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- Creator:
- Verdugo, Anael
- Description:
- This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of delay differential equations (DDEs) coupled to ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as becomes large.
- Resource Type:
- Article
- Campus Tesim:
- Fullerton
- Department:
- Department of Mathematics

- Creator:
- Verdugo, Anael
- Description:
- The repressilator is a genetic network that exhibits oscillations. The net-work is formed of three genes, each of which represses each other cyclically, creating a negative feedback loop with nonlinear interactions. In this work we present a computational bifurcation analysis of the mathematical model of the repressilator. We show that the steady state undergoes a transition from stable to unstable giving rise to a stable limit-cycle in a Hopf bifurcation. The nonlinear analysis involves a center manifold reduction on the six-dimensional system, which yields closed form expressions for the frequency and amplitude of the oscillation born at the Hopf. A parameter study then shows how the dynamics of the system are influenced for different parameter values and their associated biological significance.
- Resource Type:
- Article
- Campus Tesim:
- Fullerton
- Department:
- Department of Mathematics

- Creator:
- Rathbun, Matt
- Description:
- A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in F, the genus 2 Heegaard surface for S3. Primitive/primitive and primitive/Seifert knots lie in F in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in F. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in F.
- Resource Type:
- Article
- Campus Tesim:
- Fullerton
- Department:
- Department of Mathematics

- Creator:
- Rathbun, Matt
- Description:
- We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called generalized Hopf banding. We further characterize generalized crossing changes between fibered links, and the resulting changes in monodromy.
- Resource Type:
- Article
- Campus Tesim:
- Fullerton
- Department:
- Department of Mathematics

