Mathematics Commons^™

An Exploration Of Combinatorial Interpretations For Fibonomial Coefficients, Richard Shapley

HMC Senior Theses

We can define Fibonomial coefficients as an analogue to binomial coefficients as F(n,k) = F_nF_n-1 … F­_n-k+1 / F­_kF_k-­1…F₁, where F_n represents the nth Fibonacci number. Like binomial coefficients, there are many identities for Fibonomial coefficients that have been proven algebraically. However, most of these identities have eluded combinatorial proofs.

Sagan and Savage (2010) first presented a combinatorial interpretation for these Fibonomial coefficients. More recently, Bennett et al. (2018) provided yet another interpretation, that is perhaps more tractable. However, there still has been little progress towards using these interpretations …

A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo

HMC Senior Theses

Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …

The Wedge Family Of The Cohomology Of The C-Motivic Steenrod Algebra, Hieu Trung Thai

Wayne State University Dissertations

Computing the stable homotopy groups of the sphere spectrum is one of the most important problems of stable homotopy theory. Focusing on the 2-complete stable homotopy groups instead of the integral homotopy groups, the Adams spectral sequence appears to be one of the most effective tools to compute the homotopy groups. The spectral sequence has been studied by J. F. Adams, M. Mahowald, M. Tangora, J. P. May and others.

In 1999, Morel and Voevodsky introduced motivic homotopy theory. One of its consequences is the realization that almost any object studied in classical algebraic topology could be given a motivic …

Variational Analysis In Second-Order Cone Programming And Applications, Hang Thi Van Nguyen

Wayne State University Dissertations

This dissertation conducts a second-order variational analysis for an important class on nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone. These second-order cone programs (SOCPs) are mathematically challenging due to the nonpolyhedrality of the underlying second-order cone while being important for various applications. The two main devices in our study are second epi-derivative and graphical derivative of the normal cone mapping which are proved to accumulate vital second-order information of functions/constraint systems under investigation. Our main contribution is threefold:

- proving the twice epi-differentiability of the indicator function of the second-order cone and of the augmented Lagrangian associated with …

The Chebyshev Difference Equation, Tom Cuchta

Mathematics Faculty Research

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations.

A Description Of A Humans Knowledge Using Artificial Intelligence, Dj Price

Mahurin Honors College Capstone Experience/Thesis Projects

There currently does not exist a way to easily view the relationships between a collection of written items (e.g. sports articles, diary entries, research papers). In recent years, novel machine learning methods have been developed which are very good at extracting semantic relationships from large numbers of documents. One of them is the (unsupervised) machine learning model Doc2Vec which constructs vectors for documents. The research project detailed in this paper uses this and other already existing algorithms to analyze the relationship between pieces of text. We set forth a broader ambition for this project before discussing the use and need …

The Mathematics, Computer Science, And Data Science Student Research Showcase, Seton Hall University

Petersheim Academic Exposition

No abstract provided.

Petersheim Academic Exposition 2020 Math & Computer Science Abstracts, Seton Hall University

Petersheim Academic Exposition

No abstract provided.

Algebraic And Combinatorial Approaches For Counting Cycles Arising In Population Biology, Brian Chau

Honors Undergraduate Theses

Within population biology, models are often analyzed for the net reproduction number or other generalized target reproduction numbers, which describe the growth or decline of the population based on specific mechanisms. This is useful in determining the strength and efficiency of control measures for inhibiting or enhancing population growth. The literature contains many algebraic and combinatorial approaches for deriving the net reproduction number and generalized target reproduction numbers from digraphs and associated matrices. Finding, categorizing, and counting the permutations of disjoint cycles, or cycles unions is a requirement of the Cycle Union approach by Lewis et al. (2019). These cycles …

Preliminary Study Of Highway Pavement And Materials, Omer Eljairi

CGU Theses & Dissertations

This preliminary study covered (a) the effects of in-place air voids and other factors on fatigue cracking using Long-Term Pavement Performance data, (b) fracture properties of asphalt concrete in a semicircular bend (SCB) test using a noncontact camera and crosshead movement, and (c) hot applied modified-binder-chip-seal field performance in California. The objective is to improve pavement performance and life, establish a quality assurance/quality control (QA/QC) tests of fracture properties of asphalt mixtures, and save millions of dollars on maintenance and rehabilitation. Chapter 1 investigated the effect of in-place air voids (AV), asphalt content (AC), bulk-specific gravity (BSG), and maximum specific …

Comparative Survival Of Asian And White Metastatic Castration-Resistant Prostate Cancer Men Treated With Docetaxel, Susan Halabi, Sandipan Dutta, Catherine M. Tangen, Mark Rosenthal, Daniel P. Petrylak, Ian M. Thompson Jr., Kim N. Chi, Johann S. De Bono, John C. Araujo, Christopher Logothetis, Mario A. Eisenberger, David I. Quinn, Karim Fizazi, Michael J. Morris, Celestia S. Higano, Ian F. Tannock, Eric J. Small, William Kevin Kelly

Mathematics & Statistics Faculty Publications

There are few data regarding disparities in overall survival (OS) between Asian and white men with metastatic castration-resistant prostate cancer (mCRPC). We compared OS of Asian and white mCRPC men treated in phase III clinical trials with docetaxel and prednisone (DP) or a DP-containing regimen. Individual participant data from 8820 men with mCRPC randomly assigned on nine phase III trials to receive DP or a DP-containing regimen were combined. Men enrolled in these trials had a diagnosis of prostate adenocarcinoma. The median overall survival was 18.8 months (95% confidence interval [CI] = 17.4 to 22.1 months) and 21.2 months (95% …

Quantitative Analysis Of A Stochastic Seitr Epidemic Model With Multiple Stages Of Infection And Treatment, Olusegun M. Otunuga, Mobolaji O. Ogunsolu

Mathematics Faculty Research

We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining R_T,n and ℛ_T,n as the basic deterministic and stochastic reproduction …

Dynamic Gompertz Model, Tom Cuchta, Sabrina Streipert

Mathematics Faculty Research

After a brief introduction that includes some fundamentals of time scales, we lay the foundation for dynamic Gompertz models. We derive their unique solutions, present examples in the discrete, quantum, and mixed time scale settings, and we compare its behavior to the solution in the continuous time setting. A discussion of the results and open problems are addressed in the conclusion.

Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes

All Graduate Theses, Dissertations, and Other Capstone Projects

This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.

Fuzzy Logistic Regression For Detecting Differential Dna Methylation Regions, Tarek M. Bubaker Bennaser

Doctoral Dissertations

“Epigenetics is the study of changes in gene activity or function that are not related to a change in the DNA sequence. DNA methylation is one of the main types of epigenetic modifications, that occur when a methyl chemical group attaches to a cytosine on the DNA sequence. Although the sequence does not change, the addition of a methyl group can change the way genes are expressed and produce different phenotypes. DNA methylation is involved in many biological processes and has important implications in the fields of biomedicine and agriculture.

Statistical methods have been developed to compare DNA methylation at …

Investigating The Numerical Stability Of Using An Impedance Boundary Condition To Model Broadband Noise Scattering With Acoustic Liners, Michelle E. Rodio, Fang Q. Hu, Douglas M. Nark

Mathematics & Statistics Faculty Publications

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance …

Math Across The Curriculum, Kelsey Mason

Masters Essays

No abstract provided.

Stochastic Approximation And Applications To Networked Systems, Thu Thi Le Nguyen

Wayne State University Dissertations

This dissertation focuses on a class of SA algorithms with applications to networked systems and is based on the published works that have been done jointly during my Ph.D. training. The networked systems are fundamentally characterized by interaction among control, communications, and computing, with applications in a vast array of emerging technologies such as smart grids, intelligent transportation systems, social networks, smart city, to name just a few. Networked systems encounter many environment uncertainties that are inherently stochastic. Besides the aforementioned advantages, the framework of SA can also accommodate multiple random processes and diversified system dynamics, even random and distributed …

A Mathematical Modeling Of Infrared Neural Stimulation, Cesil S. Alex

Undergraduate Journal of Mathematical Modeling: One + Two

Electrical stimulation is the gold standard for artificial neural stimulation. The greatest disadvantage with electrical stimulation is that it scatters in space and it is difficult to achieve specific point stimulation. Recently, infrared stimulation attracted attention to address this issue. Infrared stimulation works on the principle of heating the tissue, exploiting the energy of infrared lasers to heat the cellular aqueous solution that helps transfer the energy to the cell membrane without direct contact, and provides a discrete localization of stimulation as it does not spread in space like electric fields. In the present study, a heat transfer model for …

Calculating The Area Within The Orbit Of Arrokoth, Aurielle Collins

Undergraduate Journal of Mathematical Modeling: One + Two

The area within the orbit of the Kuiper Belt object (KBO) Arrokoth (or 2014MU₆₉) is approximated using the corresponding ellipse. The fact that the Sun lies at a focus of the ellipse is of interest. Attention is made to the known deviations of orbits from their models. It is at least of interest that the ellipse model is still essentially valid. Theoretically, if there were only the Sun and Arrokoth in the Universe, the orbit would be precisely described by the ellipse model.