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2016

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Full-Text Articles in Mathematics

Topological Speedups, Drew Daehnhardt Ash Jan 2016

Topological Speedups, Drew Daehnhardt Ash

Electronic Theses and Dissertations

Given a dynamical system T:X rightarrow X one can define a speedup of (X,T) as another dynamical system conjugate to S:X rightarrow X where S(x)=T^{p(x)}(x) for some function p:X rightarrowZ^{+}. In 1985 Arnoux, Ornstein, and Weiss showed that any aperiodic measure preserving system is isomorphic to a speedup of any ergodic measure preserving system. In this thesis we study speedups in the topological category. Specifically, we consider minimal homeomorphisms on Cantor spaces. Our main theorem gives conditions on when one such system is a speedup of another. Moreover, the main theorem serves as a topological analogue of the Arnoux, Ornstein, …

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Improving Middle Grades Stem Teacher Content Knowledge And Pedagogical Practices Through A School-University Partnership, Cherie Mccollough, Tonya Jeffery, Kim Moore, Joe Champion Jan 2016

Improving Middle Grades Stem Teacher Content Knowledge And Pedagogical Practices Through A School-University Partnership, Cherie Mccollough, Tonya Jeffery, Kim Moore, Joe Champion

Mathematics Faculty Publications and Presentations

This paper outlines a University-School District partnership with the intent to increase the number of middle grades mathematics and science teachers. This externally funded initiative includes onsite, authentically situated professional development for pre- and in-service teachers at three different urban, low-socioeconomic schools with a majority Hispanic population of students. Program objectives include increasing mathematics and science content knowledge, increasing self-efficacy in teaching math and science, building and incorporating a success-driven school culture and infrastructure to increase student performance in a well-articulated, scalable and transformable model. Program components include site based common planning times, STEM Thursdays where science and mathematics lessons …

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Arithmagons And Geometrically Invariant Multiplicative Integer Partitions, J. A. Franco, J. Champion, J. W. Lyons Jan 2016

Arithmagons And Geometrically Invariant Multiplicative Integer Partitions, J. A. Franco, J. Champion, J. W. Lyons

Mathematics Faculty Publications and Presentations

In this article, we introduce a formal definition for integral arithmagons. Informally, an arithmagon is a polygonal figure with integer labeled vertices and edges in which, under a binary operation, adjacent vertices equal the included edge. By considering the group of automorphisms for the associated graph, we count the number of integral arithmagons whose exterior sum or product equals a fixed number.

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Computing With Functions In Spherical And Polar Geometries I. The Sphere, Alex Townsend, Heather Wilber, Grady B. Wright Jan 2016

Computing With Functions In Spherical And Polar Geometries I. The Sphere, Alex Townsend, Heather Wilber, Grady B. Wright

Mathematics Faculty Publications and Presentations

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. …

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The Minimal Length Of The Lagrangian Cobordism Between Legendrians, Joshua M. Sabloff, Lisa Traynor Jan 2016

The Minimal Length Of The Lagrangian Cobordism Between Legendrians, Joshua M. Sabloff, Lisa Traynor

Mathematics Faculty Research and Scholarship

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we study the minimal length of such a cobordism, which is a 1-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian contact homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). …

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Arithmetic Properties Of Fredholm Series For P-Adic Modular Forms, John Bergdall, Robert Pollack Jan 2016

Arithmetic Properties Of Fredholm Series For P-Adic Modular Forms, John Bergdall, Robert Pollack

Mathematics Faculty Research and Scholarship

We study the relationship between recent conjectures on slopes of overconvergent p ‐adic modular forms ‘near the boundary’ of p ‐adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the Up operator never vanish modulo p , a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non‐zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

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Population Density, Not Host Competence, Drives Patterns Of Disease In An Invaded Community, C. L. Searle Jan 2016

Population Density, Not Host Competence, Drives Patterns Of Disease In An Invaded Community, C. L. Searle

Mathematics and Statistics Faculty Presentations

Generalist parasites can strongly influence interactions between native and invasive species. Host competence can be used to predict how an invasive species will affect community disease dynamics; the addition of a highly competent, invasive host is predicted to increase disease. However, densities of invasive and native species can also influence the impacts of invasive species on community disease dynamics. We examined whether information on host competence alone could be used to accurately predict the effects of an invasive host on disease in native hosts. We first characterized the relative competence of an invasive species and a native host species to …

