Mathematics Commons^™

Analysis Of A Posteriori Error Estimates Of The Discontinuous Galerkin Method For Nonlinear Ordinary Differential Equations, Mahboub Baccouch

Mathematics Faculty Publications

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(h^p+1) convergence rate in the L²-norm when p-degree piece-wise polynomials with p ≥1 are used. We further prove that the DG solution is O(h^2p+1) superconvergent at the downwind points. We …

How Does The Structure Of A College Chemistry Examination Affect Pedagogy, Rajeev R. Pandey, John Mayberry, Jace Hargis

College of the Pacific Faculty Articles

This study examines variations of assessment and connections to active learning methods, which may enhance both the accuracy of assessment, engagement and retention. Correlation data relating instruction and assessment in a multiple dimensions are presented. Multiple choice (MC) and free response (FR) exams were provided and students were also given the option to provide FR answers to the MC items. This study suggests there is little overall difference in mean or median student scores on the MC vs. FR portions of the exam, but that there is some evidence to believe that student scores on MC portions are more variable …

Participation In The Virtual Environment Of Blended College Courses: An Activity Study Of Student Performance, Cathy Cavanaugh, Jace Hargis, John Mayberry

College of the Pacific Faculty Articles

This paper describes a study of success factors in the introductory semester of liberal studies blended courses offered at the bachelor of science level. The influence of student participation in the online course environment was examined, as measured by the number of times students logged into the learning management system (LMS) and average session length. These measures were correlated with final course grades to increase understanding of the participation patterns of successful students. The resulting patterns and their implications are identified. We observe that students with an intermediate number of logins and average session length tended to exhibit the optimal …

The Martian Surface Radiation Environment- A Comparison Of Models And Msl/Rad Measurements, Daniel Matthiä, Bent Ehresmann, Henning Lohf, Jan Köhler, Cary Zeitlin, Jan Appel, John W. Wilson

Mathematics & Statistics Faculty Publications

Context: The Radiation Assessment Detector (RAD) on the Mars Science Laboratory (MSL) has been measuring the radiation environment on the surface of Mars since August 6th 2012. MSL-RAD is the first instrument to provide detailed information about charged and neutral particle spectra and dose rates on the Martian surface, and one of the primary objectives of the RAD investigation is to help improve and validate current radiation transport models.

Aims: Applying different numerical transport models with boundary conditions derived from the MSL-RAD environment the goal of this work was to both provide predictions for the particle spectra and the radiation …

Stability And Data Dependence Results For The Jungck--Khan Iterative Scheme, Abdul Rahim Khan, Fai̇k Gürsoy, Vivek Kumar

Turkish Journal of Mathematics

The Jungck--Khan iterative scheme for a pair of nonself operators contains as a special case Jungck--Ishikawa and Jungck--Mann iterative schemes. In this paper, we establish improved results about convergence, stability, and data dependence for the Jungck--Khan iterative scheme.

On The Asymptotic Criterion For The Zero-Free Regions Of Certain $L$-Functions, Almasa Odzak

Turkish Journal of Mathematics

We investigate relations between zero-free regions of certain $L$-functions and the asymptotic behavior of corresponding generalized Li coefficients. Precisely, we prove that violation of the $\tau/2$-generalized Riemann hypothesis implies oscillations of corresponding $\tau$-Li coefficients with exponentially growing amplitudes. Results are obtained for class $\shfs$ that contains the Selberg class, the class of all automorphic $L$-functions, the Rankin--Selberg $L$-functions, and products of suitable shifts of the mentioned functions.

Limits4, Ruth Dover

Limits

An introduction to limits as something goes to infinity.

Approximations 4, Ruth Dover

Integrals

Trapezoidal Rule.

A Bound On The Number Of Spanning Trees In Bipartite Graphs, Cheng Wai Koo

HMC Senior Theses

Richard Ehrenborg conjectured that in a bipartite graph G with parts X and Y, the number of spanning trees is at most the product of the vertex degrees divided by |X|⋅|Y|. We make two main contributions. First, using techniques from spectral graph theory, we show that the conjecture holds for sufficiently dense graphs containing a cut vertex of degree 2. Second, using electrical network analysis, we show that the conjecture holds under the operation of removing an edge whose endpoints have sufficiently large degrees.

Our other results are combinatorial proofs that the conjecture holds for …

An Analysis Of Charles Fefferman's Proof Of The Fundamental Theorem Of Algebra, Kyle O. Linford

Senior Honors Theses and Projects

Many peoples' first exploration into more rigorous and formalized mathematics is with their early explorations in algebra. Much of their time and effort is dedicated to finding roots of polynomials-a challenge that becomes more increasingly difficult as the degree of the polynomials increases, especially if no real number roots exist. The Fundamental Theorem of Algebra is used to show that there exists a root, particularly a complex root, for any nth degree polynomial. After struggling to prove this statement for over 3 centuries, Carl Friedrich Gauss offered the first fairly complete proof of the theorem in 1799. Further proofs of …

Existence Of Positive Solutions For A Semipositone P-Laplacian Problem, Alfonso Castro, Djairo G. De Figueredo, Emer Lopera

All HMC Faculty Publications and Research

We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.

