Mathematics Commons^™

Contributions To Quandle Theory: A Study Of F-Quandles, Extensions, And Cohomology, Indu Rasika U. Churchill

USF Tampa Graduate Theses and Dissertations

Quandles are distributive algebraic structures that were introduced by David Joyce [24] in his Ph.D. dissertation in 1979 and at the same time in separate work by Matveev [34]. Quandles can be used to construct invariants of the knots in the 3-dimensional space and knotted surfaces in 4-dimensional space. Quandles can also be studied on their own right as any non-associative algebraic structures.

In this dissertation, we introduce f-quandles which are a generalization of usual quandles. In the first part of this dissertation, we present the definitions of f-quandles together with examples, and properties. Also, we provide a …

Mortgage Transition Model Based On Loanperformance Data, Shuyao Yang

Arts & Sciences Graduate Student Theses and Dissertations

The unexpected increase in loan default on the mortgage market is widely considered to be one of the main cause behind the economic crisis. To provide some insight on loan delinquency and default, I analyze the mortgage performance data from Fannie Mae website and investigate how economic factors and individual loan and borrower information affect the events of default and prepaid. Various delinquency status including default and prepaid are treated as discrete states of a Markov chain. One-step transition probabilities are estimated via multinomial logistic models. We find that in general current loan-to-value ratio, credit score, unemployment rate, and interest …

Algorithmic Factorization Of Polynomials Over Number Fields, Christian Schulz

Mathematical Sciences Technical Reports (MSTR)

The problem of exact polynomial factorization, in other words expressing a polynomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used …

Measuring The Value Of Accurate Link Prediction For Network Seeding, Yijin Wei, Gwen Spencer

Mathematics Sciences: Faculty Publications

Merging two classic questions

The influence-maximization literature seeks small sets of individuals whose structural placement in the social network can drive large cascades of behavior. Optimization efforts to find the best seed set often assume perfect knowledge of the network topology. Unfortunately, social network links are rarely known in an exact way. When do seeding strategies based on less-than-accurate link prediction provide valuable insight?

Physiological Health Parameters Among College Students To Promote Chronic Disease Prevention And Health Promotion, David R. Black, Daniel C. Coster, Samantha R. Paige

Mathematics and Statistics Faculty Publications

This study aimed to provide physiologic health risk parameters by gender and age among college students enrolled in a U.S. Midwestern University to promote chronic disease prevention and ameliorate health. A total of 2615 college students between 18 and 25 years old were recruited annually using a series of cross-sectional designs during the spring semester over an 8-year period. Physiologic parameters measured included body mass index (BMI), percentage body fat (%BF), blood serum cholesterol (BSC), and systolic (SBP) and diastolic (DBP) blood pressure. These measures were compared to data from NHANES to identify differences in physiologic parameters among 18-25 year …

Revolution In Ideology: Crafting A Holistic Scientific Dialectic, Nathan Neill

Dialogue & Nexus

Ideology drives scientific research far more than is acknowledged. Since science itself is conducted by individuals, each scientist has a biased conception of themselves and their surroundings relative to the rest of the universe, even if it is never explicated. This sense of relation to the greater universe is what defines the ideology of the individual. It is this sense of relation and self that creates the individual, who goes on to investigate the natural world by the scientific method. In this paper I will examine extant scientific ideology, particularly in Western science, and propose changes that could be helpful.

A Priori Bounds And Existence Of Positive Solutions For Semilinear Elliptic Systems, Nsoki Mavinga, R. Pardo

Mathematics & Statistics Faculty Works

We provide a-priori L∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v^p and u^q for any (p,q) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system.

Connectivity Bounds And S-Partitions For Triangulated Manifolds, Alexandru Ilarian Papiu

Arts & Sciences Graduate Student Theses and Dissertations

Two of the fundamental results in the theory of convex polytopes are Balinski’s Theorem on connectivity and Bruggesser and Mani’s theorem on shellability. Here we present results that attempt to generalize both results to triangulated manifolds. We obtain new connectivity bounds for complexes with certain missing faces and introduce a way to measure how far a manifold is from being shellable using S-partitions and the Stanley-Reisner Ring.