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Calculus Students’ Ideas About Functions: Identifying Opportunities To Support Teacher Learning, Laurie Overman Cavey, Patrick R. Lowenthal Jan 2016

Calculus Students’ Ideas About Functions: Identifying Opportunities To Support Teacher Learning, Laurie Overman Cavey, Patrick R. Lowenthal

Mathematics Faculty Publications and Presentations

We describe the first phase of a study aimed at developing video-based instructional modules for secondary mathematics teachers. We began by consulting the literature on figural pattern tasks (c.f. Rivera, 2010) and teachers’ ability to interpret student work (c.f. El Mouhayar & Jurdak, 2012). Interpreting student work on figural pattern tasks requires awareness of different problem solving strategies, such as recursive and constructive, and how students might use them with tasks that require different levels of generalization (El Mouhayar & Jurdak, 2012).

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A Lower Bound For Stanley Depth Of Squarefree Monomial Ideals, Guangjun Zhu Jan 2016

A Lower Bound For Stanley Depth Of Squarefree Monomial Ideals, Guangjun Zhu

Turkish Journal of Mathematics

Let $S=K[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $K$ in $n$ variables and $I$ a squarefree monomial ideal of $S$ with Schmitt--Vogel number $sv(I)$. In this paper, we show that $\mbox{sdepth}\,(I)\geq \mbox{max}\,\{1, n-1-\lfloor \frac{sv(I)}{2}\rfloor\},$ which improves the lower bound obtained by Herzog, Vladoiu, and Zheng. As some applications, we show that Stanley's conjecture holds for the edge ideals of some special $n$-cyclic graphs with a common edge.

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Fourth-Order Birkhoff Regular Problems With Eigenvalue Parameter Dependent Boundary Conditions, Bertin Zinsou Jan 2016

Fourth-Order Birkhoff Regular Problems With Eigenvalue Parameter Dependent Boundary Conditions, Bertin Zinsou

Turkish Journal of Mathematics

A regular fourth-order differential equation that depends quadratically on the eigenvalue parameter $\lambda$ is considered with classes of separable boundary conditions independent of $\lambda$ or depending on $\lambda$ linearly. Conditions are given for the problems to be Birkhoff regular.

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The Effect Of Regional Analgesia On Vascular Tone In Hip Arthroplasty Patients, Enrique A. Goytizolo, Ottokar Stundner, Sandra M. Hurtado Rua, Dorothy Marcello, Valeria Buschiazzo, Ansara M. Vaz, Stavros G. Memtsoudis Jan 2016

The Effect Of Regional Analgesia On Vascular Tone In Hip Arthroplasty Patients, Enrique A. Goytizolo, Ottokar Stundner, Sandra M. Hurtado Rua, Dorothy Marcello, Valeria Buschiazzo, Ansara M. Vaz, Stavros G. Memtsoudis

Mathematics and Statistics Faculty Publications

Background: While it is assumed that neuraxial analgesia and pain management may beneficially influence perioperative hemodynamics, few studies provided data quantifying such effects and none have assessed the potential contribution of the addition of a nerve block. Questions/purposes: This clinical trial compared the visual analog scale (VAS) scores and measurement of arterial tone using augmentation index of patients who received combined spinal-epidural (CSE) only to patients who received both CSE and lumbar plexus block. Methods: After obtaining written consent, 92 patients undergoing total hip arthroplasty were randomized to receive either CSE or CSE with lumbar plexus block (LPB). Perioperative pain …

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Characterization Of Completely K-Magic Regular Graphs, Arnold A. Eniego, Ian June L. Garces Jan 2016

Characterization Of Completely K-Magic Regular Graphs, Arnold A. Eniego, Ian June L. Garces

Mathematics Faculty Publications

Let k ∈ N and c ∈ Zk. A graph G is said to be c-sum k-magic if there is a labeling ` : E(G) → Zk \ {0} such that P u∈N(v) `(uv) ≡ c (mod k) for every vertex v of G, where N(v) is the neighborhood of v in G. We say that G is completely k-magic whenever it is c-sum k-magic for every c ∈ Zk. In this paper, we characterize all completely k-magic regular graphs.

