Mathematics Commons^™

A New Public-Key Cryptosystem, Christopher James Hettinger

Theses and Dissertations

Public key cryptosystems offer important advantages over symmetric methods, but the most important such systems rely on the difficulty of integer factorization (or the related discrete logarithm problem). Advances in quantum computing threaten to render such systems useless. In addition, public-key systems tend to be slower than symmetric systems because of their use of number-theoretic algorithms. I propose a new public key system which may be secure against both classical and quantum attacks, while remaining simple and very fast. The system's action is best described in terms of linear algebra, while its security is more naturally explained in the context …

Eigenvalue Estimates Of Laplacians Defined By Fractal Measures, Sze-Man Ngai

Mathematical Sciences: Faculty Presentations (1991-2022)

We study various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Euclidean spaces. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. By using properties of self-similar measures, such as Strichartz's second-order self-similar identities, we improve some of the eigenvalue estimates.

Evaluation Of Bias-Variance Trade-Off For Commonly Used Post-Summarizing Normalization Procedures In Large-Scale Gene Expression Studies, Zhixin Wu, Rui Hu, Xing Qiu

Mathematics Faculty Publications

Normalization procedures are widely used in high-throughput genomic data analyses to remove various technological noise and variations. They are known to have profound impact to the subsequent gene differential expression analysis. Although there has been some research in evaluating different normalization procedures, few attempts have been made to systematically evaluate the gene detection performances of normalization procedures from the bias-variance trade-off point of view, especially with strong gene differentiation effects and large sample size. In this paper, we conduct a thorough study to evaluate the effects of normalization procedures combined with several commonly used statistical tests and MTPs under different …

The Control Of The False Discovery Rate Under Structured Hypotheses, Gavin Lynch

Dissertations

The hypotheses in many multiple testing problems often have some inherent structure based on prior information such as Gene Ontology in gene expression data. However, few false discovery rate (FDR) controlling procedures take advantage of this inherent structure. In this dissertation, we develop FDR controlling methods which exploit the structural information of the hypotheses.

First, we study the fixed sequence structure where the testing order of the hypotheses has been pre-specified. We are motivated to study this structure since it is the most basic of structures, yet, it has been largely ignored in the literature on large scale multiple testing. …

Electro-Deformation Of A Moving Boundary: A Drop Interface And A Lipid Bilayer Membrane, Herve Nganguia

Dissertations

This dissertation focuses on the deformation of a viscous drop and a vesicle immersed in a (leaky) dielectric fluid under an electric field. A number of mathematical tools, both analytical and numerical, are developed for these investigations. The dissertation is divided into three parts. First, a large-deformation model is developed to capture the equilibrium deformation of a viscous spheroidal drop covered with non-diffusing insoluble surfactant under a uniform direct current (DC) electric field. The large- deformation model predicts the dependence of equilibrium spheroidal drop shape on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Results from the …

Data Analysis And Simulation For Warranties And Golf Handicaps, Sonia Bandha

Dissertations

In this dissertation, we discuss the application of data analysis and numerical simulation in order to gain insight into problems related to warranty cost management and the effectiveness of the golf handicap system. Despite the commonalities of the approaches, we will discuss these problems in turn.

For many moderately high value items with a substantial sales volume (such as automobiles), a warranty is used as an important element of marketing products as a better warranty typically signals higher product quality to customers. Much recent research on modeling and optimization of servicing costs for Non-Renewing Free Replacement Warranties (NR-FRW) assumes that …

Fwer Controlling Procedures For Testing Multiple Hypotheses With Hierarchical Structure And Applications In Clinical Trials, Zhiying Qiu

Dissertations

In applications of clinical trials, the hypotheses to be tested often exhibit a hierarchical structure, and are usually hierarchically ordered based on their importance, clinical relevance, or dose concentration, etc. Thus, they are tested in a pre-defined fixed sequence. In some more complex cases, the hypotheses to be tested are hierarchically grouped into several families, and thus the families are tested in a sequential order. Although such problems of structured multiple testing have received much attention and several popular FWER controlling procedures, such as conventional fixed sequence procedure, fallback procedure and gatekeeping procedure have been introduced, not much progress has …

Inverse Methods For Sound Speed Estimation In The Ocean, Tao Lin

Dissertations

This dissertation presents theoretical and computational approaches for estimating sound-speed in the ocean under different conditions. The first part of the dissertation discusses a fast approach for solving the inverse problem of estimating sediment sound-speed based on the Deift-Trubowitz trace formula. Under certain assumptions, this algorithm can recover the sound speed profile in the seabed using pressure field measurements in the water column at low frequencies. The inversion algorithm requires solving a non-linear integral equation. In the past, Stickler and Zhou employed a first order Born approximation for solving the integral equation. This work introduces two new methods. The first …

The Tame-Wild Principle For Discriminant Relations For Number Fields, John W. Jones, David P. Roberts

Mathematics Publications

Consider tuples ( K₁ , … , K_r ) of separable algebras over a common local or global number field F₁, with the K_i related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of K_i ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.

