Mathematics Commons^™

Students' Perceptions Of Sense Of Community In Abstract Algebra: Contributing Factors And Benefits, Hortensia Soto-Johnson, Nissa Yestness, Casey Dalton

Mathematical Sciences Faculty Publications

In this phenomenological study, we explore how multiple assessments contribute to creating a sense of community (SOC) in an undergraduate abstract algebra course. Strike (2004) describes community as a process rather than a feeling and outlines four characteristics of community: coherence, cohesion, care, and contact. In this report, we describe contributing factors to and perceived benefits of SOC that students provided in an open-ended interview. Our findings indicate students viewed the teacher and the classroom environment as the primary sources of creating a SOC. Our findings also suggest students believed the SOC of the classroom increased classroom interaction and opened …

Pricing The Asian Call Option, Vinh Xuan Dang, Scott Glasgow, Harrison Potter, Stephen Taylor

Undergraduate Mathematics Day: Past Content

Background material on measure-theoretic probability theory and stochastic calculus is provided in order to clarify notation and inform the reader unfamiliar with these concepts. These fields are then employed in exploring two distinct but related approaches to fair option pricing: developing a partial differential equation whose solution, given specified boundary conditions, is the desired fair option price and evaluating a riskneutral conditional expectation whose value is the fair option price. Both approaches are illustrated by example before being applied to the Asian call option. .

Two results are obtained by applying the latter option pricing approach to the Asian call …

An Application Of Analytic Geometry To Designing Machine Parts--And Dresses, Karl Hess

Undergraduate Mathematics Day: Past Content

This paper presents the solution of an engineering problem that the author was asked to solve. The problem involves creating a flat pattern that could be cut from a piece of sheet metal and rolled to form a tube whose top edge would be contained in a plane that is not perpendicular to the central axis of the tube. A piece of this nature needs to be fabricated whenever two sheet metal tubes must be joined at any angle other than a straight angle.

Rearrangement On Conditionally Convergent Integrals In Analogy To Series, Edward J. Timko

Undergraduate Mathematics Day: Past Content

Rearrangements on conditionally convergent series suggests the existence of a similar process for integrals, here also referred to as rearrangement. In this document, a general theorem concerning rearrangement for conditionally convergent integrals is presented, as well as supporting theorems and a corollary to the general theorem.

Editorial: Expanding Spheres Of Influence- The Zenith, The Nadir And Everything In-Between, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Faculty Research and Creative Activity

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince

Faculty Publications & Research

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs K_s,t for a fixed s and large t. Myers proved upper bounds on D(K_s,t ) and made a conjecture on the order of magnitude of D(K_s,t ) for a fixed s and …

The Influence Of Dose Grid Resolution On Beam Selection Strategies In Radiotherapy Treatment Design, Ryan Acosta, Matthias Ehrgott, Allen G. Holder, Daniel Nevin, Josh Reese, Bill Salter

Mathematics Faculty Research

The design of a radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of a fluence pattern for each selected beam angle (intensity problem), and finding a sequence of configurations of a multilef collimator to deliver the treatment (realization problem). While many mathematical optimization models and algorithms have been proposed for the intensity problem and (to a lesser extent) the realization problem, this is not the case for the geometry problem. In clinical practice, beam directions are manually selected by a clinician and are typically based on the clinician’s experience. Solving the beam selection problem optimally is …

Foundations Of Geometry, Lawrence Michael Clarke

Theses Digitization Project

In this paper, a brief introduction to the history, and development of Euclidean geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.

Operations In Hilbert Space, Dennis Michael Gumaer

Theses Digitization Project

This thesis reviews some of the major topics in elementary Hilbert space theory. The theory of operators is developed by providing details regarding several types of operators, in particular compact operators. This study of compact operators is the start of the refinement of bounded linear operators to those which are also members of the Schatten p-class operators.

Studies In Free Module And It's Basis, Hsu-Chia Chen

Theses Digitization Project

The purpose of this project was to study some basic properties of free modules over a ring. A module with a basis is called a free module and a free module over a division ring (or field) is called a vector space. We show every vector has a basis and any two bases of a vector space have same cardinality. However, a free module over an arbitrary ring (with identity) does not have this property.

Tessellations Of The Hyperbolic Plane, Roberto Carlos Soto

Theses Digitization Project

In this thesis, the two models of hyperbolic geometry, properties of hyperbolic geometry, fundamental regions created by Fuchsian groups, and the tessellations that arise from such groups are discussed.

Local Geometry Of Zero Sets Of Holomorphic Functions Nears The Torus, Jim Agler, John E. Mccarthy, Mark Stankus

Mathematics

We call a holomorphic function f on a domain in Cⁿ locally toral at the point P in the n-torus if the intersection of the zero set of f with the n-torus has dimension n−1 at P. We study the interplay between the structure of locally toral functions and the geometry of their zero sets.

Sheaf Theoretic Formulation Of Entanglement, Goro C. Kato

Mathematics

A formulation in terms of sheaf theoretic (or categorical) notions for quantum entanglement is given with direct experimental consequences. The notions from sheaf theory and category theory give structural theory, i.e., qualitative theory, as a candidate for quantum gravity. Its advantage is the following: it provides not only space-time background independent, but also scale independent.This theory is called the theory of temporal topos (or simply t-topos theory).