- Creator:
- Homier, Samantha
- Description:
- Pretend that you cannot remember where you parked your car in the parking lot of the grocery store, but you do remember some of the cars parked near you. One could construct a graph based on your memory of the cars and then use the idea of the metric dimension to find your car. The metric dimension was introduced by PJ Slater in 1975 and has since been applied in fields such as chemistry, optimization, navigation, and more. There is no general/standard metric dimension for every graph, however, there are known metric dimensions for families of graphs. In this paper we study the metric dimension of Cayley graphs, which are graphs based on groups that have convenient algebraic properties. Our main goal is to find the metric dimension of the Cayley graph associated with the symmetric group $S_4$ and its set of transpositions $T_4$.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Ignatius, Darian
- Description:
- Compartmental models have been used in epidemiology for many years to study the spread of infectious diseases throughout the world. In this thesis, we are recreating and extending the work done by others in [1] to apply one of these models, the SEIZ model, to the spread of news and rumors on Twitter. After deriving the model and discussing its background, we obtained data regarding 6 events, 3 real news stories and 3 rumors. We showed that the method used to minimize the error between the model and the actual data was quite accurate, and that the model was able to work very early on in a story or with limited information. We also attempted to find several combinations of parameters which could distinguish the stories between news and rumors, but no consistent results were found. Finally, we restricted the amount of data fed into the model, and took a look at its ability to estimate the number of tweets in the future.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Avalos Galvez, Diego Gerardo Andree
- Description:
- The eigenvalues of an $n\times n$ real nonzero skew-symmetric matrix $S$ are purely imaginary or zero. Let the list of distinct purely imaginary eigenvalues of $S$ be $\pm\theta_1i,\dots ,\pm\theta_pi$ such that each $\theta_j>0$. We algebraically demonstrate that the exponential $e^S$ can be expressed in terms of the powers $I_n,S,\dots,S^{n-1}$, where the coefficients are in terms of the distinct values $\theta_j$, by using the method by Gallier and Xu \cite{Gallier and Xu}. Furthermore, the formulas of $e^S$ (in terms of the $\theta_j$s) depend solely on the number of distinct eigenpairs $\pm\theta_ji$ of $S$ and whether zero is an eigenvalue of $S$, but are independent of their algebraic multiplicities. Only the formulas of the $\theta_j$s (in terms of the entries of $S$) depend on the multiplicities of the $\theta_ji$s in the characteristic polynomial of $S$. This allows us to determine that if $n$ is even, then $e^S$ has $n-1$ different cases, and $\frac{n-1}{2}$ cases if $n$ is odd. In this thesis, we calculate all the closed form formulas of $e^S$ for $2\leq n\leq 9$ because we can obtain the eigenvalues of $S$ in terms of its entries up to the case $n=9$ using the linear, quadratic, cubic, and quartic formulas. Nevertheless, the theory allows us to calculate the closed formula of $e^S$ for any arbitrary $n$ assuming the eigenvalues of $S$ are known. Lastly, we implement the formulas obtained in this thesis on our Matlab function \texttt{skewexpm} and compare the orthogonality errors using our formulas on randomly generated skew-symmetric matrices to those obtained by applying Matlab's \texttt{expm}. It turns out that our formulas give a smaller error than \texttt{expm} for over $97\%$ of the time up to size $n=5$, over $92\%$ of the time up to size $n=7$, and for over $ 60\%$ of the time for sizes $n=8$ and $9$ (see Table 4.1). Finally, if we allow the entries of a skew-symmetric matrix to range from $-10^{15}$ to $10^{15}$, we can rely that our closed formulas will have a far better and acceptable error than \texttt{expm}, as our example with $n=9$ illustrates in Figure 5.9.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Qi, Xin
- Description:
- The most important reason that the UnitedStates has dominated the world is advanced technology. This is inseparable from the popularization of education. Education is one of the most powerful tools to reduce poverty and sets the foundation for sustained economic growth. Education is very important to our lives; therefore, it is necessary to know how school revenues, expenses, and school enrollment have changed over the years. Educational fnancesin the U.S. are really important as theyrepresenta vitalfactorin people’s choices of elementary and high schools for their kids. Usually, good districts are associated with high-incomefamilies, high educationalfamily background, and high house prices. Therefore, people prefer to choose high quality districts for their kids based on their maximum economic ability since qualityof educationmay determinestheir kids’ future. Since education revenues largely depend on the overall U.S. market economy, knowledge of how both revenues and the stock market change over time is crucial. My graduate thesis focuses on the issue of U.S educational fnances, on revenue and expenditure for U.S. grade schools,by year andby state;I examine the impactof school expense and revenue on school enrollment rates, and how revenue per student change. My project examines all 50 states from 1993 to 2015, analyzing data across this 23-year period. This thesis has three aims; carry out basic exploratory data analysis, examine simple relationships between variables, and then use advanced methods (e.g., longitudinal models) to predict and examine relationships among the enrollment, states, years, revenues, and expenses. My results indicates that state revenue and local revenue each account for 45% of total educational revenue with federal revenue accounting for about 10%. Although California gets the highest educational fnancial support in terms of absolute amount, the overwhelming numbersof students make foralow per capita revenue. NewYork and New Jersey are the top two revenue-per-capita states. As time goes by, the sub-class revenue per student increasesevery year, hittinga peakin 2011 and thenfalling. This mirrors the rise andfallin the stock marketover the some period.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Almond, Isaiah
- Description:
- Statistics is a widely popular field when it comes to organizations attempting to answer business questions and forecast performance in the near or long-term future. In order for this to be accomplished, data must be collected and utilized. Data is not always plentiful and also may be very complex to analyze. This is where the bootstrap method can be used as an alternative to estimating population parameters of interest when conventional methods would have difficulty in doing so. This thesis will cover bootstrap methods for the general case as well as extend its application to hypothesis testing and simple linear regression. Alternative methods of bootstrapping along with strengths and weaknesses will be covered briefly as well.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Si, Jiaxin
- Description:
- More than 35,000 medals have been awarded at the Olympics since 1896. The IOC (International Olympic Committee) retrospectively awarded gold, silver, and bronze to athletes based on their rankings. The dataset we used covers Summer Olympics from 1896to2012andWinter Olympicsfrom 1924-2014;eachyear includesarowforevery Olympic athlete that haswon a medal since the frstgames. Also, this dataset has each IOC country’s population and GDP in 2012 to 2014. This report has four main analysis parts. The frst part introduces the base information about Olympics. In the second part, we explore the basic analysis about Summer and WinterOlympics.Thethirdpart consistsofjointanalysisofboththeSummerandWinter the Olympic Games. The fourth part, involving the number of medals in the Summer Olympics and the Winter Olympics, will be explored. At the same time, in order to reveal the relationship between the number of medals and the basic characteristics of each country average high temperature in winter and GDP per capita were introduced in 2012 and 2014. The relationship between the average high temperature in winter and GDP per capita affects the number of medals a country obtains.
- Resource Type:
- Thesis
- Campus Tesim:
- Pomona
- Department:
- Department of Mathematics

- Creator:
- Suceavă, Bogdan
- Description:
- This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2018 makes available to a wide audience many pieces not easily found anywhere else―and you don’t need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice―and taking readers behind the scenes of today’s hottest mathematical debates.
James Grime shows how to build subtly mischievous dice for playing slightly unfair games and Michael Barany traces how our appreciation of the societal importance of mathematics has developed since World War II. In other essays, Francis Su extolls the inherent values of learning, doing, and sharing mathematics, and Margaret Wertheim takes us on a mathematical exploration of the mind and the world―with glimpses at science, philosophy, music, art, and even crocheting. And there’s much, much more.
In addition to presenting the year’s most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable pieces on mathematics.
This is a must-read for anyone interested in where math has taken us―and where it is headed.
- Resource Type:
- Book
- Identifier:
- 9780691182766
- Campus Tesim:
- Fullerton
- Department:
- Department of Mathematics