Mathematics Commons^™

Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

Theses

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the …

Parameter Estimation In Nonlinear Regression: Exploring Confidence Intervals For Estimated Coefficients, Brandon Williams

Departmental Honors & Graduate Capstone Projects

Previously, we explored generating data using four different curved source functions with normally distributed errors and then fitting various curved models to this generated data. This research aimed to study the reliability of the goodness-of-fit measures, such as R² and AIC, when the data originates from a curved source model. Ultimately, we found that, for logarithmic source models, AIC picks the correct source model most often, while R² selects either the logarithmic or quadratic source models. For quadratic source models, AIC picks the power source model most often, whereas R² selects the correct source model. Lastly, we …

Qualitative Analysis Of Corequisite Instruction In A Quantitative Reasoning Course, Zachary Beamer

Inquiry: The Journal of the Virginia Community Colleges

In corequisite models of instruction, marginally prepared students are placed directly into college-level coursework, taught with a paired support course. Initial research suggests that such models yield significant improvements in the number of students passing credit-level mathematics when compared to previous models of prerequisite remediation. The present study employs qualitative methods to investigate methods of instruction at one community colleges to understand how instructors identify and respond to student needs. It concludes with recommendations for practice and highlights advantages of small format corequisite classes taught by the same instructor.

Simulations Of An Attack On Rsa, Makayla Ferrell

Departmental Honors & Graduate Capstone Projects

We introduce a variation of Shanks’ baby-step/giant-step (BSGS) to carry out a chosen plaintext attack on RSA. The original usage of BSGS is to find a solution to the discrete log problem. Although BSGS is guaranteed to find a solution, it is not guaranteed to find the matching RSA private key. Our variation of BSGS finds the matching private key d by using the order of the plaintext message to generate a set of solutions. We then use the length of this set to determine the most likely candidate for φ(n). We use a large data set to test the …

Krein Reproducing Kernel Modules In Clifford Analysis, Daniel Alpay, Paula Cerejeiras, Uwe Kähler

Mathematics, Physics, and Computer Science Faculty Articles and Research

Classic hypercomplex analysis is intimately linked with elliptic operators, such as the Laplacian or the Dirac operator, and positive quadratic forms. But there are many applications like the crystallographic X-ray transform or the ultrahyperbolic Dirac operator which are closely connected with indefinite quadratic forms. Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry. In this paper we are going to show that Clifford-Krein modules are naturally appearing in this context. Even taking into account the difficulties, e.g., the existence of different inner products …

Second-Order Semi-Discretized Schemes For Solving Stochastic Quenching Models On Arbitrary Spatial Grids, Nina Garcia-Montoya, Julienne Kabre, Jorge E. Macías-Díaz, Qin Sheng

Mathematics Faculty Articles

Reaction-diffusion-advection equations provide precise interpretations for many important phenomena in complex interactions between natural and artificial systems. This paper studies second-order semi-discretizations for the numerical solution of reaction-diffusion-advection equations modeling quenching types of singularities occurring in numerous applications. Our investigations particularly focus at cases where nonuniform spatial grids are utilized. Detailed derivations and analysis are accomplished. Easy-to-use and highly effective second-order schemes are acquired. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and capability of the new methods for solving singular quenching problems on arbitrary grid platforms.

Time Series Forecasting Of Covid-19 Deaths In Massachusetts, Andrew Disher

Honors Program Theses and Projects

The aim of this study was to use data provided by the Department of Public Health in the state of Massachusetts on its online dashboard to produce a time series model to accurately forecast the number of new confirmed deaths that have resulted from the spread of CoViD-19. Multiple different time series models were created, which can be classified as either an Auto-Regressive Integrated Moving Average (ARIMA) model or a Regression Model with ARIMA Errors. Two ARIMA models were created to provide a baseline forecasting performance for comparison with the Regression Model with ARIMA Errors, which used the number of …

Application Of Randomness In Finance, Jose Sanchez, Daanial Ahmad, Satyanand Singh

Publications and Research

Brownian Motion which is also considered to be a Wiener process and can be thought of as a random walk. In our project we had briefly discussed the fluctuations of financial indices and related it to Brownian Motion and the modeling of Stock prices.

Interfacial Dynamics And Ionic Transport Of Radiologic Contrast Media In Carbohydrate Matrix: Utility And Limits Of X-Ray Imaging, Lin Mousa, Hayley Sanchez, Subhendra Sarkar, Zoya Vinokur

Publications and Research

Hello, our names are Lin Mousa and Hayley Sanchez, this semester we participated in a research project dedicated to analyzing the interactions of contrast media with the molecular components of fruits to compare how they would react with the human brain. This project involved the injection of fruits with varying contrasts and the imaging of the diffusion and interactions of the contrast within the fruits with X-rays. With setup technical parameters on the x-ray equipment images were taken with identical setups at an hourly rate for several days. The final results of this experiment indicated that contrasts such as Gadolinium …

Exact Solutions To Optimal Control Problems For Wiener Processes With Exponential Jumps, Mario Lefebvre

Journal of Stochastic Analysis

No abstract provided.

