Digital Commons Network^™

Multilevel Optimization With Dropout For Neural Networks, Gary Joseph Saavedra

Mathematics & Statistics ETDs

Large neural networks have become ubiquitous in machine learning. Despite their widespread use, the optimization process for training a neural network remains com-putationally expensive and does not necessarily create networks that generalize well to unseen data. In addition, the difficulty of training increases as the size of the neural network grows. In this thesis, we introduce the novel MGDrop and SMGDrop algorithms which use a multigrid optimization scheme with a dropout coarsening operator to train neural networks. In contrast to other standard neural network training schemes, MGDrop explicitly utilizes information from smaller sub-networks which act as approximations of the full …

Nash Blowups Of Toric Varieties In Prime Characteristic, Daniel Duarte, Jack Jeffries, Luis Núñez-Betancourt

Department of Mathematics: Faculty Publications

We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.

Matroid Generalizations Of Some Graph Results, Cameron Crenshaw

LSU Doctoral Dissertations

The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.

For a graph G, the ratio of |E(G)| to the minimum degree of G has a natural lower bound. For a matroid M that is representable over a finite field, we generalize this to a lower bound on …

Commutators On Fock Spaces, Daniel Alpay, Paula Cerejeiras, Uwe Kähler, Trevor Kling

Mathematics, Physics, and Computer Science Faculty Articles and Research

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson

LSU Doctoral Dissertations

We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …

Applying Hallgren’S Algorithm For Solving Pell’S Equation To Finding The Irrational Slope Of The Launch Of A Billiard Ball, Sangheon Choi

Mathematical Sciences Technical Reports (MSTR)

This thesis is an exploration of Quantum Computing applied to Pell’s equation in an attempt to find solutions to the Billiard Ball Problem. Pell’s equation is a Diophantine equation in the form of x² − ny² = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgren’s algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard …

A New Approach To Proper Orthogonal Decomposition With Difference Quotients, Sarah Locke Eskew, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

In a Recent Work (Koc Et Al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the Authors Showed that Including Difference Quotients (DQs) is Necessary in Order to Prove Optimal Pointwise in Time Error Bounds for Proper Orthogonal Decomposition (POD) Reduced Order Models of the Heat Equation. in This Work, We Introduce a New Approach to Including DQs in the POD Procedure. Instead of Computing the POD Modes using All of the Snapshot Data and DQs, We Only Use the First Snapshot Along with All of the DQs and Special POD Weights. We Show that This Approach Retains All of the …

Rank-Based Inference For Survey Sampling Data, Akim Adekpedjou, Huybrechts F. Bindele

Mathematics and Statistics Faculty Research & Creative Works

For regression models where data are obtained from sampling surveies, the statistical analysis is often based on approaches that are either non-robust or inefficient. The handling of survey data requires more appropriate techniques, as the classical methods usually result in biased and inefficient estimates of the underlying model parameters. This article is concerned with the development of a new approach of obtaining robust and efficient estimates of regression model parameters when dealing with survey sampling data. Asymptotic properties of such estimators are established under mild regularity conditions. To demonstrate the performance of the proposed method, Monte Carlo simulation experiments are …

Numerical Range Of Strictly Triangular Matrices Over Finite Fields, Ariel Russell

Mathematics Student Projects

In this paper, we investigate the numerical range of matrices over finite fields, particularly triangular matrices. We conjecture that all strictly triangular matrices over finite fields of dimension 3 or greater have a numerical range encompassing the entirety of the finite field. We use both algebraic and computational methods to support this claim, making some concrete progress towards the algebraic proof. Further, we conjecture that all matrices over finite fields have a numerical range falling into one of five potential categories, providing an extensive appendix of randomly generated computational examples which seems to support this conjecture.

Match Results And The Equity Value Of Publicly Traded Soccer Teams, Ryan Bebyn

Honors Projects in Mathematics

This study aims to examine the relationship between the match performance of publicly traded European soccer clubs and their stock price returns. It is important for investors, owners, and fans to understand how competitive factors both on and off the field affect the equity value of European club soccer teams. Competitive factors such as basic match results (Win, Loss, or Draw), rivalry matches, and international matches will be evaluated against the daily stock prices of publicly traded soccer clubs. A sample of teams traded on public exchanges will be used to analyze the effects of their matches on the daily …

Causality: Hypergraphs, Matter Of Degree, Foundations Of Cosmology, Cliff Joslyn, Andres Ortiz-Muñoz, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The notion of causality is very important in many applications areas. Because of this importance, there are several formalizations of this notion in physics and in AI. Most of these definitions describe causality as a crisp ("yes"-"no") relation between two events or two processes -- cause and effect. However, such descriptions do not fully capture the intuitive idea of causality: first, often, several conditions are needed to be present for an effect to occur, and, second, the effect is often a matter of degree. In this paper, we show how to modify the current description of causality so as to …

Foundations Of Neural Networks Explain The Empirical Success Of The "Surrogate" Approach To Ordinal Regression -- And Recommend What Next, Salvador Robles, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Recently, a new efficient semi-heuristic statistical method -- called Surrogate Approach -- has been proposed for dealing with regression problems. How can we explain this empirical success? And since this method is only an approximation to reality, what can we recommend if there is a need for a more accurate approximation? In this paper, we show that this empirical success can be explained by the same arguments that explain the empirical success of neural networks -- and these arguments can also provide us with possible more general techniques (that will hopefully lead to more accurate approximation to real-life phenomena).

