Mathematics Commons^™

Ranking Score Vectors Of Tournaments, Sebrina Ruth Cropper

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Given , a tournament on vertices, Landau derived a method to determine how close is to being transitive or regular. This comparison is based on the tournament’s hierarchy number, ̅, a value derived from its score vector ̅ ( ). Let be the set of all score vectors of tournaments on vertices with the entries listed in non-decreasing order. A partial order, poset, exists on the set using the following binary relation. Given ̅ ̅ such that ̅ ̅, let ̅ ̅ if Σ Σ for and Σ Σ . Let this poset be represented as ( ) ( ) …

Convolution Equations In Spaces Of Distributions Supported By Cones, Alex Meril, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We describe some examples of surjective convolutors on D'(T), for T a closed convex cone in Rn. We also give necessary and suffficient conditions on Si,..., Sm in S'(T) to be generators of the whole convolution algebra S'(F).

Capturing Low Probability Of Disease Dynamics In Coupled Populations, Jackson Burton

Theses, Dissertations and Culminating Projects

Researchers using mathematical models have made significant contributions to the field of epidemiology in recent years. These models have both explanatory and predicative power to describe disease dynamics. More recent work has considered multi-population models and the effects vaccinations have on the population as a whole. One such example can be seen in the West African country of Cameroon, which has two distinct patterns of measles outbreaks. By considering Cameroon as two subpopulations, a deterministic model is developed that includes the effect of vaccinations. Stability analysis is then performed on the model over a range of coupling and vaccination rates …

Continuous Blooming Of Convex Polyhedra, Erik D. Demaine, Martin L. Demaine, Vi Hart, Joan Iacono, Stefan Langerman, Joseph O'Rourke

Computer Science: Faculty Publications

We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.

Row Reduction Of Macaulay Matrices, Lorrin Debenport

Honors Theses

A computer can use a matrix to represent a system of non-linear multivariate polynomial equations. The fastest known ways to transform this system into a form with desirable computational properties rely on transforming its matrix into upper-triangular form [8, 9]. The matrix for such a system will have mostly zero entries, which we call sparse [7]. We propose to analyze several methods of performing row-reduction, the process by which matrices are reduced into upper-triangular form [2].

What is special about row-reducing matrices in this context? When row-reducing a matrix, swapping rows or columns is typically acceptable. However, if the order …

“Drawing” Upon Your Students’ Creativity: Teaching (Your Subject Here) With Comic Books, Bruce Kessler

Mathematics Faculty Publications

During Spring 2009, Dr. Kessler created and published a comic book series that embedded math content into the story for 4th-6th grade students. The comics were well received in the classrooms at Cumberland Trace Elementary. Dr. Kessler contends that this approach to teaching and learning can be used in any content area, and is useful for engaging students who might not be as interested otherwise. This session will explore ways of utilizing the skills of your students to construct learning comics in your classes, regardless of the funds, technology, and artistic experience at your disposal. The session will include a …

2011 Sonia Kovalevsky Math For Girls Day Flyer, Association For Women In Mathematics, Lincoln University Of Missouri, Donna L. Stallings

Math for Girls Day Documents

6th Annual Lincoln University Sonia Kovalevsky Math for Girls Day program flyer on April 29, 2011.

2011 Sonia Kovalevsky Math For Girls Day Evaluation Forms, Association For Women In Mathematics, Lincoln University Of Missouri, Donna L. Stallings

Math for Girls Day Documents

Presenter, student and teacher evaluation forms for the 6th Annual Lincoln University Sonia Kovalevsky Math for Girls Day program flyer on April 29, 2011.

Public Policy, Long Run Growth And Economic Transition From Agriculture To Industrial Mass Production., Rudrani Bhattacharya Dr.

Doctoral Theses

It has been recognized that the heart of the development process lies on the transformation of an economy from traditional activities in agriculture to industrial activities. As emphasized by Lewis (1954), the movement of labor from primary sector to industry is conducive to the rise of economy’s savings and investment rate and thus fostering economic growth. The core of this paradigm is the historically observed rapid growth in today’s developed countries associated with expansion of industrial activities.It is well-documented by now that at least over a few centuries leading up to 1800 AD there was little change in the world …

Integer Optimization And Computational Algebraic Topology, Bala Krishnamoorthy

Systems Science Friday Noon Seminar Series

We present recently discovered connections between integer optimization, or integer programming (IP), and homology. Under reasonable assumptions, these results lead to efficient solutions of several otherwise hard-to-solve problems from computational topology and geometric analysis. The main result equates the total unimodularity of the boundary matrix of a simplicial complex to an algebraic topological condition on the complex (absence of relative torsion), which is often satisfied in real-life applications . When the boundary matrix is totally unimodular, the problem of finding the shortest chain homologous under Z (ring of integers) to a given chain, which is inherently an integer program, can …

Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the intersection of two classical and fundamental areas in graph theory:  graph packing and degree sequences.  The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set.  The most significant result in this area is Kundu's k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence.  We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors.  

