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- All Subjects: Mathematics
- Creators: School of Mathematical and Statistical Sciences
Description
In this project we focus on COVID-19 in a university setting. Arizona State University has a very large population on the Tempe Campus. With the emergence of diseases such as COVID-19, it is very important to track how such a disease spreads within that type of community. This is vital for containment measures and the safety of everyone involved. We found in the literature several epidemiology models that utilize differential equations for tracking a spread of a disease. However, our goal is to provide a granular look at how disease may spread through contact in a classroom. This thesis models a single ASU classroom and tracks the spread of a disease. It is important to note that our variables and declarations are not aligned with COVID-19 or any other specific disease but are chosen to exemplify the impact of some key parameters on the epidemic size. We found that a smaller transmissibility alongside a more spread-out classroom of agents resulted in fewer infections overall. There are many extensions to this model that are needed in order to take what we have demonstrated and align those ideas with COVID-19 and it’s spread at ASU. However, this model successfully demonstrates a spread of disease through single-classroom interaction, which is the key component for any university campus disease transmission model.
ContributorsJoseph, Mariam (Author) / Bartko, Ezri (Co-author) / Sabuwala, Sana (Co-author) / Milner, Fabio (Thesis director) / O'Keefe, Kelly (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Division of Teacher Preparation (Contributor)
Created2022-12
Description
This research project dug into mathematics in music, exploring the various ways a number series was used in the 20th century to create musical compositions. The Fibonacci Series (FS) is an infinite number series that is created by taking the two previous numbers to create the next, excluding 0 and 1 at the very start of the series. As the numbers grow larger, the ratios between the numbers of the FS approach the value of another mathematical concept known as the Golden Mean (GM). The GM is so closely related to the series that it is used interchangeably in terms of proportions and overall structure of musical pieces. This is similar to how both the FS and GM are found in aspects of nature, like to all too well-known conch shell spiral.
The FS in music was used in a variety of ways throughout the 20th century, primarily focusing on durations and overall structure in its use. Examples of this are found in Béla Bartók’s Music for Strings, Percussion, and Celeste (1936), Allegro barbaro (1911), Karlheinz Stockhausen’s Klavierstück IX (1955), and Luigi Nono’s il canto sospeso (1955). These works are analyzed in detail within my research, and I found every example to have a natural feel to them even if its use of the FS is carefully planned out by the composer. Bartók’s works are the least precise of my examples but perhaps the most natural ones. This imprecision in composition may be considered a more natural use of the FS in music, since nature is not always perfect either. However, in works such as Stockhausen’s, the structure is meticulously formatted in such that the precision is masked by a cycle as to appear more natural.
The conclusion of my research was a commissioned work for my instrument, the viola. I provided my research to composer Jacob Miller Smith, a DMA Music Composition student at ASU, and together we built the framework for the piece he wrote for me. We utilized the life cycle of the Black-Eyed Susan, a flower that uses the FS in its number of petals. The life cycle of a flower is in seven parts, so the piece was written to have seven separate sections in a palindrome within an overall ABA’ format. To utilize the FS, Smith used Fibonacci number durations for rests between notes, note/gesture groupings, and a mapping of 12358 as the set (01247). I worked with Smith during the process to make sure that the piece was technically suitable for my capabilities and the instrument, and I premiered the work in my defense.
The Fibonacci Series and Golden Mean in music provides a natural feel to the music it is present in, even if it is carefully planned out by the composer. More work is still to be done to develop the FS’s use in music, but the examples presented in this project lay down a framework for it to take a natural place in music composition.
The FS in music was used in a variety of ways throughout the 20th century, primarily focusing on durations and overall structure in its use. Examples of this are found in Béla Bartók’s Music for Strings, Percussion, and Celeste (1936), Allegro barbaro (1911), Karlheinz Stockhausen’s Klavierstück IX (1955), and Luigi Nono’s il canto sospeso (1955). These works are analyzed in detail within my research, and I found every example to have a natural feel to them even if its use of the FS is carefully planned out by the composer. Bartók’s works are the least precise of my examples but perhaps the most natural ones. This imprecision in composition may be considered a more natural use of the FS in music, since nature is not always perfect either. However, in works such as Stockhausen’s, the structure is meticulously formatted in such that the precision is masked by a cycle as to appear more natural.
