Mathematics Commons^™

The Miracle Of Applied Mathematics, Frank Blume

The Mathematics Enthusiast

We provide an outline of a college-level lecture on systems of differential equations that is intended to offer an alternative, integrative view of the subject. Particular emphasis is placed on the fact that very elementary concepts and insights can be employed to gain access to a very wide range of important applications, including the Schr¨odinger equation, the Klein-Gordon equation, the Laplace equation, and the Poisson equation.

Generalizing Cantor-Schroeder-Bernstein: Counterexamples In Standard Settings, Tien Chih

The Mathematics Enthusiast

The Cantor-Schroeder-Bernstein theorem states that any two sets that have injections into each other have the same cardinality, i.e. there is a bijection between them. Another way to phrase this is if two sets A;B have monomorphisms from A to B and B to A, then they are isomorphic in the setting of sets. One naturally wonders if this may be extended to other commonly studied systems of sets with structure and functions which preserve that structure. Given two objects with injective structure preserving maps between them are the structures of these objects the same? In other words, would these …

A Study On Malaysian Mathematicians’ Way Of Knowing, Lim Chap Sam

The Mathematics Enthusiast

“Can you name me a mathematician?” “Einstein??”

“Do you wish to become a mathematician one day?”

If these questions were asked to the public or any school students, the most likely answer to both questions might be a big ‘No!’. Why is this so?

The director of the Public Understanding of Mathematics Forum, Gene Kloz (1996) claims that the mathematics profession is the most misunderstood in all of academia. According to him, the public thinks that mathematicians contemplate ancient proofs and work as lonely recluses. Moreover, the most common public image of a mathematician has been furnished by a physicist …

Reasoning-And-Proving Within Ireland’S Reform-Oriented National Syllabi, Jon Davis

The Mathematics Enthusiast

As educational systems around the world attempt to reform their mathematics programs to increase students’ opportunities to engage in processes central to the practice of mathematics such as proof, it is important to understand how this mathematical act is portrayed in national curriculum documents that drive that change. This study examined the presence of reasoning-and-proving (RP) in Ireland’s national reform-oriented secondary syllabi for junior cycle (ages 12-15) and senior cycle (ages 15-18) students. The analyses reveal that there were no differences among direct and indirect RP learning outcomes within each syllabus, but statistically significant differences did exist across syllabi in …

Difficulties In Solving Context-Based Pisa Mathematics Tasks: An Analysis Of Students’ Errors, Ariyadi Wijaya, Marja Van Den Heuvel-Panhuizen, Michiel Doorman, Alexander Robitzsch

The Mathematics Enthusiast

The intention of this study was to clarify students’ difficulties in solving context-based mathematics tasks as used in the Programme for International Student Assessment (PISA). The study was carried out with 362 Indonesian ninth- and tenth-grade students. In the study we used 34 released PISA mathematics tasks including three task types: reproduction, connection, and reflection. Students’ difficulties were identified by using Newman’s error categories, which were connected to the modeling process described by Blum and Leiss and to the PISA stages of mathematization, including (1) comprehending a task, (2) transforming the task into a mathematical problem, (3) processing mathematical procedures, …

Scholastic Standards In The United States – The Discussion Concerning The ‘Common Core’, Alan H. Schoenfeld, Günter Törner

The Mathematics Enthusiast

Preface: This article has been developed based on a personal discussion between the German author Günter Törner and Alan Schoenfeld, who is an expert in the field of mathematical didactics. Basically there are three reasons for us to share our insights with the public:

(1) Readers, having subscribed to Jerry Becker’s e-mail information network, have received numerous messages over the past few months; what do we need to know about this fact in Germany?