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The Characterization Of Vectors In RN With The Haar Property, Terrence Kelleher Jan 2016

The Characterization Of Vectors In RN With The Haar Property, Terrence Kelleher

Undergraduate Review

Introduction:
As modern communication becomes more digital than ever, infrastructure tries to maintain an equal pace. Unfortunately, this is not always possible. Therefore, computer scientists and mathematicians alike have endeavored to invent ways to store, send and retrieve data, even if transmitted signals are severely damaged. One such way is by using what is called an error-correcting code, or an ECC. An ECC is a method of encoding information such that a signal that possesses a message can be significantly altered and still be decoded. There are many different types of ECCs. The type that is of interest in this …

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Graph Cohomology, Matthew Lin Jan 2016

Graph Cohomology, Matthew Lin

HMC Senior Theses

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the …

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Convexity Of Neural Codes, Robert Amzi Jeffs Jan 2016

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

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On The Similarity Of Ab And Ba For Normal And Other Matrices, Stephan Ramon Garcia, David Sherman, Gary Weiss Jan 2016

On The Similarity Of Ab And Ba For Normal And Other Matrices, Stephan Ramon Garcia, David Sherman, Gary Weiss

Pomona Faculty Publications and Research

It is known that AB and BA are similar when A and B are Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if A is Hermitian and B is normal. Perhaps surprisingly, similarity does hold when A is positive semidefinite and B is normal.

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Methods For Quantized Compressed Sensing, Hao-Jun Michael Shi, Mindy Case, Xiaoyi Gu, Shenyinying Tu, Deanna Needell Jan 2016

Methods For Quantized Compressed Sensing, Hao-Jun Michael Shi, Mindy Case, Xiaoyi Gu, Shenyinying Tu, Deanna Needell

CMC Faculty Publications and Research

In this paper, we compare and catalog the performance of various greedy quantized compressed sensing algorithms that reconstruct sparse signals from quantized compressed measurements. We also introduce two new greedy approaches for reconstruction: Quantized Compressed Sampling Matching Pursuit (QCoSaMP) and Adaptive Outlier Pursuit for Quantized Iterative Hard Thresholding (AOP-QIHT). We compare the performance of greedy quantized compressed sensing algorithms for a given bit-depth, sparsity, and noise level.

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Lattices From Tight Equiangular Frames, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell Jan 2016

Lattices From Tight Equiangular Frames, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell

CMC Faculty Publications and Research

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n=k+1 and that there are infinitely many k such that a lattice emerges for n=2k. We dispose of all cases in dimensions k at most 9. In …

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Lattice Theory And Toeplitz Determinants, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj Jan 2016

Lattice Theory And Toeplitz Determinants, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

CMC Faculty Publications and Research

This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher-Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lattices under consideration states that they always have a basis of minimal vectors, while our results in the other direction concern exact and asymptotic formulas for perturbed Toeplitz determinants. The survey is a …

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On Arithmetic Lattices In The Plane, Lenny Fukshansky, Pavel Guerzhoy, Florian Luca Jan 2016

On Arithmetic Lattices In The Plane, Lenny Fukshansky, Pavel Guerzhoy, Florian Luca

CMC Faculty Publications and Research

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices.

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Decomposition Of Finite Schmidt Rank Bounded Operators On The Tensor Product Of Separable Hilbert Spaces, Abdelkrim Bourouihiya Jan 2016

Decomposition Of Finite Schmidt Rank Bounded Operators On The Tensor Product Of Separable Hilbert Spaces, Abdelkrim Bourouihiya

Mathematics Faculty Articles

Inverse formulas for the tensor product are used to develop an algorithm to compute Schmidt decompositions of Finite Schmidt Rank (FSR) bounded operators on the tensor product of separable Hilbert spaces. The algorithm is then applied to solve inverse problems related to the tensor product of bounded operators. In particular, we show how properties of a FSR bounded operator are reflected by the operators involved in its Schmidt decomposition. These properties include compactness of FSR bounded operators and convergence of sequences whose terms are FSR bounded operators.

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Energy-Conserving Numerical Scheme For The Poisson-Nerst-Plank Equations, Julienne Kabre Jan 2016

Energy-Conserving Numerical Scheme For The Poisson-Nerst-Plank Equations, Julienne Kabre

Mathematics Faculty Proceedings, Presentations, Speeches, Lectures

Preliminary report.
The Poisson-Nernst-Planck equations are a system of nonlinear partial differential equations that describe flow of charged particles in solution. In particular, we are interested in the transport of ions in the biological membrane proteins (ion channels). This work is about the design of numerical schemes that preserve exactly (up to roundoff error) a discretized form of the energy dynamics of the system. We will present a scheme that achieves the goal of preserving the energy dissipation law and some preliminary numerical results.