The conclusion of my research was a commissioned work for my instrument, the viola. I provided my research to composer Jacob Miller Smith, a DMA Music Composition student at ASU, and together we built the framework for the piece he wrote for me. We utilized the life cycle of the Black-Eyed Susan, a flower that uses the FS in its number of petals. The life cycle of a flower is in seven parts, so the piece was written to have seven separate sections in a palindrome within an overall ABA’ format. To utilize the FS, Smith used Fibonacci number durations for rests between notes, note/gesture groupings, and a mapping of 12358 as the set (01247). I worked with Smith during the process to make sure that the piece was technically suitable for my capabilities and the instrument, and I premiered the work in my defense.
The Fibonacci Series and Golden Mean in music provides a natural feel to the music it is present in, even if it is carefully planned out by the composer. More work is still to be done to develop the FS’s use in music, but the examples presented in this project lay down a framework for it to take a natural place in music composition.
ContributorsFerry, Courtney (Author) / Knowles, Kristina (Thesis director) / Buck, Nancy (Committee member) / School of Music (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-12
Description
There exists a notable gender gap in the field of economics. One explanation for this gap is the low supply of women entering the economics labor market. To understand the shortage of female economics students, I observe students at the undergraduate and graduate level. My data consists of a sample of current undergraduate students and a sample of past Ph.D. applicants at Arizona State University. The gender gaps in these samples, both at the undergraduate and graduate level, can largely be explained by the variation in mathematical preparation of the students. The data reveals that undergraduate male economics students are more frequently enrolled in higher level math courses compared to female undergraduate students. Likewise, a higher number of male Ph.D. applicants have stronger mathematical backgrounds relative to female Ph.D. applicants. This common factor might explain the higher supply of male students who apply and get accepted to postgraduate studies in economics, relative to female students, holding all else constant. I conclude with the following recommended interventions: make information regarding postgraduate opportunities in economics more readily available, and increase math requirements for a bachelor’s degree in economics at ASU.
ContributorsZafari, Zorah (Author) / Datta, Manjira (Thesis director) / Zafar, Basit (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / School of Social Transformation (Contributor, Contributor) / Department of Economics (Contributor) / Dean, W.P. Carey School of Business (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The goal of this thesis is to explore and present a range of approaches to “algorithmic choreography.” In the context of this thesis, algorithmic choreography is defined as choreography with computational influence or elements. Traditionally, algorithmic choreography, despite containing works that use computation in a variety of ways, has been used as an umbrella term for all works that involve computation.
This thesis intends to show that the diversity of algorithmic choreography can be reduced into more specific categories. As algorithmic choreography is fundamentally intertwined with the concept of computation, it is natural to propose that algorithmic choreography works be separated based on a spectrum that is defined by the extent of the involvement of computation within each piece.
This thesis seeks to specifically outline three primary categories that algorithmic works can fall into: pieces that involve minimal computational influence, entirely computationally generated pieces, and pieces that lie in between. Three original works were created to reflect each of these categories. These works provide examples of the various methods by which computation can influence and enhance choreography.
The first piece, entitled Rαinwater, displays a minimal amount of computational influence. The use of space in the piece was limited to random, computationally generated paths. The dancers extracted a narrative element from the random paths. This iteration resulted in a piece that explores the dancers’ emotional interaction within the context of a rainy environment. The second piece, entitled Mymec, utilizes an intermediary amount of computation. The piece sees a dancer interact with a projected display of an Ant Colony Optimization (ACO) algorithm. The dancer is to take direct inspiration from the movement of the virtual ants and embody the visualization of the algorithm. The final piece, entitled nSkeleton, exhibited maximal computational influence. Kinect position data was manipulated using iterative methods from computational mathematics to create computer-generated movement to be performed by a dancer on-stage.