(2) Scholastic standards – a keyword that sounds very familiar to us in terms of educational policy… But it is also a hot topic in other …

Development Of The Binary Number System And The Foundations Of Computer Science, Daniel R. Lande

The Mathematics Enthusiast

This paper discusses the formalization of the binary number system and the groundwork that was laid for the future of digital circuitry, computers, and the field of computer science. The goal of this paper is to show how Gottfried Leibniz formalized the binary number system and solidified his thoughts through an analysis of the Chinese I Ching. In addition, Leib-niz’s work in logic and with computing machines is presented. This work laid the foundation for Boolean algebra and digital circuitry which was continued by George Boole, Augustus De Mor-gan, and Claude Shannon in the centuries following. Some have coined Leibniz …

The Use Of Variable-Bagging And The Cross-Validation Selector In The Prediction Of Alzheimer’S Using The Adni Database., Michael Wayne Godbey

Electronic Theses and Dissertations

Dimensionality plays a huge part in the modeling process. If there are more elements in a data set than variables in each element then there are very few restrictions in selection of an algorithm. Bagging, bootstrap aggregating (Breiman, 1994), may also be used to improve a model’s prediction capability. On the other hand, if there more variables in each observation than the number of observations in the dataset, the number of usable algorithms is greatly reduced. The recently developed algorithm, support vector machines, was designed for such situations, in comparison to algorithms such as logistic regression which have instability issues …

Mixed Variational Formulation For The Wellposedness And Numerical Approximation Of A Pde Model Arising In A 3-D Fluid-Structure Interaction, George Avalos, Tom Clark

Faculty Work Comprehensive List

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O coupled to a fourth order plate equation, possibly with rotational inertia parameter ρ>0. This plate PDE evolves on a flat portion Ω of the boundary of O. The coupling on Ω is implemented via the Dirichlet trace of the Stokes system …

Hybrid Meshless Method For Numerical Solution Of Partial Differential Equations, Jeanette Marie Monroe

Dissertations

A meshless method for solving partial differential equations (PDEs) which combines the method of fundamental solutions (MFS) and the method of particular solutions (MPS) is formulated and tested. The hybrid method finds a numerical approximation by solving only one system of equations as opposed to the two-stage method of fundamental solutions and method of particular solutions. This new approach, denoted MFS-MPS, one-stage MFS-MPS, or hybrid method, can be applied to a wide variety of PDEs including PDEs with variable coefficients. The MFS-MPS can simplify Helmholtz-type differential operators to Laplacian-type differential operators providing flexibility and simplification to calculating particular solutions and …

Gradient-Based Compressive Sensing For Noise Image And Video Reconstruction, Huihuang Zhao, Yaonan Wang, Xiaojiang Peng, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this study, a fast gradient-based compressive sensing (FGB-CS) for noise image and video is proposed. Given a noise image or video, the authors first make it sparse by orthogonal transformation, and then reconstruct it by solving a convex optimisation problem with a novel gradient-based method. The main contribution is twofold. Firstly, they deal with the noise signal reconstruction as a convex minimisation problem, and propose a new compressive sensing based on gradient-based method for noise image and video. Secondly, to improve the computational efficiency of gradient-based compressive sensing, they formulate the convex optimisation of noise signal reconstruction under Lipschitz …

Pro-Covering Fibrations Of The Hawaiian Earring, Nickolas Brenten Callor

Theses and Dissertations

Let H be the Hawaiian Earring, and let H denote its fundamental group. Assume (Bi) is an inverse system of bouquets of circles whose inverse limit is H. We give an explicit bijection between finite normal covering spaces of H and finite normal covering spaces of Bi. This bijection induces a correspondence between a certain family of inverse sequences of these covering spaces. The correspondence preserves the inverse limit of these sequences, thus offering two methods of constructing the same limit. Finally, we characterize all spaces that can be obtained in this fashion as a particular type of fibrations of …

Two Dimensional Mathematical Model Of Fluid Flow In A Growing Solid Tumor, Adriana Gracia

Theses and Dissertations - UTB/UTPA

We investigate the problem of steady and unsteady fluid flow in a growing solid tumor. We develop a mathematical model for the two dimensional fluid flow in a spherical tumor where the spatial variations of the interstitial velocity, interstitial pressure and the drug concentration within the tumor are, in general, with respect to the radial distance and the latitudinal angle in the spherical coordinates. The expressions for radial and latitudinal variations of the interstitial velocity, interstitial pressure, and the two investigated drug concentrations were determined analytically. We calculated these quantities in the tumor as well as in a corresponding normal …

