Mathematics Commons^™

Computational Utilities For The Game Of Simplicial Nim, Nelson Penn

Theses and Dissertations--Computer Science

Simplicial nim games, a class of impartial games, have very interesting mathematical properties. Winning strategies on a simplicial nim game can be determined by the set of positions in the game whose Sprague-Grundy values are zero (also zero positions). In this work, I provide two major contributions to the study of simplicial nim games. First, I provide a modern and efficient implementation of the Sprague-Grundy function for an arbitrary simplicial complex, and discuss its performance and scope of viability. Secondly, I provide a method to find a simple mathematical expression to model that function if it exists. I show the …

Simiode Expo 2021 - Minicourse M-R2 - Delay Differential Equations In Epidemiology, Nsoki Mavinga

Mathematics & Statistics Faculty Works

Many problems in epidemiology give rise to delay differential equations (DDEs). These are differential equations in which the current rate of change of the system depends not only on the current state but also on the history of the system; i.e. the system has memory. In this minicourse, we will discuss some key tools necessary to understand the applications involving DDEs. We will go through an example that illustrates the need and implementation of DDE in certain infectious diseases.

Glycinergic Axonal Inhibition Subserves Acute Spatial Sensitivity To Sudden Increases In Sound Intensity, T. P. Franken, B. J. Bondy, D. B. Haimes, Joshua H. Goldwyn, N. L. Golding, P. H. Smith, P. X. Joris

Mathematics & Statistics Faculty Works

Locomotion generates adventitious sounds which enable detection and localization of predators and prey. Such sounds contain brisk changes or transients in amplitude. We investigated the hypothesis that ill-understood temporal specializations in binaural circuits subserve lateralization of such sound transients, based on different time of arrival at the ears (interaural time differences, ITDs). We find that Lateral Superior Olive (LSO) neurons show exquisite ITD-sensitivity, reflecting extreme precision and reliability of excitatory and inhibitory postsynaptic potentials, in contrast to Medial Superior Olive neurons, traditionally viewed as the ultimate ITD-detectors. In vivo, inhibition blocks LSO excitation over an extremely short window, which, in …

Dynamics Of The Auditory Continuity Illusion, Qianyi Cao , '22, Noah Parks , '22, Joshua H. Goldwyn

Mathematics & Statistics Faculty Works

Illusions give intriguing insights into perceptual and neural dynamics. In the auditory continuity illusion, two brief tones separated by a silent gap may be heard as one continuous tone if a noise burst with appropriate characteristics fills the gap. This illusion probes the conditions under which listeners link related sounds across time and maintain perceptual continuity in the face of sudden changes in sound mixtures. Conceptual explanations of this illusion have been proposed, but its neural basis is still being investigated. In this work we provide a dynamical systems framework, grounded in principles of neural dynamics, to explain the continuity …

Branched Covers Bounding Rational Homology Balls, P. Aceto, J. Meier, Allison N. Miller, M. Miller, Jh Park, A. I. Stipsicz

Mathematics & Statistics Faculty Works

Prime power–fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. We give a new construction of nonslice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.

A Mathematical Analysis Of The Wind Triangle Problem And An Inquiry Of True Airspeed Calculations In Supersonic Flight, Leonard T. Huang, Lisa I. Cummings

International Journal of Aviation, Aeronautics, and Aerospace

In the first half of this paper, we present a fresh perspective toward the Wind Triangle Problem in aerial navigation by deriving necessary and sufficient conditions, which we call "go/no-go conditions", for the existence/non-existence of a solution of the problem. Although our derivation is based on simple trigonometry and basic properties of quadratic functions, it is mathematically rigorous. We also offer examples to demonstrate how easy it is to check these conditions graphically. In the second half of this paper, we use function theory to re-examine another problem in aerial navigation, namely, that of computing true airspeed — even in …