Perhaps the most fundamental result in degree sequence theory is the Erdos-Gallai Theorem, characterizing …

Extremal Trees And Reconstruction, Andrew Ray

Department of Mathematics: Dissertations, Theses, and Student Research

Problems in two areas of graph theory will be considered.

First, I will consider extremal problems for trees. In these questions we examine the trees that maximize or minimize various invariants. For instance the number of independent sets, the number of matchings, the number of subtrees, the sum of pairwise distances, the spectral radius, and the number of homomorphisms to a fixed graph. I have two general approaches to these problems. To find the extremal trees in the collection of trees on n vertices with a fixed degree bound I use the certificate method. The certificate is a branch invariant, …

Circle The Cat: Developing Intelligent Strategies For A Simple Two-Player Game, Kirk Baly, Micheal Lay, Owen Prough

Symposium on Research and Creative Expression (SORCE)

Circle the Cat is an adaptation of the classic mathematical puzzle Qudraphage in which a cat attempts to escape a hexagonal board, but after each move one hexagon is blocked. We have coded a simulation of this classic puzzle. The result of our development allows for the implementation of varying strategies to computationally examine their effectiveness. The code itself, though extensive, is open enough for additions/extensions to be added fairly easily. Analysis of simpler strategies has allowed for improvement of both Cat and Player AI systems. We are currently looking at implementing more advanced systems, perhaps turning to machine learning.

Lazy Golfing, Didar Ashkerbekov, Max Hougas

Symposium on Research and Creative Expression (SORCE)

Walking golfers are expected to leave their golf bags just outside the edge of the green while they are putting the ball. We construct a simple model of this situation, in which the green is assumed to be a circle with the hole at the center and the golfer one-putts the ball. (He or she is a great golfer!) We construct a function to determine the distance that the golfer walks from his or her approach shot on the green and then to the next tee box. Given this model, we determine the optimal location for leaving the golf bag …

Annihilators Of Local Cohomology Modules, Laura Lynch

Department of Mathematics: Dissertations, Theses, and Student Research

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.

Adviser: Thomas Marley

Applications Of Hidden Markov Models In Microarray Gene Expression Data, Huimin Geng, Xutao Deng, Hesham Ali

Computer Science Faculty Publications

Hidden Markov models (HMMs) are well developed statistical models to capture hidden information from observable sequential symbols. They were first used in speech recognition in 1970s and have been successfully applied to the analysis of biological sequences since late 1980s as in finding protein secondary structure, CpG islands and families of related DNA or protein sequences [1]. In a HMM, the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to determine the hidden parameters from the observable parameters. In this chapter, we described two applications using HMMs to predict gene functions …

Multiple Solutions For An Elliptic Problem Related To Vortex Pairs, Yi Li, Shuangjie Peng

Mathematics and Statistics Faculty Publications

Let Ω be a bounded domain in RN (N⩾2), φ a harmonic function in Ω¯. In this paper we study the existence of solutions to the following problem arising in the study of vortex pairs(Pλ){−Δu=λ(u−φ)+p−1,x∈Ω,u=0,x∈∂Ω. The set Ωp={x∈Ω,u(x)>φ} is called “vortex core”. Existence of solutions whose “vortex core” consists of one component and asymptotic behavior of “vortex core” were studied by many authors for large λ recently. Under the condition that φ has k strictly local minimum points on the boundary ∂Ω, we obtain in this paper that for λ large enough, (Pλ) has a solution with “vortex core” …

A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger

All HMC Faculty Publications and Research

We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.

A Direct Limit For Limit Hilbert-Kunz Multiplicity For Smooth Projective Curves, Holger Brenner, Jinjia Li, Claudia Miller

Mathematics - All Scholarship

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic …

Elliptic Pairs Of Primes In Cryptography And Their Effects On Rsa Security, Suzanne Craig, Liljana Babinkostova

College of Arts and Sciences Poster Presentations

Cryptography is a constantly evolving field – elliptic curve groups have been a major part of this field since 1987 when Koblitz and Miller separately proposed their use in cryptosystems such as RSA.