Mathematical Modeling Of Immune Responses To Hepatitis C Virus Infection, Ivan Ramirez

Electronic Theses and Dissertations

An existing mathematical model of ordinary differential equations was studied to better understand the interactions between hepatitis C virus (HCV) and the immune system cells in the human body. Three possible qualitative scenarios were explored: dominant CTL response, dominant antibody response, and coexistence. Additionally, a sensitivity analysis was carried out to rank model parameters for each of these scenarios. Therapy was addressed as an optimal control problem. Numerical solutions of optimal controls were computed using a forward-backward sweep scheme for each scenario. Model parameters were estimated using ordinary least squares fitting from longitudinal data (serum HCV RNA measurements) given in …

Truckload Shipment Planning And Procurement, Neo Nguyen

Graduate Theses and Dissertations

This dissertation presents three issues encountered by a shipper in the context of truckload transportation. In all of the studies, we utilize optimization techniques to model and solve the problems. Each study is inspired from the real world and much of the data used in the experiments is real data or representative of real data.

The first topic is about the freight consolidation in truckload transportation. We integrate it with a purchase incentive program to increase truckload utilization and maximize profit. The second topic is about supporting decision making collaboration among departments of a manufacturer. It is a bi-objective optimization …

Connections Of Zero Curvature And Applications To Nonlinear Partial Differential Equations, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Bäcklund tranformations can be formulated for such equations that occur as zero curvature terms.

Radio Number For Fourth Power Paths, Linda V. Alegria

Electronic Theses, Projects, and Dissertations

A path on n vertices, denoted by Pn, is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the order. A fourth power path, Pn4, is obtained from Pn by adding edges between any two vertices, u and v, whose distance in Pn, denoted by dPn(u,v), is less than or equal to four. The diameter of a graph G, denoted diam(G) is the greatest distance between any two distinct vertices of G. A radio labeling of a graph G is a function f that assigns to each …

Modeling And Analysis Of Pedestrian Flows, Romesh Khaddar

UNLV Theses, Dissertations, Professional Papers, and Capstones

According to the Traveler Opinion and Perception Survey of 2005, about 107.4 million Americans regularly use walking as a mode of transport during their commute, which amounts for 51% of the total American population. In 2009, 4092 pedestrian fatalities were reported nationwide, out of 59,000 pedestrian crashes. This amounts for 12% of the fatalities in the total traffic accidents recorded, and shows an over-representation of pedestrians incidents. Thus, it is imperative to understand the causes behind such statistics, and conduct a comprehensive research on pedestrian walking behavior and their interaction with surroundings.

A lot of researches on pedestrian flows have …

A Survey Of Ekeland's Variational Principle And Related Theorems And Applications, Jessica Robinson

UNLV Theses, Dissertations, Professional Papers, and Capstones

Ekeland's Variational Principle has been a key result used in various areas of analysis such as fixed point analysis, optimization, and optimal control theory. In this paper, the application of Ekeland's Variational Principle to Caristi's Fixed Point Theorem, Clarke's Fixed Point Theorem, and Takahashi's Minimization theorem is the focus. In addition, Ekeland produced a version of the classical Pontryagin Mini- mum Principle where his variational principle can be applied. A further look at this proof and discussion of his approach will be contrasted with the classical method of Pontryagin. With an understanding of how Ekeland's Variational Princple is used in …

Exact Controllability Of The Lazer-Mckenna Suspension Bridge Equation, Lanxuan Yu

UNLV Theses, Dissertations, Professional Papers, and Capstones

It is well known that suspension bridges may display certain oscillations under external aerodynamic forces. Since the collapse of the Tacoma Narrows suspension bridge in 1940, suspension bridge models have been studied by many researchers. Based upon the fundamental nonlinearity in suspension bridges that the stays connecting the supporting cables and the roadbed resist expansion, but do not resist compression, new models describing oscillations in suspension bridges have been developed by Lazer and McKenna [Lazer and McKenna (1990)]. Except for a paper by Leiva [Leiva (2005)], there have been very few work on controls of the Lazer-McKenna suspension bridge models …