Potential Effects Of Autonomous Vehicles On The Insurance Industry, Zachery Trump

Senior Honors Theses

The implementation of autonomous vehicles, or self-driving cars, promises to radically change much of the normal way of life. While it may seem inconsequential to start small with a vehicle of relatively low level of automation, there are many factors to consider. Some of these factors include security, moral dilemmas, and even the insurance field. One can look back at previous implementations of new technology, such as air bags, and see that it can be difficult to predict consequences and adapt. However, actuaries have been suggesting solutions to make autonomous vehicles a safe reality. While the solutions may vary, one …

Revolt Pimenov Taught Us How To Be Scientists, Vladik Kreinovich

Departmental Technical Reports (CS)

In 2021, we are celebrating the 90th birthday of Revolt Pimenov, a specialist in space-time geometry. He was my teacher. In this article, I am trying to summarize what he taught to his students.

A Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte

Dissertations

Krylov Subspace Spectral (KSS) Methods have demonstrated to be highly scalable methods for PDEs. However, a current limitation of these methods is the requirement of a rectangular or box-shaped domain. Smooth Extension Embedding Methods (SEEM) use fictitious domain methods to extend a general domain to a simple, rectangular or box-shaped domain. This dissertation describes how these methods can be combined to extend the applicability of KSS methods, while also providing a component-wise approach for solving the systems of equations produced with SEEM.

Why Kappa Regression?, Julio C. Urenda, Orsolya Csiszár, József Dombi, György Eigner, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent book provide examples that a new class of probability distributions and membership functions -- called kappa-regression distributions and membership functions -- leads to better data processing results than using previously known classes. In this paper, we provide a theoretical explanation for this empirical success -- namely, we show that these distributions are the only ones that satisfy reasonable invariance requirements.

Fuzzy Techniques, Laplace Indeterminacy Principle, And Maximum Entropy Approach Explain Lindy Effect And Help Avoid Meaningless Infinities In Physics, Julio C. Urenda, Sean R. Aguilar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, the only information that we have about some quantity S is a lower bound T ≤ S. In such a situation, what is a reasonable estimate for S? For example, we know that a company has survived for T years, and based on this information, we want to predict for how long it will continue surviving. At first glance, this is a type of a problem to which we can apply the usual fuzzy methodology -- but unfortunately, a straightforward use of this methodology leads to a counter-intuitive infinite estimate for S. There is an empirical …

What Is Wrong With Micromanagement: Economic View, Sean R. Aguilar, Olga Kosheleva

Departmental Technical Reports (CS)

Purpose: It is well known that micromanagement -- excessive control of employees -- is detrimental to the employees' morale and thus, decreases their productivity. But what if the managers keep people happy -- will there still be negative consequences of micromanagement? This is the problem analyzed in this paper.

Design/methodology/approach: To analyze our problem, we use general -- but simplified -- mathematical models of how productivity depends on the working rate.

Findings: We show that even in the absence of psychological discomfort, micromanagement is still detrimental to productivity. Interestingly, the negative effect of micromanagement increases as the population becomes more …

Is Our World Becoming Less Quantum?, Lidice Castro, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the general idea of quantization, all physical dependencies are only approximately deterministic, and all physical "constants" are actually varying. A natural conclusion -- that some physicists made -- is that Planck's constant (that determines the magnitude of quantum effects) can also vary. In this paper, we use another general physics idea -- the second law of thermodynamics -- to conclude that with time, this constant can only decrease. Thus, with time (we are talking cosmological scales, of course), our world is becoming less quantum.

Order Relations Are Ubiquitously Fundamental: Alexandrov(-Zeeman) Theorem Extended From Space-Time Physics To Logical Reasoning And Decision Making, Vladik Kreinovich, Olga Kosheleva, Laxman Bokati, Laura Berrout

Departmental Technical Reports (CS)

In all areas of human activity, there are natural ordering relations: causality in space-time physics, preference in decision making, and logical inference in reasoning. In space-time physics, a 1950 theorem by A. D. Alexandrov proved that causality relation is fundamental: many other features, including numerical characteristics of time and space, can be reconstructed from this relation. In this paper, we provide simple proofs that, similarly, the corresponding ordering relations are fundamental in decision making and in logical reasoning.

Shall We Ignore All Intermediate Grades?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most European universities, the overall student's grade for a course is determined exclusively by this student's performance on the final exam. All intermediate grades -- on homework, quizzes, and previous texts -- are, in effect, ignored. This arrangement helps gauge the student's performance by the knowledge that the student shows at the end of the course. The main drawback of this approach is that some students do not start studying until later, thinking that they can catch up and even get an excellent grade -- and this hurts their performance. To motivate students to study hard throughout the semester, …

Extension To Multidimensional Problems Of A Fuzzy-Based Explainable & Noise-Resilient Algorithm, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

While Deep Neural Networks (DNNs) have shown incredible performance in a variety of data, they are brittle and opaque: easily fooled by the presence of noise, and difficult to understand the underlying reasoning for their predictions or choices. This focus on accuracy at the expense of interpretability and robustness caused little concern since, until recently, DNNs were employed primarily for scientific and limited commercial work. An increasing, widespread use of artificial intelligence and growing emphasis on user data protections, however, motivates the need for robust solutions with explainable methods and results. In this work, we extend a novel fuzzy based …