Proper 3-Colorings Of Cycles And Hypercubes, Emily Cairncross

Honors Papers

In this paper, we look at two families of graphs, cycles and hypercubes, and compare how their sets of proper 3-colorings differ as the graphs get arbitrarily large. In particular, we find the probability of pairs of vertices at various distances being the same color in order to understand the range and scale of interactions between them. As we look at larger and larger cycles, larger and larger hypercubes, patterns begin to emerge. While the colors of vertices fixed fractions of the cycle away from each other are independent, a random 3-coloring of the hypercube is essentially a 2-coloring. This …

Collaborative Game Mosaics, Xiaoyun Gong

Honors Papers

Imagine two players are playing a strategy board game. With one player holding black stones and the other player holding white stones, they take turns to place their pieces. After a while, they start to recognize some patterns. With three colors in front of them, black, white, and the color of the game board, they are curious about if they can form a recognizable image. Instead of playing the game to win, they start to collaborate with each other to form a mosaic. Later, they invite more players and provide them their own stones in different shades of gray. The …

Higher Homotopy Invariants For Spaces And Maps, David Blanc, Mark W. Johnson

University Faculty Publications and Creative Works

For a pointed topological space X, we use an inductive construction of a simplicial resolution of X by wedges of spheres to construct a “higher homotopy structure” for X (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover X up to weak equivalence. It can also be used to distinguish between different maps f : X → Y which induce the same morphism f∗ : π∗X → π∗Y.

Proctoring And Apps In College Algebra, Cynthia M. Shelton

Theses and Dissertations--Education Sciences

The pandemic forced more instructors and students to move to online learning. For the first time, many experienced a loosening of the reigns and were forced to allow students to submit non-proctored work. Many may have questioned what students really learned in the year 2020. Many college math course competencies emphasize procedures. Now that apps can do that for students, where does that leave math instructors? Additionally, online instruction has exploded over the last decade and has challenged the teaching of college mathematics. While online instruction opens the door to access, it does beg the question of whether students complete …

Occam Software (And Manual) For Reconstructability Analysis, Martin Zwick, Kenneth Willett, Joe Fusion, Heather Alexander

Complex Systems Faculty Publications and Presentations

OCCAM is a Discrete Multivariate Modeling (DMM) tool based on the methodology of Reconstructability Analysis (RA). As an acronym it stands for Organizational Complexity Computation and Modeling, and the name is also a reference (with a non-standard spelling) to Ockham’s Razor. The principal programmers of its current version have been Kenneth Willett, Joe Fusion, and Heather Alexander. Ken Willett totally rewrote earlier versions of OCCAM. His version was originally called “OCCAM3” to distinguish it from these earlier OCCAM incarnations; the “3” has finally been dropped.

OCCAM’s typical use is to analyze data involving a large number of discrete variables, but …

Integer-Valued Polynomials Over Discrete Valuation Domains, Rattiya Meesa

Chulalongkorn University Theses and Dissertations (Chula ETD)

The classical theorem of Lucas states that binomial polynomials, which form a basis for integer-valud polynomials, satisfy a congruence relation, modulo a prime, related to their digits in the base prime representation. In this thesis, we define the Lucas property in the setting of discrete-valued structures and investigate when and where the Lucas property holds. General criteria are derived for bases of integervalued polynomials in this setting to satisfy the Lucas property. Examples of bases including those of Lagrange type and of Carlitz-like polynomials are worked out. In addition, one of the best known properties of binomial polynomials in the …

Detailing The Connection Between A Family Of Polar Graphs And Tremain Equiangular Tight Frames, Nicholas Brown

Electronic Theses and Dissertations

The relationship between strongly regular graphs and equiangular tight frames has been known for several years, and this relationship has been used to construct many of the most recent examples of new strongly regular graphs. In this paper, we present an explicit construction of a family of equiangular tight frames using the geometry of a quadratic space over the field of four elements. We observe that these frames give rise to a strongly regular graph on a subset of points of a quadratic space over the field with 4 elements. We then demonstrate an isomorphism between this graph and a …

Teacher Perceptions Of Using Standards-Based Rubrics For Monitoring Student Growth In Teacher Evaluation, Jennette Susan Winters

Walden Dissertations and Doctoral Studies

In the United States, national and state legislative mandates have forced school districts to include student growth measures in teacher evaluation systems. However, statistical models for monitoring student growth on standardized tests have not been found to foster teachers’ reflective practice or pedagogical content knowledge and goal-based models have been found to lack adequate structure for supporting implementation. This basic qualitative inquiry explored how teachers perceive using standards-based rubrics to monitor student growth for teacher evaluation influences their reflective practice and pedagogical content knowledge in mathematics. Nine teachers who have used standards-based rubrics to monitor student growth were recruited through …

Supporting Student Success And Persistence In Stem With Active Learning Approaches In Emerging Scholars Classrooms, David Miller, Jessica Deshler, Tim Mceldowney, John Stewart, Edgar Fuller, Matt Pascal, Lynnette Michaluk

Faculty & Staff Scholarship

Over the last several decades, Emerging Scholars Programs (ESPs) have incorporated active learning strategies and challenging problems into collegiate mathematics, resulting in students, underrepresented minority (URM) students in particular, earning at least half of a letter grade higher than other students in Calculus. In 2009, West Virginia University (WVU) adapted ESP models for use in Calculus I in an effort to support the success and retention of URM STEM students by embedding group and inquiry-based learning into a designated section of Calculus I. Seats in the class were reserved for URM and first- generation students. We anticipated that supporting students …

Exploring Plane Partitions, Nicholas Heim

Williams Honors College, Honors Research Projects

The combinatorial theory of partitions has a number of applications including the representation theory of the symmetric group. A particularly important result counts the number of standard Young tableau of a given partition in terms of the hook lengths of the partition. In this paper we explore the analog of the hook length formula for plane partitions, the three-dimensional analog of ordinary partitions. We show that equality does not always hold but we conjecture that a certain inequality holds. Using a computer program, we verify this conjectured inequality for all 1982 plane partitions up to n = 11.

Predicting Carcass Cut Yields In Cattle From Digitalimages Using Artificial Intelligence, Darragh Matthews

Theses

Beef carcass classification in Europe is predicated on the EUROP grid for both fatness and conformation. Although this system performs well for grouping visually similar carcasses, it cannot be used to accurately predict meat yields from these groups, especially when considered on an individual cut level. Deep Learning (DL) has proven to be a successful tool for many image classification problems but has yet to be fully proven in a regression scenario using carcass images. Here we have trained DL models to predict carcass cut yields and compared predictions to more standard machine learning (ML) methods. Three approaches were undertaken …

Tilings With Congruent Edge Coronae, Ma. Louise Antonette N. De Las Peñas, Mark D. Tomenes

Mathematics Faculty Publications

In this paper, we discuss properties of a normal tiling of the Euclidean plane with congruent edge coronae. We prove that the congruence of the first edge coronae is enough to say that the tiling is isotoxal.

Using Mobile Technology To Promote Higher-Order Thinking Skills In Elementary Mathematics, Debbie Marie Versoza, Ma. Louise Antonette N. De Las Peñas, Jumela F. Sarmiento, Maria Alva Q. Aberin, Mark Anthony C. Tolentino, Mark L. Loyola

Mathematics Faculty Publications

The problem of rote-based learning in mathematics is well documented. Mobile technology can provide a potential solution, especially when application (app) design is based on sound pedagogical principles and gamification elements. However, an inventory of available mobile apps for mathematics reveals that many of the available apps are guided by a behaviorist perspective that favors repetition over meaningful learning. This paper reports on the design of mobile mathematics apps that harness gamification techniques to promote higher-order thinking skills (HOTS) even in basic elementary school concepts such as number comparison, and addition and subtraction. The integration of these apps in the …

Quandles With Orbit Series Conditions, Alissa Crans

Mathematics, Statistics and Data Science Faculty Works

We introduce the notion of an orbit series in a quandle. Using this notion we define four families of quandles based on finiteness conditions on their orbit series. Intuitively, the classes tOS and tOSn correspond to finitary compositions of trivial quandles while the classes OS and OSn correspond to finitary compositions of connected quandles. We study properties of these four families of quandles and explore their relationships with several previously studied families of quandles: reductive, n-reductive, locally reductive, n-locally reductive, and solvable quandles