The Spruce Budworm And Forest: A Qualitative Comparison Of Ode And Boolean Models, Raina Robeva, David Murrugarra

Mathematics Faculty Publications

Boolean and polynomial models of biological systems have emerged recently as viable companions to differential equations models. It is not immediately clear however whether such models are capable of capturing the multi-stable behaviour of certain biological systems: this behaviour is often sensitive to changes in the values of the model parameters, while Boolean and polynomial models are qualitative in nature. In the past few years, Boolean models of gene regulatory systems have been shown to capture multi-stability at the molecular level, confirming that such models can be used to obtain information about the system’s qualitative dynamics when precise information regarding …

Electrical Efficiency Of A Solar Cell, Johnnie Cairns

Undergraduate Journal of Mathematical Modeling: One + Two

Our goal was to determine the electrical efficiency of a solar cell, specifically a CuInGaSe2 solar cell. Solar cells work by converting energy from the sun in the form of photons to electrical energy in the form of electrons. However, the solar energy converted into electrical energy is limited by a property of solar cells called the quantum efficiency. The quantum efficiency of a solar cell is the fraction of photons hitting the cell that are converted into electrons; quantum efficiency varies as a function of wavelength. Another key component which also varies as a function of wavelength is spectral …

Analysis Of Adiabatic Batch Reactor, Erald Gjonaj

Undergraduate Journal of Mathematical Modeling: One + Two

A mixture of acetic anhydride is reacted with excess water in an adiabatic batch reactor to form an exothermic reaction. The concentration of acetic anhydride and the temperature inside the adiabatic batch reactor are calculated with an initial temperature of 20°C, an initial temperature of 30°C, and with a cooling jacket maintaining the temperature at a constant of 20°C. The graphs of the three different scenarios show that the highest temperatures will cause the reaction to occur faster.

Hydrolysis Of Acetic Anhydride In A Cstr, Veronica N. Coraci

Undergraduate Journal of Mathematical Modeling: One + Two

To find the optimal reactor volume and temperature for the hydrolysis of acetic anhydride at the lowest possible cost with a 90% conversion of acetic anhydride, a formula for the total cost of the reaction was created. Then, the first derivative was taken to find a value for the temperature. This value was then inputted into the second derivative of the equation to find the sign of the value which would indicate whether that point was a minima or maxima value. The minima value would then be the lowest total cost for the optimum reaction to take place.

Optimization Of An Agitated Extractor, Evan Zapf

Undergraduate Journal of Mathematical Modeling: One + Two

In this scenario, we use calculus to determine the optimal operating specifications of a chemical extraction process. The results are achieved by first developing an expression that yields the total annual cost of the process. Factoring in electricity, vessel, agitator, and solvent costs, an annual cost expression was fabricated. AGIEX company manufactures a certain amount of a product, A, which has an impurity that it wishes be reduced. By engineering an optimally sized vessel to perform the extraction of impurity, the company will save costs by ensuring no inefficient spending on power or excess solvent occurs. A specific amount of …

Temperature Of The Central Processing Unit, Ivan Lavrov

Undergraduate Journal of Mathematical Modeling: One + Two

Heat is inevitably generated in the semiconductors during operation. Cooling in a computer, and in its main part – the Central Processing Unit (CPU), is crucial, allowing the proper functioning without overheating, malfunctioning, and damage. In order to estimate the temperature as a function of time, it is important to solve the differential equations describing the heat flow and to understand how it depends on the physical properties of the system. This project aims to answer these questions by considering a simplified model of the CPU + heat sink. A similarity with the electrical circuit and certain methods from electrical …

Size And Concentration Analysis Of Gold Nanoparticles With Ultraviolet-Visible Spectroscopy, Sanim Rahman

Undergraduate Journal of Mathematical Modeling: One + Two

Spherical gold nanoparticles (GNPs) are synthesized by Turkevich Method and PEL 35 UV-VIS spectrophotometer recorded the wavelength and absorption of the Surface Plasmon Resonance (SPR) peak. The diameter and concentration of solute GNPs are calculated. The concentration of GNPs is done with Beer’s law. The average diameter of GNPs is done via the ratio of SPR peak to the absorbance at 450 nm. The diameters are compared to SEM scan of synthesized GNPs and Sigma-Aldrich values.

Maximum Efficiency Of A Wind Turbine, Marisa Blackwood

Undergraduate Journal of Mathematical Modeling: One + Two

The concept of a wind turbine is not new technology, however, in a day and age where renewable energy is popular, the interest in wind turbines has increased dramatically. This project investigates multiple aspects of a wind turbine and will derive the maximum power efficiency of an ideal wind turbine, first introduced by Albert Betz in 1919. This project will also calculate the size of a wind turbine at maximum efficiency given certain parameters and determine the optimum outlet velocity as a function of wind speed to maximize the mechanical energy produced. These equations will be derived by using concepts …

Spintronic Circuits: The Building Blocks Of Spin-Based Computation, Roshan Warman

Undergraduate Journal of Mathematical Modeling: One + Two

In the most general situation, binary computation is implemented by means of microscopic logical gates known as transistors. According to Moore’s Law, the size of transistors will half every two years, and as these transistors reach their fundamental size limit, the quantum effects of the electrons passing through the transistors will be observed. Due to the inherent randomness of these quantum fluctuations, the basic binary logic will become uncontrollable. This project describes the basic principle governing quantum spin-based computing devices, which may provide an alternative to the conventional solid-state computing devices and circumvent the technological limitations of the current implementation …