Each original piece was originally intended to be presented to the public as part of an evening-length show. However, due to the rise of the COVID-19 pandemic caused by the novel coronavirus, all public campus events have been canceled and the government has recommended that gatherings with more than 10 people be entirely avoided. Thus, the pieces will instead be presented in the form of a video published online. This video will encompass information about the creation of each piece as well as clips of choreography.
This thesis intends to show that the diversity of algorithmic choreography can be reduced into more specific categories. As algorithmic choreography is fundamentally intertwined with the concept of computation, it is natural to propose that algorithmic choreography works be separated based on a spectrum that is defined by the extent of the involvement of computation within each piece.
This thesis seeks to specifically outline three primary categories that algorithmic works can fall into: pieces that involve minimal computational influence, entirely computationally generated pieces, and pieces that lie in between. Three original works were created to reflect each of these categories. These works provide examples of the various methods by which computation can influence and enhance choreography.
The first piece, entitled Rαinwater, displays a minimal amount of computational influence. The use of space in the piece was limited to random, computationally generated paths. The dancers extracted a narrative element from the random paths. This iteration resulted in a piece that explores the dancers’ emotional interaction within the context of a rainy environment. The second piece, entitled Mymec, utilizes an intermediary amount of computation. The piece sees a dancer interact with a projected display of an Ant Colony Optimization (ACO) algorithm. The dancer is to take direct inspiration from the movement of the virtual ants and embody the visualization of the algorithm. The final piece, entitled nSkeleton, exhibited maximal computational influence. Kinect position data was manipulated using iterative methods from computational mathematics to create computer-generated movement to be performed by a dancer on-stage.
Each original piece was originally intended to be presented to the public as part of an evening-length show. However, due to the rise of the COVID-19 pandemic caused by the novel coronavirus, all public campus events have been canceled and the government has recommended that gatherings with more than 10 people be entirely avoided. Thus, the pieces will instead be presented in the form of a video published online. This video will encompass information about the creation of each piece as well as clips of choreography.
ContributorsJawaid, Zeeshan (Co-author, Co-author) / Jackson, Naomi (Thesis director) / Curry, Nicole (Committee member) / Espanol, Malena (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Dean, W.P. Carey School of Business (Contributor) / School of Film, Dance and Theatre (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
In the modern world with the ever growing importance of technology, the challenge of information security is of increasing importance. Cryptographic algorithms used to encode information stored and transmitted over the internet must be constantly improving as methodology and technology for cyber attacks improve. RSA and Elliptic Curve cryptosystems such as El Gamal or Diffie-Hellman key exchange are often used as secure asymmetric cryptographic algorithms. However, quantum computing threatens the security of these algorithms. A relatively new algorithm that is based on isogenies between elliptic curves has been proposed in response to this threat. The new algorithm is thought to be quantum resistant as it uses isogeny walks instead of point addition to generate a shared secret key. In this paper we will analyze this algorithm in an attempt to understand the theory behind it. A main goal is to create isogeny graphs to visualize degree 2 and 3 isogeny walks that can be taken between supersingular elliptic curves over small fields to get a better understanding of the workings and security of the algorithm.
ContributorsLoucks, Sara J (Author) / Jones, John (Thesis director) / Bremner, Andrew (Committee member) / Computer Science and Engineering Program (Contributor) / School of Film, Dance and Theatre (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
A one-way function (OWF) is a function that is computationally feasible to compute in one direction, but infeasible to invert. Many current cryptosystems make use of properties of OWFs to provide ways to send secure messages. This paper reviews some simple OWFs and examines their use in contemporary cryptosystems and other cryptographic applications. This paper also discusses the broader implications of OWF-based cryptography, including its relevance to fields such as complexity theory and quantum computing, and considers the importance of OWFs in future cryptographic development
ContributorsMcdowell, Jeremiah Tenney (Author) / Hines, Taylor (Thesis director) / Foy, Joseph (Committee member) / Sprung, Florian (Committee member) / School of Mathematical and Statistical Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Ion channels in the membranes of cells in the body allow for the creation of action potentials from external stimuli, allowing us to sense our surroundings. One particular channel, TRPM8, is a trans-membrane ion channel believed to be the primary cold sensor in humans. Despite this important biological role and intense study of the channel, TRPM8 is not fully understood mechanistically and has not been accurately modeled. Existing models of TRPM8 fail to account for menthol activation of the channel. In this paper we re-implement an established whole cell model for TRPM8 with gating by both voltage and temperature. Using experimental data obtained from the Van Horn lab at Arizona State University, we refined the model to represent more accurately the dynamics of the human TRPM8 channel and incorporate the channel activation through menthol agonist binding. Our new model provides a large improvement over preexisting models, and serves as a basis for future incorporation of other channel activators of TRPM8 and for the modeling of other channels in the TRP family.
ContributorsAckerman, David (Author) / Crook, Sharon (Thesis director) / Van Horn, Wade (Committee member) / School of Earth and Space Exploration (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The objective of this paper is to find and describe trends in the fast Fourier transformed accelerometer data that can be used to predict the mechanical failure of large vacuum pumps used in industrial settings, such as providing drinking water. Using three-dimensional plots of the data, this paper suggests how a model can be developed to predict the mechanical failure of vacuum pumps.
ContributorsHalver, Grant (Author) / Taylor, Tom (Thesis director) / Konstantinos, Tsakalis (Committee member) / Fricks, John (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
This honors thesis explores and models the flow of air around a cylindrical arrow that is rotating as it moves through the air. This model represents the airflow around an archery arrow after it is released from the bow and rotates while it flies through the air. This situation is important in archery because an understanding of the airflow allows archers to predict the flight of the arrow. As a result, archers can improve their accuracy and ability to hit targets. However, not many computational fluid dynamic simulations modeling the airflow around a rotating archery arrow exist. This thesis attempts to further the understanding of the airflow around a rotating archery arrow by creating a mathematical model to numerically simulate the airflow around the arrow in the presence of this rotation. This thesis uses a linearized approximation of the Navier Stokes equations to model the airflow around the arrow and explains the reasoning for using this simplification of the fully nonlinear Navier Stokes equations. This thesis continues to describe the discretization of these linearized equations using the finite difference method and the boundary conditions used for these equations. A MATLAB code solves the resulting system of equations in order to obtain a numerical simulation of this airflow around the rotating arrow. The results of the simulation for each velocity component and the pressure distribution are displayed. This thesis then discusses the results of the simulation, and the MATLAB code is analyzed to verify the convergence of the solution. Appendix A includes the full MATLAB code used for the flow simulation. Finally, this thesis explains potential future research topics, ideas, and improvements to the code that can help further the understanding and create more realistic simulations of the airflow around a flying archery arrow.
ContributorsCholinski, Christopher John (Author) / Tang, Wenbo (Thesis director) / Herrmann, Marcus (Committee member) / Mechanical and Aerospace Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The Jordan curve theorem states that any homeomorphic copy of a circle into R2 divides the plane into two distinct regions. This paper reconstructs one proof of the Jordan curve theorem before turning its attention toward generalizations of the theorem and their proofs and counterexamples. We begin with an introduction to elementary topology and the different notions of the connectedness of a space before constructing the first proof of the Jordan curve theorem. We then turn our attention to algebraic topology which we utilize in our discussion of the Jordan curve theorem’s generalizations. We end with a proof of the Jordan-Brouwer theorems, extensions of the Jordan curve theorem to higher dimensions.
ContributorsClark, Kacey (Author) / Kawski, Matthias (Thesis director) / Paupert, Julien (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05