Towards Decision Making Under Interval Uncertainty, Juan A. Lopez, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, we need to make a decision. In many cases, we know the optimal decision in situations when we know the exact value of the corresponding quantity x. However, often, we do not know the exact value of this quantity, we only know the bounds on the value x -- i.e., we know the interval containing $x$. In this case, we need to select a decision corresponding to some value from this interval. The selected value will, in general, be different from the actual (unknown) value of this quantity. As a result, the quality of our decision …

Low-Probability High-Impact Events Are Even More Important Than It Is Usually Assumed, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A large proportion of undesirable events like earthquakes, floods, tornados occur in zones where these events are frequent. However, a significant number of such events occur in other zones, where such events are rare. For example, while most major earthquakes occur in a vicinity of major faults, i.e., on the border between two tectonic plates, some strong earthquakes also occur inside plates. We want to mitigate all undesirable events, but our resources are limited. So, to allocate these resources, we need to decide which ones are more important. For this decision, a natural idea is to use the product of …

What Do Goedel's Theorem And Arrow's Theorem Have In Common: A Possible Answer To Arrow's Question, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Kenneth Arrow, the renowned author of the Impossibility Theorem that explains the difficulty of group decision making, noticed that there is some commonsense similarity between his result and Goedel's theorem about incompleteness of axiomatic systems. Arrow asked if it is possible to describe this similarity in more precise terms. In this paper, we make the first step towards this description. We show that in both cases, the impossibility result disappears if we take into account probabilities. Namely, we take into account that we can consider probabilistic situations, that we can make probabilistic conclusions, and that we can make probabilistic decisions …

State Gross Domestic Product Predictions Using Hierarchical Clustering And Multivariate Time Series, Austin Dae Nietfeld

Mathematics Senior Capstone Papers

This research was conducted to determine the weight certain taxes and expenditures have over state Gross Domestic Product(GDP) as well as how accurately these predictors can predict future GDP. The motivation behind this project comes from a desire to find the most efficient way to increase the GDP of states with poorer economies. This will improve the quality of life of citizens of these states. To come to a consensus as to what predictors are most influential, Hierarchical Clustering will be used to split the states into four groups. The average of each tax, expenditure and GDP from 2015-2020 will …

Extending The Radius Of Convergence Of The Drinfeld Module Logarithm, Ethan Clapp

Mathematics Senior Capstone Papers

Drinfeld modules are, in essence, a way to create a function field analogue to complex multiplication. With these modules, we can define a logarithm function analogous to the natural logarithm by virtue of a power series. Similarly to how we extend the radius of convergence of the natural logarithm with complex numbers, we can also extend the radius of convergence for these Drinfeld module logarithms. While there are proofs for extending the radius of convergence of the Carlitz module, the simplest form of Drinfeld module, there is no proof for a generalized Drinfeld. In this paper, a method of extending …

Regression Analysis Of Injuries On Nfl Quarterbacks, Julie Weems

Mathematics Senior Capstone Papers

Risk assessment is an important aspect of many careers such as first responders and the military. This is no different for people who play sports, especially people who are in contact sports such as football. These players’ lives can be changed forever with one bad hit. The goal of this research is to analyze the probability of an injury for the National Football League’s (NFL) quarterbacks. It is hard to predict when, what, and where an injury will occur, because of this very little work has been done on the subject matter in a general form. The goal of this …

Effects Of Topography On Tornado Paths Using Navier Stokes Equations, Kayleigh Smith

Mathematics Senior Capstone Papers

Tornadoes are relatively common in Louisiana with an average of 55 tornadoes per year. Predicting tornado paths has been extremely challenging due to the many factors that play into the formation of a tornado. According to NASA/ADS topography can have significant influence on tornado direction. The main goal of this research is to analyze tornado patterns and to determine if local topography has an effect on tornadic activity. Navier-Stokes partial differential equations will be used to model the data that is collected from the national weather service and a finite difference method will be used to solve the equations. Data …

Firefighter Safety, Haynes Mandino

Mathematics Senior Capstone Papers

There are close to 1.2 million career and volunteer firefighters across the United States. In the year 2020 alone 62 of these firefighters died and 64,875 were injured. The following research was performed to determine if the firefighter profession has become safer due to new standards and regulations. Each year the National Fire Protection Agency(NFPA) and the Federal Emergency Management Agency(FEMA) collect data on the number of firefighter deaths and injuries, in order to determine if the standards and regulations are keeping firefighters safe. Statistical hypothesis testing and linear regression were performed on the data to show if in fact …