CO-Characterization Of Symplectic And Contact Embeddings And Lagrangian Rigidity, Stefan Müller

Mathematical Sciences: Faculty Publications

We present a novel C0-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by J.-C. Sikorav and Y. Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of C0-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from J-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a C0-characterization of contact embeddings …

An Operational Drought Prediction Framework With Application Of Vine Copula Functions, Mahkameh Zarekarizi

Student Research Symposium

Early and accurate drought predictions can benefit water resources and emergency managers by enhancing drought preparedness. Soil moisture memory is shown to contain helpful information for prediction of future values. This study uses the soil moisture memory to predict their future states via multivariate statistical modeling. We present a drought forecasting framework which issues monthly and seasonal drought forecasts. This framework estimates droughts with different lead times and updates the forecasts when more data become available. Forecasts are generated by conditioning future soil moisture values on antecedent drought status. The statistical model is initialized by soil moisture simulations retrieved from …

Cox Processes For Visual Object Counting, Yongming Ma

Student Research Symposium

We present a model that utilizes Cox processes and CNN classifiers in order to count the number of instances of an object in an image. Poisson processes are well suited to events that occur randomly in space, like the location of objects in an image, as well as to the task of counting. Mixed Poisson processes also offer increased flexibility, however they do not easily scale with image size: they typically require O(n3) computation time and O(n2) storage, where n is the number of pixels. To mitigate this problem, we employ Kronecker algebra which takes advantage of the direct product …

Extremal Graph Theory: Turán’S Theorem, Vincent Vascimini

Honors Program Theses and Projects

No abstract provided.

Symmetric Full Spark Frames, Brian Sheehan

Honors Program Theses and Projects

A full-spark frame of an n-dimensional vector space is a finite collection of m vectors (m ≥ n) with the following property: every subset of cardinality n of the given collection is a basis for the vector space. In this thesis, we realize the symmetric group S_n as a matrix group of invertible matrices with n² entries for n > 2: This representation induces a natural linear action on the vector space ℂⁿ and we prove that S_n admits an orbit which is a full-spark frame if and only if n ≤ 3:

On Tarski's Axiomatic Foundations Of The Calculus Of Relations, Hajnal Andréka, Steven Givant, Peter Jipsen, István Németi

Mathematics, Physics, and Computer Science Faculty Articles and Research

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if …

Tropical Algebra, Graph Theory, & Foreign Exchange Arbitrage, Bradley A. Mason

Senior Honors Projects, 2010-2019

We answer the question, given n currencies and k trades, how can a maximal arbitrage opportunity be found and what is its value? To answer this question, we use techniques from graph theory and employ a max-plus algebra (commonly known as tropical algebra). Further, we show how the tropical eigenvalue of a foreign exchange rate matrix relates to arbitrage among the currencies and can be found algorithmically. We finish by employing time series techniques to study the stability of maximal, high-currency arbitrage opportunities.

From Simplest Recursion To The Recursion Of Generalizations Of Cross Polytope Numbers, Yutong Yang

KSU Journey Honors College Capstones and Theses

My research project involves investigations in the mathematical field of combinatorics. The research study will be based on the results of Professors Steven Edwards and William Griffiths, who recently found a new formula for the cross-polytope numbers. My topic will be focused on "Generalizations of cross-polytope numbers". It will include the proofs of the combinatorics results in Dr. Edwards and Dr. Griffiths' recently published paper. $E(n,m)$ and $O(n,m)$, the even terms and odd terms for Dr. Edward's original combinatorial expression, are two distinct combinatorial expressions that are in fact equal. But there is no obvious algebraic evidence to show that …

Integrating Non-Euclidean Geometry Into High School, John Buda

Honors Thesis

The purpose of this project is to provide the framework for integrating the study of non-Euclidean geometry into a high school math class in such a way that both aligns with the Common Core State Standards and makes use of research-based practices to enhance the learning of traditional geometry. Traditionally, Euclidean geometry has been the only strand of geometry taught in high schools, even though mathematicians have developed several other strands. The non-Euclidean geometry that I focus on in this project is what is known as taxicab geometry. With the Common Core Standards for Math Practice pushing students to “model …

Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, Daniel Alpay, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give a generalization of the Beurling–Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.

Network Modeling Of Infectious Disease: Transmission, Control And Prevention, Christina M. Chandler

Honors College Theses

Many factors come into play when it comes to the transmission of infectious diseases. In disease control and prevention, it is inevitable to consider the general population and the relationships between individuals as a whole, which calls for advanced mathematical modeling approaches.

We will use the concept of network flow and the modified Ford-Fulkerson algorithm to demonstrate the transmission of infectious diseases over a given period of time. Through our model one can observe what possible measures should be taken or improved upon in the case of an epidemic. We identify key nodes and edges in the resulted network, which …

Oscillation Criteria For Third-Order Nonlinear Functional Difference Equations With Damping, Martin Bohner, C. Dharuman, R. Srinivasan, Ethiraju Thandapani

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we obtain some new criteria for the oscillation of certain third-order difference equations using comparison principles with a suitable couple of first-order difference equations. The presented results improve and extend the earlier ones. Examples are provided to illustrate the main results.

Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We always teach students that cramming for a test is a bad idea, that they should study at the same speed throughout the semester – but many still cram. We ourselves are not that different: when we prepare papers for a conference, we often “cram” in the last days before the deadline instead of working with a regular speed for the whole time before the conference. The ubiquity of cramming makes us think that maybe it is not necessarily always a bad idea. And indeed, a simple model of a study process shows that an optimal solution often involve some …

No Idea Is A Bad Idea: A Theoretical Explanation, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Many business publications state that no idea is a bad idea, that even if the idea is, at first glance, not helpful, there are usually some aspects of this idea which are helpful Usually, this statement is based on the experience of the author, and it is given without any theoretical explanation. In this paper, we provide a theoretical explanation for this statement.

An Exploration Of A Quantitative Reasoning Instructional Approach To Linear Equations In Two Variables With Community College Students, Paul T. Belue, Laurie Overman Cavey, Margaret T. Kinzel

Mathematics Faculty Publications and Presentations

In this exploratory study, we examined the effects of a quantitative reasoning instructional approach to linear equations in two variables on community college students’ conceptual understanding, procedural fluency, and reasoning ability. This was done in comparison to the use of a traditional procedural approach for instruction on the same topic. Data were gathered from a common unit assessment that included procedural and conceptual questions. Results demonstrate that small changes in instruction focused on quantitative reasoning can lead to significant differences in students’ ability to demonstrate conceptual understanding compared to a procedural approach. The results also indicate that a quantitative reasoning …

Complementary Coffee Cups, Brandon Sams

Mathematics Undergraduate Theses

In a 2006 paper [1], mathematician Thomas Banchoff posed a new calculus problem, when he filled two cups with coffee. One was convex, while the other was concave, but in such a way so as to fit together perfectly. These cups were thus, complementary. The next natural question he asked was "Which cup has more volume?", so that he could be gracious enough to give the larger of the two to his wife. Banchoff decided to explore this topic, but he left some unanswered questions along the way. Under what conditions would two Complementary Coffee Cups have the same volume? …

Three-Player Gen On Groups, Jon Blomquist

All College Thesis Program, 2016-2019

The three-player game of GEN involves taking turns selecting elements of a group to add to a common pool of elements. Each element can only be selected once, and all players share the pool of elements. The object of the game is to be the player that adds the final element to the pool that will generate the entire group. We will identify and develop the strategies players must follow in order to win the game when playing with cyclic groups, dihedral groups, and nilpotent groups.

Optical Tomography On Graphs, Francis J. Chung, Anna C. Gilbert, Jeremy G. Hoskins, John C. Schotland

Mathematics Faculty Publications

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

Attraction-Repulsion Forces Between Biological Cells: A Theoretical Explanation Of Empirical Formulas, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Biological calls attract and repulse each other: if they get too close to each other, they repulse, and if they get too far away from each other, they attract. There are empirical formulas that describe the dependence of the corresponding forces on the distance between the cells. In this paper, we provide a theoretical explanation for these empirical formulas.

How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical implication is a somewhat counter-intuitive notion. For students, it is difficult to understand why a false statement implies everything. In this paper, we present a simple pedagogical way to make logical implication more intuitive.

Weighted Upwinding Compact Scheme For Shock Capturing, Hassan Abd Al-Dujaly

Mathematics Dissertations - Archive

The Weighted Upwinding Compact Scheme in this dissertation has been constructed due to dissipation and dispersion analysis at each stencil. The new scheme is applied to many one-dimensional typical problems involving discontinuities and shock waves, and it maintains a 7th order of accuracy in smooth areas. Additionally, when using the technology of decoupling the system of WUCS, the global dependency problem of the compact scheme is transferred to a local dependency problem in shock regions. As a result of the decoupling method, the shocks are captured sharply with fewer points compared to the related schemes. Furthermore, high order, high resolution, …

Nonparametric Adaptive Distribution-Free Procedure For Crossover Design With Repeated Measures, Afshan Boodhwani

Mathematics Dissertations - Archive

We propose to apply adaptive nonparametric procedures (Hill, Padmanabhan, & Puri, 1988) on 2x2 crossover design with repeated measures. We will derive the test-statistics (based on function of ranks) and find their asymptotic distributions. These test-statistics will be used to test (a) equality of carryover effects; (b) equality of direct treatment effects; (c) equality of carryover effects over time (repeated measures); and (d) equality of direct treatment effects over time (repeated measures), as suggested by Johnson and Grender (Johnson & Grender, 1993). We will be testing these hypotheses using modified versions of the test statistics derived by Johnson and Grender …