A Positive Mass Theorem On Asymptotically Hyperbolic Manifolds With Corners Along A Hypersurface, Vincent Bonini, Jie Qing

Mathematics

In this paper we take an approach similar to that in [13] to establish a positive mass theorem for spin asymptotically hyperbolic manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics.

A Lost Theorem: Definite Integrals In An Asymptotic Setting, Ray Cavalcante, Todor D. Todorov

Mathematics

No abstract provided.

Microcosm To Macrocosm Via The Notion Of A Sheaf (Observers In Terms Of T-Topos), Goro Kato

Mathematics

The fundamental approach toward matter, space and time is that particles (either objects of macrocosm or microcosm), space and time are all presheafified. Namely, the concept of a presheaf is most fundamental for matter, space and time. An observation of a particle is represented by a morphism from the observed particle (its associated presheaf) to the observer (its associated presheaf) over a specified object (called a generalized time period) of a t-site (i.e. a category with a Grothendieck topology). This formulation provides a scale independent and background space-time free theory (since, for the t-topos theoretic formulation, space and time are …

Chinese Remainder Theorem And Its Applications, Jacquelyn Ha Lac

Theses Digitization Project

No abstract provided.

Poincaré Duality, Christopher Michael Duran

Theses Digitization Project

This project is an expository study of the Poincaré duality theorem. Homology, cohomology groups of manifolds and other aglebraic and topological preliminaires are discussed.

Simulating Spatial Partial Differential Equations With Cellular Automata, Brian Paul Strader

Theses Digitization Project

The purpose of this project was to define the relationship and show how an important subset of spatial differential equations can be transformed into cellular automata. Contains source code.

Factorization, Di Phan Reagan

Theses Digitization Project

The purpose of this thesis will focus on the two most efficient algorithms which are quadratic sieve and number field sieve. Background information such as definitions and theorems are given to help understand the concepts behind each method.

The Two Variable Substitution Problem For Free Products Of Groups, Leo P. Comerford, Charles C. Edmunds

Faculty Research and Creative Activity

We consider equations of the form W(x,y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x,y) = U has only finitely many solutions, up to the action of the stabilizer of W(x,y) in Aut().

C^1 Actions Of The Mapping Class Group On The Circle, Kamlesh Parwani

Faculty Research and Creative Activity

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

On The Asymptotic Integration Of Nonlinear Dynamic Equations, Elvan Akin, Smail Djebali, Toufik Moussaoui, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases

Transition To Turbulence, Small Disturbances, And Sensitivity Analysis Ii: The Navier-Stokes Equations, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

Recent research has shown that small disturbances in the linearized Navier-Stokes equations cause large energy growth in solutions. Although many researchers believe that this interaction triggers transition to turbulence in flow systems, the role of the nonlinearity in this process has not been thoroughly investigated. This paper is the second of a two part work in which sensitivity analysis is used to study the effects of small disturbances on the transition process. In the first part, sensitivity analysis was used to predict the effects of a small disturbance on solutions of a motivating problem, a highly sensitive one dimensional Burgers' …

Transition To Turbulence, Small Disturbances, And Sensitivity Analysis I: A Motivating Problem, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new "mostly linear" theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity …

Multiscale Deformable Registration Of Noisy Medical Images, Dana C. Paquin, Doron Levy, Lei Xing

Mathematics

Multiscale image registration techniques are presented for the registration of medical images using deformable registration models. The techniques are particularly effective for registration problems in which one or both of the images to be registered contains significant levels of noise. A brief overview of existing deformable registration techniques is presented, and experiments using B-spline free-form deformation registration models demonstrate that ordinary deformable registration techniques fail to produce accurate results in the presence of significant levels of noise. The hierarchical multiscale image decomposition described in E. Tadmor, S. Nezzar, and L. Vese’s, ”A multiscale image representation using hierarchical (BV, L^{2 …}

Solitary Wave Families In Two Non-Integrable Models Using Reversible Systems Theory, Jonathan Leto

Electronic Theses and Dissertations

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal …

Mathematics Of Voting, Darryl H. Yong

All HMC Faculty Publications and Research

Voting theory is a fascinating area of research involving mathematics, political scientists, and economists. The American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics chose mathematics and voting as the theme for Mathematics Awareness Month 2008. There is more information on mathematics and voting at www.mathaware.org/mam/08/. It is a mathematical topic that is rich yet accessible to students, pertinent to their lives, especially during this election year, and has the potential to draw students who may not have a strong affinity for mathematics to become interested in mathematics.

Mathematicians Playing A Role In Math Education: What We Learned At The Ime/Mime Workshop, Anna Bargagliotti, Rama Chidambaram, Gizem Karaali

Pomona Faculty Publications and Research

In Hollywood, some actors are regularly cast as mean, others as sweet and endearing, and some typically play innocent big-eyed youths who inevitably succeed after awakening to the particular facts of life that their producer wants them to awaken to. It is unusual and difficult for actors to cross the bridge between different types on a regular basis. However, there are always exceptions to the rule.

In the seemingly unrelated world of academics, mathematics faculty may find themselves playing different roles. People with different skills and interests strive to balance their careers in ways that will be uniquely fulfilling to …