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Articles 1 - 21 of 21
Full-Text Articles in Applied Mathematics
Buckling Loads Of A Graphene Layer Interacting With Rigid Substrates, Bradley Beckwith
Buckling Loads Of A Graphene Layer Interacting With Rigid Substrates, Bradley Beckwith
Williams Honors College, Honors Research Projects
The goal of this project is to formulate a model that can predict the buckling of a graphene layer between two rigid substrates. The model will predict the buckling of the graphene layer when it is parallel to the substrates and an edge load is applied to the ends of the layer. Our main focus is to use the model to predict buckling loads given different assumptions for interaction forces between the graphene layer and the substrates. For this project continuum modeling will be used to create a model for the graphene buckling problem. This modeling leads to a total …
Supercritical And Subcritical Pitchfork Bifurcations In A Buckling Problem For A Graphene Sheet Between Two Rigid Substrates, Jake Grdadolnik
Supercritical And Subcritical Pitchfork Bifurcations In A Buckling Problem For A Graphene Sheet Between Two Rigid Substrates, Jake Grdadolnik
Williams Honors College, Honors Research Projects
In this paper we study a model of the buckling of a sheet of graphene between two rigid substrates. We seek to understand the buckling of the sheet as the substrate separation is varied with a fixed load on each end of the sheet. We write down the expression for total energy of the system and from it derive a 2-point nonlinear boundary-value problem whose solutions are equilibrium configurations of the sheet. We cannot get an explicit solution. Instead, we perform a bifurcation analysis by using asymptotics to approximate solutions on the bifurcating branches near the bifurcation points. The bifurcating …
An Enumeration Of Nested Networks, Nathan Cornelius
An Enumeration Of Nested Networks, Nathan Cornelius
Williams Honors College, Honors Research Projects
Nested networks have several applications in phylogenetics and electrical circuit theory. In many cases, there may exist more than one distinct network which correctly models a given data set. This proposes a combinatorial problem to determine all possible network solutions. In this paper, we partially solve this problem by developing exponential generating functions which enumerate all 1-nested and 2-nested unicyclic networks. We also describe our procedure to directly count all 1-nested and 2-nested networks and provide all 1-nested networks with 7, 8, and 9 terminal nodes.
Design Project: Smart Headband, John Michel, Jack Durkin, Noah Lewis
Design Project: Smart Headband, John Michel, Jack Durkin, Noah Lewis
Williams Honors College, Honors Research Projects
Concussion in sports is a prevalent medical issue. It can be difficult for medical professionals to diagnose concussions. With the fast pace nature of many sports, and the damaging effects of concussions, it is important that any concussion risks are assessed immediately. There is a growing trend of wearable technology that collects data such as steps and provides the wearer with in-depth information regarding their performance. The Smart Headband project created a wearable that can record impact data and provide the wearer with a detailed analysis on their risk of sustaining a concussion. The Smart Headband uses accelerometers and gyroscopes …
Obstructive Wiring Patterns To Circular Planarity In Electrical Networks, Hannah Lebo
Obstructive Wiring Patterns To Circular Planarity In Electrical Networks, Hannah Lebo
Williams Honors College, Honors Research Projects
In order for an electrical network to be printed on a flat surface without changing the network’s input or output, it is important to consider if any wires will cross and if this problem can be avoided. If a circular network can be printed so that no wires cross, the network is said to be circular planar. In this paper, we identify a number of wiring patterns that make circular planarity impossible. We find exactly 3 wiring patterns using circular pairs with sets of two nodes, and we find exactly 78 wiring patterns using circular pairs with sets of three …
Phylogenetic Networks And Functions That Relate Them, Drew Scalzo
Phylogenetic Networks And Functions That Relate Them, Drew Scalzo
Williams Honors College, Honors Research Projects
Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate …
Equilibrium Structures And Thermal Fluctuations In Interacting Monolayers, Emmanuel Rivera
Equilibrium Structures And Thermal Fluctuations In Interacting Monolayers, Emmanuel Rivera
Williams Honors College, Honors Research Projects
Coherency strains appear in interacting atomic monolayers due to differing bond lengths, which can arise from different materials or geometries. Examples include extended monolayers interacting with a substrate and the interacting walls of a multi-walled carbon nanotube. These strains can induce various equilibrium configurations, which we will analyze by means of a phenomenological model that incorporates forces from bond stretching and bending within each layer and the weak van der Waals interactions connecting the separate layers. We vary the strengths of each interaction to explore their effects on equilibrium structures, and the specific case of a two-walled carbon nanotube is …
Understanding The Ntru Cryptosystem, Benjamin Clark
Understanding The Ntru Cryptosystem, Benjamin Clark
Williams Honors College, Honors Research Projects
In this paper, we will examine the NTRU Public Key Cryptosystem. The NTRU cryptosystem was created by Joseph Silverman, Jeffery Hoffstein, and Jill Pipher in 1996. This system uses truncated polynomial rings to encrypt and decrypt data. It was recently released into the public domain in 2013. This paper will describe how this cryptosystem works and give a basic understanding on how to encrypt and decrypt using this system.
Understanding The Nature Of Nanoscale Wetting Through All-Atom Simulations, Oliver Evans
Understanding The Nature Of Nanoscale Wetting Through All-Atom Simulations, Oliver Evans
Williams Honors College, Honors Research Projects
The spreading behavior of spherical and cylindrical water droplets between 30Å and 100Å in radius on a sapphire surface is investigated using all-atom molecular dynamics simulations for durations on the order of tens of nanoseconds. A monolayer film develops rapidly and wets the surface, while the bulk of the droplet spreads on top of the monolayer, maintaining the shape of a spherical cap. Unlike previous simulations in the literature, the bulk radius is found to increase to a maximum value and receed as the monolayer continues to expand. Simple time and droplet size dependence is observed for monolayer radius and …
Using A Coupled Bio-Economic Model To Find The Optimal Phosphorus Load In Lake Tainter, Wi, Mackenzie Jones
Using A Coupled Bio-Economic Model To Find The Optimal Phosphorus Load In Lake Tainter, Wi, Mackenzie Jones
Williams Honors College, Honors Research Projects
In Dunn County, Wisconsin the lakes suffer from algae blooms due to excess phosphorus runoff. A coupled bio-economic model is studied with the intent of finding the optimal level of phosphorus that should be allowed into the lake depending on certain biologic and economic parameters. We model the algae and phosphorus concentration in the lake over time based off the phosphorus input. Community welfare is modeled by comparing the costs and benefits of phosphorus fertilizer. This model is proposed to find the phosphorus level that maximizes community welfare and then determine how certain environmental and social change initiatives will affect …
A Mathematical Model Of A Corrosion System Containing Inhibitors, Abigael Frey
A Mathematical Model Of A Corrosion System Containing Inhibitors, Abigael Frey
Williams Honors College, Honors Research Projects
A two dimensional model is developed to describe how organic and inorganic inhibitors slows down the corrosion damage of a coated metal plate that contains a defect. The model contains a metal covered on one side by a coating that contains organic and inorganic inhibitors, electrolytes that are on the outside of the coating, and a small defect in the coating. The defect is an area where the coating is more porous and allows the electrolytes to leak in faster. In this model the organic inhibitor is presumed to be dissolved into the coating and the inorganic inhibitor is released …
Using Mathematical Research Methods To Solve A Problem In Music Theory Instruction, Specifically, The Teaching Of Secondary Dominant Chords, Angela Ulrich
Williams Honors College, Honors Research Projects
The mathematical method for research is used to find a solution to a problem in music theory: understanding and identifying secondary dominant chords. By reviewing and assessing the teaching methods of university professors and theory textbooks, and comparing those findings with student reviews, a new method for teaching the concept is developed. The proposed system incorporates aural, visual, and kinetic exercises to serve every learner. The literature review and sample unit plan are followed by a possible procedure for testing the effectiveness of the new method.
Diffusive Behavior Of Physically Cross-Linked Hydrogels, Morgan A. Stilke
Diffusive Behavior Of Physically Cross-Linked Hydrogels, Morgan A. Stilke
Williams Honors College, Honors Research Projects
A hydrogel is a water swollen polymeric network that has the potential to transform a variety of biomedical applications because of its biocompatibility and water content. Unfortunately, hydrogels typically have poor mechanical robustness. The majority of the content is water, while the remaining structure consists of a physical network, a covalent network, or a combination of both, yielding strength and toughness. There are several strategies attempted to improve mechanical properties including double network hydrogels comprised of two interpenetrating networks: one being a highly crosslinked sacrificial network and the other being a weakly crosslinked network that maintains reversibility after deformation. However, …
Effects Of Invasion Timing In A One-Dimensional Model Of Competing Species With An Infectious Disease, Eliza Jacops
Effects Of Invasion Timing In A One-Dimensional Model Of Competing Species With An Infectious Disease, Eliza Jacops
Williams Honors College, Honors Research Projects
In combining two classes of models, we are able to analyze the dynamics of two species that compete for the same resources while fighting a disease. The native species is the disease host and the invasive species enters their habitat and encounters the disease for the first time. Their natural response is to evolve resistance to the disease, and this can assist in their invasion of the natives' habitat. We find conditions for coexistence of the two species, conditions under which an invasion would succeed and wipe out all native individuals, and conditions under which the invasion fails. We explore …
Galvanically Induced/Accelerated Crevice Corrosion, Zachary R. Roland
Galvanically Induced/Accelerated Crevice Corrosion, Zachary R. Roland
Williams Honors College, Honors Research Projects
In this thesis, a one dimensional model is developed to investigate the initial stages of corrosion in a fastener assembly consisting of a stainless steel fastener and aluminum 7075 as the plate. Differential equations are formulated and solved to determine the profiles for the potential, the oxygen concentration, and the aluminum ion concentration in the crevice, and also the potential in the bulk electrolyte. This fastener system exhibits galvanic corrosion, pitting corrosion, and crevice corrosion. It is found that the potential decreases monotonically down the length of the crevice, the oxygen concentration decreases exponentially down the length of the crevice, …
Price Signaling In A Two-Market Duopoly, Matthew Hughes
Price Signaling In A Two-Market Duopoly, Matthew Hughes
Williams Honors College, Honors Research Projects
Within any industry, firms typically produce related products over multiple subsequent periods in an attempt to build consumer loyalty and achieve continued sales. Apple releases new iPhones and car companies produce new models every year, relying on consumers believing each new product is of high quality. Firms rely on the spillover effects from previous markets, where firms are able to more easily demonstrate their product's quality to the consumers before purchase. The goal is to find a range of prices which allows the high quality firm to distinguish its type to consumers via the price pH and if spillover effects …
Subgroups Of Finite Wreath Product Groups For P=3, Jessica L. Gonda
Subgroups Of Finite Wreath Product Groups For P=3, Jessica L. Gonda
Williams Honors College, Honors Research Projects
Let M be the additive abelian group of 3-by-3 matrices whose entries are from the ring of integers modulo 9. The problem of determining all the normal subgroups of the regular wreath product group P=Z9≀(Z3 × Z3) that are contained in its base subgroup is equivalent to the problem of determining the subgroups of M that are invariant under two particular endomorphisms of M. In this thesis we give a partial solution to the latter problem by implementing a systematic approach using concepts from group theory and linear algebra.
Illustrating Pit Initiation And Evolution In Aluminum Alloys According To A 3-Dimensional Cellular Automata Based Model, Kathryn Stalker
Illustrating Pit Initiation And Evolution In Aluminum Alloys According To A 3-Dimensional Cellular Automata Based Model, Kathryn Stalker
Williams Honors College, Honors Research Projects
The objective of this work is to develop a cellular automata based model of pitting initiation and subsequent three-dimensional evolution of pit shapes. Here, a cellular automaton is a collection of cells, each of which may be in one of two states, metallic or electrolyte, arranged over a grid. Pit initiation is implemented over a 2-dimensional grid representative of the metal surface while pit propagation is resolved over a 3-dimensional grid which describes a subsection of the bulk metal. The size of the bulk metal will be on the order of millimeters, and a layer of electrolyte, in which chloride …
Parameters Estimation Of Material Constitutive Models Using Optimization Algorithms, Kiswendsida Jules Kere
Parameters Estimation Of Material Constitutive Models Using Optimization Algorithms, Kiswendsida Jules Kere
Williams Honors College, Honors Research Projects
Optimization Algorithms are very useful for solving engineering problems. Indeed, optimization algorithms can be used to optimize engineering designs in terms of safety and economy. Understanding the proprieties of materials in engineering designs is very important in order to make designs safe. Materials are not really perfectly homogeneous and there are heterogeneous distributions in most materials. In this paper, Self-OPTIM which is an inverse constitutive parameter identification framework will be used to identify parameters of a linear elastic material constitutive model. Data for Self-OPTIM will be obtained using ABAQUS simulation of a dog-bone uniaxial test. Optimization Algorithms will be used …
The Relationship Among Math Anxiety, Mathematical Performance, And Math Education In Undergraduate Nursing Students, Joshua D. Beall, Troy Roebuck, Paul Penkalsky
The Relationship Among Math Anxiety, Mathematical Performance, And Math Education In Undergraduate Nursing Students, Joshua D. Beall, Troy Roebuck, Paul Penkalsky
Williams Honors College, Honors Research Projects
Although nurses spend up to 40% of their day calculating and administering medication doses, undergraduate nursing students often perform poorly on nursing math exams. The purpose of this study was (a) to examine the relationship among mathematical education, performance, and anxiety and (b) to compare the mathematical education, performance, and anxiety in sophomore and senior baccalaureate nursing students at a public university in the Midwest. This cross-sectional, descriptive study was guided by Bandura's self-efficacy theory. Math performance was measured with an 11-item math instrument, math education was measured with number of math courses, and math anxiety was measured with Fennema–Sherman …
An Examination Of Mathematical Models For Infectious Disease, David M. Jenkins
An Examination Of Mathematical Models For Infectious Disease, David M. Jenkins
Williams Honors College, Honors Research Projects
Starting with the original 1926 formulation of the SIR (Susceptible-Infected-Removed) model for infectious diseases, mathematical epidemiology continued to grow. Many extensions such as the SEIR, MSIR, and MSEIR models were developed using SIR as a basis to model diseases in a variety of circumstances. By taking the original SIR model, and reducing the system of three first-order equations to a single first-order equation, analysis shows that the model predicts two possible situations. This analysis is followed by discussion of an alternative use of the SIR model which allows for one to track the amount of sustainable genetic variation in a …