• Elliptic Curve: A curve defined by the equation: where A and B are less than a prime p, and where the discriminant is not equal to zero. We are concerned with the curves where A = 0.
• Elliptic Curve Group (ECG): A group whose elements are defined by the elliptic curve and the operation in question. Elliptic curves naturally qualify as groups due simply to their nature. …

Linear Feedback Shift Registers: Pseudo-Random Number Generators And Their Use In Cryptosystems, Michael Perez, Marion Scheepers

College of Arts and Sciences Poster Presentations

Cryptology is the study and application of encrypting and decrypting data so that only the intended recipient and senders can view data. This is important in applications such as online banking and military operations. Linear Feedback Shift Registers, or LFSRs for short, create a pseudo-random number stream that is computationally efficient. LFSRs are used (in conjunction with other methods) to encrypt items such as DVD’s and many wireless signals, including digital TV and radio. Using random methods, a seed value can be created to create a strong random number stream.

Prospective K-8 Teachers’ Knowledge Of Relational Thinking, Marta Magiera, Leigh A. Van Den Kieboom, John Moyer

Mathematics, Statistics and Computer Science Faculty Research and Publications

The goal of this study was to examine two issues: First, pre-service teachers’ ability and inclination to think relationally prior to instruction about the role relational thinking plays in the K-8 mathematics curriculum. Second, to examine task specific variables possibly associated with pre-service teachers’ inclination to engage in relational thinking. The results revealed that preservice teachers engage in relational thinking about equality, however, their inclination to do so is rather limited. Furthermore, they tend to engage in relational thinking more frequently in the context of arithmetic than algebra-related tasks. Pre-service teachers’ inclination to engage in relational thinking appeared to also …

When Is Hyponormality For 2-Variable Weighted Shifts Invariant Under Powers?, Raul E. Curto, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, ℓ) 7→ W (h,ℓ) (α,β) := (T h 1 , T ℓ 2 ) (h, ℓ ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,ℓ) (α,β) is k-hyponormal (all h ≥ 2, ℓ ≥ 1) but W(α,β) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality …

B-Coloring Of Regular Graphs, Taole Zhu, Josh Brown-Kramer, Faculty Advisor

John Wesley Powell Student Research Conference

No abstract provided.

Axiomatic Foundations Of Mathematics, Ryan Melton

Undergraduate Research Conference

No abstract provided.

Wavelet-Based Analysis Of Neutron-Induced Photon Spectral Data, Bruce Kessler, Alexander Barzilov, Phillip Womble

Mathematics Faculty Publications

Neutron-based methods of non-destructive inter- rogation of objects for the purpose of their characterization are well-established techniques, employed in the field of bulk material analysis, contraband detection, unexploded ordnance, etc. The characteristic gamma rays produced in nuclear reactions initiated by neutrons in the volume of the irradiated object (inelastic neutron scattering, thermal neutron capture, and activation) are used for the elemental identification. In many real-world applications, an automated spectral analysis is needed, and many algorithms are used for that purpose. The Applied Physics Institute at Western Kentucky University has recently started to employ a mathematical spectrum analysis technique based on …

Gut Flora Metabolism Of Phosphatidylcholine Promotes Cardiovascular Disease, Zeneng Wang, Elizabeth Klipfell, Brian J. Bennett, Robert Koeth, Bruce S. Levison, Brandon Dugar, Ariel E. Feldstein, Earl B. Britt, Xiaoming Fu, Yoo Mi Chung, Yuping Wu, Phil Schauer, Jonathan D. Smith, Hooman Allayee, W.H. Wilson Tang, Joseph A. Didonato, Aldons J. Lusis, Stanley L. Hazen

Mathematics and Statistics Faculty Publications

Metabolomics studies hold promise for the discovery of pathways linked to disease processes. Cardiovascular disease (CVD) represents the leading cause of death and morbidity worldwide. Here we used a metabolomics approach to generate unbiased small-molecule metabolic profiles in plasma that predict risk for CVD. Three metabolites of the dietary lipid phosphatidylcholine—choline, trimethylamine N-oxide (TMAO) and betaine—were identified and then shown to predict risk for CVD in an independent large clinical cohort. Dietary supplementation of mice with choline, TMAO or betaine promoted upregulation of multiple macrophage scavenger receptors linked to atherosclerosis, and supplementation with choline or TMAO promoted atherosclerosis. Studies using …

Impulsive Semilinear Neutral Functional Differential Inclusions With Multivalued Jumps, Nadjet Abada, Ravi P. Agarwal, Mouffak Benchohra, Hadda Hammouche

Mathematics and System Engineering Faculty Publications

In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Efficient First Order Methods For Linear Composite Regularizers, Andreas Argyriou, Charles A. Micchelli, Massimiliano Pontil, Lixin Shen, Yuesheng Xu

Mathematics - All Scholarship

A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed …

On Random Topological Structures, Ravi P. Agarwal, Yeolje Cho, Reza Saadati

Mathematics and System Engineering Faculty Publications

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.