Count Like An Egyptian: A Hands-On Introduction To Ancient Mathematics (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: Reimer, David. Count LIke an Egyptian: A Hand-On Introduction to Ancient Mathematics. Princeton, NJ: Princeton University Press, 2014. 237 pages. ISBN 9780691160122

Why You Need A Summer Plan, Gizem Karaali

Pomona Faculty Publications and Research

In the last decade, for many times, I have tasted first-hand the end-of-summer blues I described above. I have spent many early spring months dreaming of all that I would be doing when the summer arrives, only to realize that it was already September and I had not much to show for for the months in between. I have also observed many of my peers going through similar things, and I just assumed for years that this was how it had to be. But then some time in the middle of the tenure track, I decided to try approaching my …

Vanishing Configurations In Network Dynamics With Asynchronous Updates, Ian H. Dinwoodie

Mathematics and Statistics Faculty Publications and Presentations

We consider Boolean dynamics for biological networks where stochasticity is introduced through asynchronous updates. An exact method is given for finding states which can reach a steady state with positive probability, and a method is given for finding states which cannot reach other steady states. These methods are based on computational commutative algebra. The algorithms are applied to dynamics of a cell survival network to determine node assignments that exclude termination in a cancerous state

A Note On Independent Sets In Graphs With Large Minimum Degree And Small Cliques, Jeremy Lyle

Faculty Publications

Graphs with large minimum degree containing no copy of a clique on r vertices (K_r) must contain relatively large independent sets. A classical result of Andrásfai, Erdös, and Sós implies that K_r-free graphs G with degree larger than ((3r − 7)/(3r − 4))|V(G)| must be (r − 1)-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order (1/(r − 1))|V(G)|. The following paper provides improved bounds on the minimum degree which would imply the …

Franklin Finds A Home : An Original Mathematical Story For Children Seven To Nine Years Of Age, Rachel Greenblat

Graduate Student Independent Studies

This original story is written for children seven to nine years of age. In the story, the reader accompanies Franklin, a fish in a fantastical underwater world, on a journey to find a new apartment. Using mathematical thinking, Franklin is able to find the perfect apartment. The mathematical concepts embedded in the story are measurement, length, width, perimeter, and area.

Dynamics Of Climate Change: Explaining Glacier Retreat Mathematically, Robert Guillette

Honors Program Theses and Projects

No abstract provided.

Free Energies For Materials With Memory In Terms Of State Functionals, John Murrough Golden

Articles

Abstract The aim of thiswork is to determinewhat free energy functionals are expressible as quadratic forms of the state functional It which is discussed in earlier papers. The single integral form is shown to include the functional wF proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a newsystematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to …

Basis Function Approaches For Two Dimensional Cochlear Models, Lihua Li

Dissertations

The human cochlea possesses the amazing ability of analyzing audio signals. The structures and mechanisms behind its characteristic response to sound stimuli has been an active area of research for decades. It has been demonstrated that mathematical cochlear modeling poses a promising alternative to discover the elusive activities in an in vivo cochlea. However, despite the successful application of numerical methods such as the Wentzel-Kramers-Brillouin (WKB) method, finite difference method (FDM) and finite element method (FEM), the critical effects of the choice of basis functions have not been studied exclusively for the numerical solutions of cochlea models. This work presents …

Generalized Mandelbrot Sets, Aaron Schlenker

Undergraduate Honors Thesis Collection

A complex point Z₀ is defined to be a member of the famous Mandelbrot set fractal when the iterative process using the function Z² stays bounded when applied to Z₀. We investigate what happens if we change the iterative process so that Z² is now composed with, for example, a Mobius transformation, indexed on a parameter a. The Mandelbrot set corresponds to a = O. What happens when we change a = 0 to other values, repeating the iterative process and then drawing the sets? Do these Generalized Mandelbrot sets have similar properties to …

Ranking World Class Chess Players Using Only Results From Head-To-Head Games, Sterling Swygert

Senior Theses

This honors thesis explores a method of ranking the world’s top ten chess grand- masters using only the outcomes of games containing only players in that very set. This method allows for players in a single era to be quickly ranked via algorithmic and numerical means, including very specific information, from a statistical stand- point. Furthermore, unlike the rating systems that are commonly used, the Elo and the Glicko systems, this method is Classicist in its statistical approach, rather than Bayesian. Finally, this ranking method also differs from others as it limits the infor- mation to games between the individuals …

Improving Image Clustering Using Sparse Text And The Wisdom Of The Crowds, Anna Ma, Arjuna Flenner, Deanna Needell, Allon G. Percus

CMC Faculty Publications and Research

We propose a method to improve image clustering using sparse text and the wisdom of the crowds. In particular, we present a method to fuse two different kinds of document features, image and text features, and use a common dictionary or “wisdom of the crowds” as the connection between the two different kinds of documents. With the proposed fusion matrix, we use topic modeling via non-negative matrix factorization to cluster documents.

Review: Truncated Toeplitz Operators Of Finite Rank, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Surface Reconstruction Using Differential Invariant Signatures, Sophors Khut

Mathematics, Statistics, and Computer Science Honors Projects

This thesis addresses the problem of reassembling a broken surface. Three di- mensional curve matching is used to determine shared edges of broken pieces. In practice, these pieces may have different orientation and position in space, so edges cannot be directly compared. Instead, a differential invariant signature is used to make the comparison. A similarity score between edge signatures determines if two pieces share an edge. The Procrustes algorithm is applied to find the translations and rotations that best fit shared edges. The method is implemented in Matlab, and tested on a broken spherical surface.

Hurdle Models And Age Effects In The Major League Baseball Draft, Justin Sims

Mathematics, Statistics, and Computer Science Honors Projects

Major League Baseball (MLB) franchises expend an abundance of resources on scouting in preparation for the June Amateur Draft. In addition to the classic "tools" assessed, another factor considered is age: younger players may get selected over older players of equal ability because of anticipated development, whereas college players may get selected over high school players due to a shortened latency before reaching the majors. Additionally, Little League rules in effect until 2006 operated on an August 1-July 31 year, meaning that, in their youth, players born on August 1 were the eldest relative to their cohort. We examine the …

Parallel Design Patterns And Program Performance, Yu Zhao

Mathematics, Statistics, and Computer Science Honors Projects

With the rapid advancement of parallel and distributed computing (PDC), three types of hardware and their corresponding software (hardware-software pairs) are becoming more and more popular: Distributed Memory Systems with the Message Passing Interface (MPI) library, Shared Memory Systems with the OpenMP library and Co-processor Systems with a general purpose parallel computing library. Alongside the development of both hardware and software aspects of PDC, the process of designing parallel programs has also improved significantly over the years. A consequence of this is that researchers have been able to describe many parallel design patterns, which are recurring solutions to well-known problems …

The Effects Of The Melting Arctic Ice Cap On Florida's Coast, Anthony Atkinson, Nicholas Joseph, Louw Scheepers

CLAS: Colby Liberal Arts Symposium

We've set out to determine the effect of arctic ice melt on the coastline of Florida, more specifically its major cities and tourist attractions. To do this, we created multiple mathematical models on different bases to ensure that our models are behaving in an appropriate manner. We constructed one statistical model based on past sea level rise data, and one theoretical model based on our interpretations of different components that add to the total sea level rise. Our data and analysis showed firm indications of extreme effects on the coastline of Florida and some of its major tourist attractions by …

Modeling The Optimization Of A Toll Booth Plaza, Liam Connell, Zachary Schutzman, Erin Hoover

CLAS: Colby Liberal Arts Symposium

When designing a toll plaza, many factors have to be considered, especially the number of toll booths to build. In this paper, we try to determine the optimal number of toll booths by creating a model for car movement at a toll plaza. We define optimal as the number of toll booths that minimizes the waiting time, particularly at a high congestion time (i.e. rush hour). To model the movement of cars through a toll booth plaza, we break the problem into two parts: time spent waiting in line and then time spent merging back into traffic. Because each of …

Mathematical Analysis Of A Nim-Like Matrix Game, Edward Chuang, Stephen Morse

CLAS: Colby Liberal Arts Symposium

This poster presentation covers our research on a mathematical game that is somewhat similar to the popular game Nim. In this game, Players play a variation of Nim where the number of piles must be a perfect square and where the piles are actually situated within a matrix. In addition, players are only allowed to subtract one from an entry on their turn, where as in Nim there are typically more options. The game is over either when a player does not have a move to make (because all entries have been reduced to zero) or the when the game …

The Highway Game: Optimizing The Number Of Tollbooths, Meaghan Lewia, Tara Chizinski, Megan Lasher

CLAS: Colby Liberal Arts Symposium

One of the biggest woes of highway drivers is the amount of tolls along the road that require each vehicle to not only slow down significantly, but also pay sometimes up to -8 to the State. This inconvenience is protested on various levels, but since its clear that the government must institute some type of toll system, we have decided to find a model that will at least optimize the time for each driver while still saving the State as much money as possible. Our report constructs a model of the ideal amount of tollbooths for any number of highway …

Oscillation Theorems For Fourth-Order Half-Linear Delay Dynamic Equations With Damping, Ravi P. Agarwal, Martin Bohner, Tongxing Li, Chenghui Zhang

Mathematics and Statistics Faculty Research & Creative Works

This article is concerned with oscillatory behavior of a class of fourth-order half-linear delay dynamic equations with damping on a time scale. Some new oscillation criteria are established. © 2013 Springer Basel.

Axioms Of Set Theory And Equivalents Of Axiom Of Choice, Farighon Abdul Rahim

Mathematics Undergraduate Theses

Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip's that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be …