Applying Emotional Analysis For Automated Content Moderation, John Shelnutt

Computer Science and Computer Engineering Undergraduate Honors Theses

The purpose of this project is to explore the effectiveness of emotional analysis as a means to automatically moderate content or flag content for manual moderation in order to reduce the workload of human moderators in moderating toxic content online. In this context, toxic content is defined as content that features excessive negativity, rudeness, or malice. This often features offensive language or slurs. The work involved in this project included creating a simple website that imitates a social media or forum with a feed of user submitted text posts, implementing an emotional analysis algorithm from a word emotions dataset, designing …

Why Too Much Interaction Between Different Parts Of The Brain Leads To Unhappiness, Ricardo Alvarez, Yamel Hernandez, Vladik Kreinovich

Departmental Technical Reports (CS)

Reasonably recent experiments show that unhappiness is strongly correlated with the excessive interaction between two parts of the brain -- amygdala and hippocampus. At first glance, in situations when outside signals are positive, additional interaction between two parts of the brain that get signals from different sensors should only reinforce the positive feeling. In this paper, we provide a simple explanation of why, instead of the expected reinforcement, we observe unhappiness.

The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall

University Scholar Projects

The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all …

Efficient Inversion Of 2.5d Electrical Resistivity Data Using The Discrete Adjoint Method, Diego Domenzain, John Bradford, Jodi Mead

Mathematics Faculty Publications and Presentations

We have developed a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We have used the adjoint method directly in the discretized Maxwell’s steady state equation to compute …

Randomized Tax Deadlines Can Help Economy, Julio C. Urenda, Olga Kosheleva

Departmental Technical Reports (CS)

Purpose: While the main purpose of reporting -- e.g., reporting for taxes -- is to gauge the economic state of a company, the fact that reporting is done at pre-determined dates distorts the reporting results. For example, to create a larger impression of their productivity, companies fire temporary workers before the reporting date and re-hire then right away. The purpose of this study is to decide how to avoid such distortion.

Design/methodology/approach: We want to make our solution applicable for all possible reasonable optimality criteria. Thus, we use a general formalism for describing and analyzing all such criteria.

Findings: We …

Godel's Proof Of Existence Of God Revisited, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his unpublished paper, the famous logician Kurt Godel provided arguments in favor of the existence of God. These arguments are presented in a very formal way, which makes them difficult to understand to many interested readers. In this paper, we describe a simplifying modification of Godel's proof which will hopefully make it easier to understand. We also describe, in clear terms, why Godel's arguments are just that -- arguments -- and not a convincing proof.

Five Revolutionary Ideas In The 1950s-70s Science: 90th Birthday Of Revolt Pimenov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

This year, Revolt Pimenov, a philosophical thinker whose main ideas were in geometry of space-time, would have turned 90. In this essay, we explain how in the 1950s-70s, when he was most productive, his were two of the five natural and important revolutionary scientific ideas -- along with fuzzy logic, constructive mathematics, and scalar-tensor theory of gravitation, ideas that, in our opinions, still have potential to change the world.

How To Teach Advanced Highly Motivated Students: Teaching Strategy Of Iosif Yakovlevich Verebeichik, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The paper describes and explains the teaching strategy of Iosif Yakovlevich Verebeichik, a successful mathematics teacher at special mathematical high schools -- schools for students interested in and skilled in mathematics. The resulting strategy seems counterintuitive and contrary to all the pedagogical advice. Our explanation is not complete: it worked well for this teacher, but others who tried to follow seemingly the same strategy did not succeed. How he made it work, how can others make it work -- this is still not clear. In the words of Verebeichik himself, while mathematics itself is a science, teaching mathematics is an …

Some Quadratic Quantum P³S With A Linear One-Dimensional Line Scheme, Ian Christopher Lim

Mathematics Dissertations - Archive

It is believed that quadratic Artin-Shelter regular (AS-regular) algebras of global dimension four (sometimes called quadratic quantum P3s can be classified using a geometry similar to that developed in the 1980’s by Artin, Tate, and Van den Bergh. Their geometry involved studying a scheme (later called the point scheme) that parametrizes the point modules over a graded algebra. The notion of line scheme (which parametrizes line modules) was introduced later by Shelton and Vancliff. It is known that “generic” quadratic quantum P3s have a finite point scheme and one-dimensional line scheme. A family of algebras with these properties is presented …

Option Pricing With Investment Strategy Under Stochastic Interest Rates, Niloofar Ghorbani

Mathematics Dissertations - Archive

Equity options are the most common types of financial derivatives that give an investor the right but not the obligation to buy or sell shares of stock at a given price in the future for a premium (option price) paid at present. The Classical Black- Scholes Formula solved a longstanding mathematical problem of finding no arbitrage option price by means of stochastic Ito calculus based on Geometric Brownian Motion dynamics of the stock price and a fixed interest rate over the option time horizon. We extend the Black-Scholes Model by adding a component of investor’s buying and selling strategies for …