Oliveira, Giácomo Balbinotto Neto

##### The Latin American and Iberian Journal of Law and Economics

This paper proposes a dynamic model based on real options to evaluate the criminal career. In the model, individuals can choose the best moment to engage in crime (illegal activity). The model proposed allows the evaluation of the impact of different risk preferences, punishment probability, punishment severity and, mainly time discount in the individual’s decision. Through model calibration it is possible to observe that the option for a criminal career depends on a high return in the illegal activity even when individuals are risk neutral and when they have a low time discount. The paper also discusses youth participation ...

Health Literacy In The Mathematics Classroom: The Iowa Core Curriculum As An Opportunity To Deepen Students’ Understanding Of Mathematics, 2017 University of Northern Iowa

#### Health Literacy In The Mathematics Classroom: The Iowa Core Curriculum As An Opportunity To Deepen Students’ Understanding Of Mathematics, Elana Joram, Susan Dobie-Roberts, Nadene Davidson

By 2012, all high schools in Iowa will be required to incorporate the new Iowa Core Curriculum, followed by elementary and middle schools in 2014 (Iowa Department of Education, 2009). The Iowa Core Curriculum addresses the question: "How is Iowa's educational system preparing our youth for successful lives in the 21st-century global environment?" (Davidson, 2009). It consists of core content standards, and identifies essential concepts and skills for content areas. The Iowa Core Curriculum also includes the ―21st Century Skills‖ of ―health, financial, technology, and civic literacy, and employability skills. These skills are to be infused into existing subject ...

Refined Inertia Of Matrix Patterns, 2017 Redeemer University College

#### Refined Inertia Of Matrix Patterns, Kevin N. Vander Meulen, Jonathan Earl, Adam Van Tuyl

##### Electronic Journal of Linear Algebra

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. The paper includes a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.

The Feferman-Vaught Theorem, 2017 Wesleyan University

#### The Feferman-Vaught Theorem, Mostafa Mirabi

##### Mostafa Mirabi

This paper aims to provide an exposition of the Feferman-Vaught theorem, closely following the presentation in Hodges [1] and Chang-Keisler [2].

2017 University of Waterloo

#### Spectral Bound For Separations In Eulerian Digraphs, Krystal Guo

##### Electronic Journal of Linear Algebra

The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices D and Y such that there are no arcs from D and Y. For a subclass of Eulerian digraphs, a bound on the size of a separation is obtained in terms of the eigenvalues of the Laplacian matrix. An infinite family of tournaments, namely, the Paley digraphs, where the bound holds with equality, is also given.

Grnsight: A Web Application And Service For Visualizing Models Of Small- To Medium-Scale Gene Regulatory Networks, 2017 Loyola Marymount University

#### Grnsight: A Web Application And Service For Visualizing Models Of Small- To Medium-Scale Gene Regulatory Networks, Kam D. Dahlquist, John David N. Dionisio, Ben G. Fitzpatrick, Nicole A. Anguiano, Anindita Varshneya, Britain J. Southwick, Mihir Samdarshi

##### Ben G. Fitzpatrick

GRNsight is a web application and service for visualizing models of gene regulatory networks (GRNs). A gene regulatory network (GRN) consists of genes, transcription factors, and the regulatory connections between them which govern the level of expression of mRNA and protein from genes. The original motivation came from our efforts to perform parameter estimation and forward simulation of the dynamics of a differential equations model of a small GRN with 21 nodes and 31 edges. We wanted a quick and easy way to visualize the weight parameters from the model which represent the direction and magnitude of the influence of ...

Ecosystem Modeling Of College Drinking: Parameter Estimation And Comparing Models To Data, 2017 University of Louisiana at Lafayette

#### Ecosystem Modeling Of College Drinking: Parameter Estimation And Comparing Models To Data, Azmy S. Ackleh, Ben G. Fitzpatrick, Richard Scribner, Neal Simonsen, Jeremy J. Thibodeaux

##### Ben G. Fitzpatrick

Recently we developed a model composed of five impulsive differential equations that describes the changes in drinking patterns (that persist at epidemic level) amongst college students. Many of the model parameters cannot be measured directly from data; thus, an inverse problem approach, which chooses the set of parameters that results in the “best” model to data fit, is crucial for using this model as a predictive tool. The purpose of this paper is to present the procedure and results of an unconventional approach to parameter estimation that we developed after more common approaches were unsuccessful for our specific problem. The ...

Optimization And Control Of Agent-Based Models In Biology: A Perspective, 2017 University of Chicago

#### Optimization And Control Of Agent-Based Models In Biology: A Perspective, G. An, B. G. Fitzpatrick, S. Christley, P. Federico, A. Kanarek, R. Miller Neilan, M. Oremland, R. Salinas, R. Laubeanbacher, S. Lenhart

##### Ben G. Fitzpatrick

Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might ...

Robustness, Weak Stability, And Stability In Distribution Of Adaptive Filteringalgorithms Under Model Mismatch, 2017 Loyola Marymount University

#### Robustness, Weak Stability, And Stability In Distribution Of Adaptive Filteringalgorithms Under Model Mismatch, Ben G. Fitzpatrick, G. Yin, Le Yi Wang

##### Ben G. Fitzpatrick

This work is concerned with robustness, convergence, and stability of adaptive filtering (AF) type algorithms in the presence of model mismatch. The algorithms under consideration are recursive and have inherent multiscale structure. They can be considered as dynamic systems, in which the `state' changes much more slowly than the perturbing noise. Beyond the existing results on adaptive algorithms, model mismatch significantly affects convergence properties of AF algorithms, raising issues of algorithm robustness. Weak convergence and weak stability (i.e., recurrence) under model mismatch are derived. Based on the limiting stochastic differential equations of suitably scaled iterates, stability in distribution is ...

Decreasing Math Anxiety Through Teaching Quadratic Equations, 2017 The College at Brockport: State University of New York

#### Decreasing Math Anxiety Through Teaching Quadratic Equations, Kaitlyn Kaufman

##### Education and Human Development Master's Theses

Math anxiety is known as having a feeling of fear that interferes with math performance. Many students today suffer from math anxiety as they push through each developmental stage in their schooling. A majority of students develop math anxiety through traditional classroom methods, such as drill and practice, assessments, memorizing, and textbooks. According to research, teachers can help decrease math anxiety in students by incorporating specific teaching styles, methods, and strategies, related to decrease math anxiety, into lessons. These teaching styles, methods, and strategies include, but not limited to, constructivist teaching, concrete-to-representation-to-abstract model, student-centered learning, and interactive lessons. Based on ...

The Enhanced Principal Rank Characteristic Sequence Over A Field Of Characteristic 2, 2017 Iowa State University

#### The Enhanced Principal Rank Characteristic Sequence Over A Field Of Characteristic 2, Xavier Martínez-Rivera

##### Electronic Journal of Linear Algebra

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over ...