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Global Regularity For Unsteady Flow Of Third Grade Fluid In An Annular Region, Saeed Ur Rahman, Tasawar Hayat, Hamed H. Alsulami Jan 2016

Global Regularity For Unsteady Flow Of Third Grade Fluid In An Annular Region, Saeed Ur Rahman, Tasawar Hayat, Hamed H. Alsulami

Turkish Journal of Mathematics

This article develops global regularity criteria for unsteady and magnetohydrodynamic flow of third grade fluid in terms of bounded mean oscillations. Uniqueness of the solution is also verified.

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The Mystery Of The Non-Transitive Grime Dice, Nicholas Pasciuto Jan 2016

The Mystery Of The Non-Transitive Grime Dice, Nicholas Pasciuto

Undergraduate Review

No abstract provided.

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4. Dragging Along, Ruth Dover Jan 2016

4. Dragging Along, Ruth Dover

Differential Equations

More information on air drag.

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Bc 1 Rate Of Change Activity Sheet Teacher Notes, Ruth Dover Jan 2016

Bc 1 Rate Of Change Activity Sheet Teacher Notes, Ruth Dover

A Simple Introduction to Rates

Before beginning this section of handouts, students will be introduced to a variety of vocabulary words often associated with calculus. These words will be used in an intuitive sense only and will not have been formally defined. Vocabulary should include graphical terms such as continuous, increasing, decreasing, maximum and minimum points, concave up, concave down, and point of inflection. In addition, discussion of the concept of "rate of change" should begin. It should be mentioned that many quantities change – population, cost, and temperature, to name just a few. All that is specifically required at this point can be related …

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The Kretschmann Scalar, Charles G. Torre Jan 2016

The Kretschmann Scalar, Charles G. Torre

How to... in 10 minutes or less

On a pseudo-Riemannian manifold with metric g, the "Kretschmann scalar" is a quadratic scalar invariant of the Riemann R tensor of g, defined by contracting all indices with g. In this worksheet we show how to calculate the Kretschmann scalar from a metric.

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What's In A Name? A Critical Review Of Definitions Of Quantitative Literacy, Numeracy, And Quantitative Reasoning, Gizem Karaali, Edwin H Villafane Hernandez '18, Jeremy Alexander Taylor '18 Jan 2016

What's In A Name? A Critical Review Of Definitions Of Quantitative Literacy, Numeracy, And Quantitative Reasoning, Gizem Karaali, Edwin H Villafane Hernandez '18, Jeremy Alexander Taylor '18

Pomona Faculty Publications and Research

This article aims to bring together various threads in the eclectic literature that make up the scholarship around the theme of Quantitative Literacy. In investigating the meanings of terms like "quantitative literacy," "quantitative reasoning," and "numeracy," we seek common ground, common themes, common goals and aspirations of a community of practitioners. A decade ago, these terms were relatively new in the public sphere; today policy makers and accrediting agencies are routinely inserting them into general education conversations. Having good, representative, and perhaps even compact and easily digestible definitions of these terms might come in handy in public relations contexts as …

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A Mathematical Model For Feral Cat Ecology With Application To Disease., Jeff Sharpe Jan 2016

A Mathematical Model For Feral Cat Ecology With Application To Disease., Jeff Sharpe

Electronic Theses and Dissertations

We formulate and analyze a mathematical model for feral cats living in an isolated colony. The model contains compartments for kittens, adult females and adult males. Kittens are born at a rate proportional to the population of adult females and mature at equal rates into adult females and adult males. Adults compete with each other in a manner analogous to Lotka-Volterra competition. This competition comes in four forms, classified by gender. Native house cats, and their effects are also considered, including additional competition and abandonment into the feral population. Control measures are also modeled in the form of per-capita removal …

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Structure-Preserving Finite Difference Methods For Linearly Damped Differential Equations, Ashish Bhatt Jan 2016

Structure-Preserving Finite Difference Methods For Linearly Damped Differential Equations, Ashish Bhatt

Electronic Theses and Dissertations

Differential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical integrators generally result in undesirable quantitative and qualitative errors . Standard numerical integrators aim to reduce quantitative errors, whereas geometric (numerical) integrators aim to reduce or eliminate qualitative errors, as well, in …

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