Modeling Reverberant Chambers Used In Time-Reversal Cavities Using Time-Frequency Methods, Jordan Savoie

Mathematics Senior Capstone Papers

The signal processing technique of time-reversal is used for various purposes in electro-magnetic interference testing, radar, communication, sonar, and medicine. This paper models the impulse response of a reverberant chamber such as might be used for a time-reversal cavity (TRC) using time-frequency methods. Reverberant chambers have chaotic delaying phase responses that cause the majority of the distortion seen in their impulse response. Time-reversal cancels this, creating a focused signal from a disperse one. Most research modeling the behavior of TRCs is more concerned with their physics, so there is relatively little commented on the form of their signals. Data of …

Does The Three Point Shot Affect Winning Percentage, Marcamus Winn

Mathematics Senior Capstone Papers

The three-point shot, introduced in the late 1970s, is a shot that occurs typically 24 feet away from the basket at the professional level. Strategically the game of basketball was originally based on two-point field goals. Recently, there has been a noticeable trend in the popularity of the three-point shot amongst professional teams. Nowadays, three point shot attempts account for more than a third of average NBA shot selection. Statistical analysis is becoming integral to athletics. Statistics has become a critical component to the development of not only on court basketball strategies, but also team structure as well. There are …

Changing Nfl Playoff Overtime Rules To Create Equal Opportunities To Win A Game, Matthew Silvia

Honors Projects in Mathematics

The NFL has attempted to create fair overtime rules over the course of the past decade; however, this study is interested in determining what playoff overtime rule (or rules) could the NFL implement to result in outcomes where both teams have a relatively equal chance of winning a game. This study aims to find which overtime rules work best at minimizing the differences between teams who possess the ball first versus teams that kick the ball off to start an overtime period. By collecting various NFL statistics from ESPN.com and FantasyOutsiders.com, this study hopes to run multiple simulations of different …

Wormholes, Superfast Computations, And Selivanov's Theorem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While modern computers are fast, there are still many practical problems that require even faster computers. It turns out that on the fundamental level, one of the main factors limiting computation speed is the fact that, according to modern physics, the speed of all processes is limited by the speed of light. Good news is that while the corresponding limitation is very severe in Euclidean geometry, it can be more relaxed in (at least some) non-Euclidean spaces, and, according to modern physics, the physical space is not Euclidean. The differences from Euclidean character are especially large on micro-level, where quantum …

Everything Is A Matter Of Degree: The Main Idea Behind Fuzzy Logic Is Useful In Geosciences And In Authorship, Christian Servin, Aaron Velasco, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

This paper presents two applications of the general principle -- the everything is a matter of degree -- the principle that underlies fuzzy techniques. The first -- qualitative -- application helps explain the fact that while most earthquakes occur close to faults (borders between tectonic plates or terranes), earthquakes have also been observed in areas which are far away from the known faults. The second -- more quantitative -- application is to the problem of which of the collaborators should be listed as authors and which should be simply thanked in the paper. We argue that the best answer to …

Conflict Situations Are Inevitable When There Are Many Participants: A Proof Based On The Analysis Of Aumann-Shapley Value, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

When collaboration of several people results in a business success, an important issue is how to fairly divide the gain between the participants. In principle, the solution to this problem is known since the 1950s: natural fairness requirements lead to the so-called Shapley value. However, the computation of Shapley value requires that we can estimate, for each subset of the set of all participants, how much gain they would have gained if they worked together without others. It is possible to perform such estimates when we have a small group of participants, but for a big company with thousands of …

Integrity First, Service Before Self, And Excellence: Core Values Of Us Air Force Naturally Follow From Decision Theory, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

By analyzing data both from peace time and from war time, the US Air Force came with three principles that determine success: integrity, service before self, and excellent. We show that these three principles naturally follow from decision theory, a theory that describes how a rational person should make decisions.

People Prefer More Information About Uncertainty, But Perform Worse When Given This Information: An Explanation Of The Paradoxical Phenomenon, Jieqiong Zhao, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In a recent experiment, decision makers were asked whether they would prefer having more information about the corresponding situation. They confirmed this preference, and such information was provided to them. However, strangely, the decisions of those who received this information were worse than the decisions of the control group -- that did not get this information. In this paper, we provide an explanation for this paradoxical situation.

How People Make Decisions Based On Prior Experience: Formulas Of Instance-Based Learning Theory (Ilbt) Follow From Scale Invariance, Palvi Aggarwal, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To better understand human behavior, we need to understand how people make decisions, how people select one of possible actions. This selection is usually based on predicting consequences of different actions, and these predictions are, in their turn, based on the past experience. For example, consequences that occur more frequently in the past are viewed as more probable. However, this is not just about frequency: recent observations are usually given more weight that past ones. Researchers have discovered semi-empirical formulas that describe our predictions reasonably well; these formulas form the basis of the Instance-Based Learning Theory (ILBT). In this paper, …

Zonality In Graphs, Andrew Bowling

Dissertations

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …