Digital Commons Network^™

The Explorer–Director Game On Graphs, Pat Devlin, E. Meger, A. Raz, Polymath Reu Participants

Mathematics & Statistics Faculty Works

The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can be described as a game where two players—Explorer and Director—determine the movement of a token that is positioned on the vertices of some given graph. At each time step, the Explorer specifies a distance that the token must move with an aim to maximize the total number of vertices ultimately visited. However, the Director adversarially chooses where to move token in an effort to minimize this number. The game ends when no new vertices can be visited. Given a graph G and a starting vertex v, the number of vertices …

A Mode Sum Regularization Prescription In Quantum Field Theory In Curved Spacetimes In Higher Dimensions And For Charged Scalar Fields, Eoin Scanlon

Academic Posters Collection

Semi-classical gravity combines classical treatment of the gravitational field with quantum mechanical treatment of matter fields. A significant challenge however is the divergence contained within the stress-energy tensor when solving the semi-classical Einstein equations. This work extends to higher dimensions an extremely efficient method for renormalizing the stress-energy tensor of a quantum scalar field in spherically-symmetric black hole spacetimes, thereby removing the divergences. The method applies to a scalar field with arbitrary field parameters. The utility of the method is demonstrated by computing the renormalized stress-energy tensor for a scalar field in the Schwarzschild black hole spacetime for odd dimensions.

Feedback, Learning Outcomes And Mathematics Anxiety In A Digital Game Based Learning Approach In Mathematics Education, André Almo

Academic Posters Collection

Feedback is a crucial part of learning, and an essential element in digital game-based learning approaches, in which digital games - known as 'serious games' - are used to deliver educational content. Feedback features respond to players' actions within the game, providing them with information and guidance, as well as potentially impacting their learning, motivation and engagement. However, these features may be designed differently, since they include various distinct characteristics and dimensions. This work proposes a new taxonomy for feedback features in serious games, with an emphasis in game design aspects, in order to provide clearer descriptions and distinctions of …

Nonlinear Two-Dimensional Water Waves With Arbitrary Vorticity, Delia Ionescu-Kruse, Rossen Ivanov

Articles

We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for …

Turbulence Phenomena In Magnetohydrodynamic Phase Transitions, Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Articles

The model developed in (Fabrizio in J. Eng. Math., 2023) and (Fabrizio in Int. J. Eng. Sci. 44:529–539, 2006), involving the use of a local Reynolds number, is applied to describe phase transitions in a fluid. Specifically, it is applied in a magnetohydrodynamics context to study the evolution of turbulence in certain phenomena. The relevant equations describing the system are those of Navier-Stokes, Ginzburg-Landau and the magnetohydrodynamic equations, all suitably interconnected.

Partially Filled Latin Squares, Mariam Abu-Adas

Scripps Senior Theses

In this thesis, we analyze various types of Latin squares, their solvability and embeddings. We examine the results by M. Hall, P. Hall, Ryser and Evans first, and apply our understandings to develop an algorithm that the determines the minimum possible embedding of an unsolvable Latin square. We also study Latin squares with missing diagonals in detail.

Computational Investigation Of The Ionization Potential Of Lead Sulfide Quantum Dots, Jessica Beyer

Scripps Senior Theses

The purpose of this work was to determine the impact of quantum dot size on ionization potential and to determine how the presence of carbonyl-based ligands affect the ionization potential of lead sulfide quantum dot systems. Ionization potential (IP) is defined as the energy required to remove an electron from an atom, molecule, or material. IP helps scientists determine how reactive the material of interest is, which is crucial information when manufacturing nanomaterials. Accurate quantum chemical calculations of ionization potential are challenging due to the computational cost associated with the numerical solution of the Dyson equation. In this work, the …

Bibliography, Bruce Kessler

Faculty/Staff Personal Papers

Bibliography of publications by Bruce Kessler.

On The Frobenius Norm Of Commutator Of Cauchy-Toeplitz Matrix And Exchange Matrix, Süleyman Solak, Mustafa Bahşi̇

Turkish Journal of Mathematics

Matrix commutator and anticommutator play an important role in mathematics, mathematical physic, and quantum physic. The commutator and anticommutator of two $n\times n$ complex matrices $A$ and $B$ are defined by $\left[ A,B% \right] =AB-BA$ and $\left( A,B\right) =AB+BA$, respectively. Cauchy-Toeplitz matrix and exchange matrix are two of the special matrices and they have excellent properties. In this study, we mainly focus on Frobenius norm of the commutator of Cauchy-Toeplitz matrix and exchange matrix. Moreover, we give upper and lower bounds for the Frobenius norm of the commutator of Cauchy-Toeplitz matrix and exchange matrix.

What Can Lattices Do For Hypersemigroups?, Niovi Kehayopulu

Turkish Journal of Mathematics

This is from Birkhoff, the "father of lattice theory" in Trends in Lattice Theory. Van Nostrand 1970: "Lattices can do things for you, no matter what kind of mathematician you are!". The aim of this paper is to show that the $le$-semigroups (lattice ordered semigroups possessing a greatest element) play the main role in studying the ordered hypersemigroups. From many results on lattice ordered semigroups corresponding results on ordered semigroups can be obtained. The converse is also possible but the beauty and simplicity of "order" makes it easier to investigate the lattice ordered semigroup at first. After getting the results …

Generalized Far-Difference Representations, Prakod Ngamlamai

HMC Senior Theses

Integers are often represented as a base-$b$ representation by the sum $\sum c_ib^i$. Lekkerkerker and Zeckendorf later provided the rules for representing integers as the sum of Fibonacci numbers. Hannah Alpert then introduced the far-difference representation by providing rules for writing an integer with both positive and negative multiples of Fibonacci numbers. Our work aims to generalize her work to a broader family of linear recurrences. To do so, we describe desired properties of the representations, such as lexicographic ordering, and provide a family of algorithms for each linear recurrence that generate unique representations for any integer. We then prove …

Beginner's Analysis Of Financial Stochastic Process Models, David Garcia

HMC Senior Theses

This thesis explores the use of geometric Brownian motion (GBM) as a financial model for predicting stock prices. The model is first introduced and its assumptions and limitations are discussed. Then, it is shown how to simulate GBM in order to predict stock price values. The performance of the GBM model is then evaluated in two different periods of time to determine whether it's accuracy has changed before and after March 23, 2020.

Permutations, Representations, And Partition Algebras: A Random Walk Through Algebraic Statistics, Ian Shors

HMC Senior Theses

My thesis examines a class of functions on the symmetric group called permutation statistics using tools from representation theory. In 2014, Axel Hultman gave formulas for computing expected values of permutation statistics sampled via random walks. I present analogous formulas for computing variances of these statistics involving Kronecker coefficients – certain numbers that arise in the representation theory of the symmetric group. I also explore deep connections between the study of moments of permutation statistics and the representation theory of the partition algebras, a family of algebras introduced by Paul Martin in 1991. By harnessing these partition algebras, I derive …

On The Modelling Of Short And Intermediate Water Waves, Rossen Ivanov

Articles

Most of the model equations for water waves are approximations for the long-wave propagation regimes, since most of the energy of the wave motion is concentrated in these waves. Long waves (or shallow-water waves) are defined usually as the depth to wavelength ratio δ = h/λ < 0.05. Several famous integrable nonlinear equations, like the K d V equation [1,2], are models for long waves of small amplitude. The short waves (or waves over deep water) are usually defined with δ > 0.5, and the intermediate waves (or transitional waves) - with 0.05 < δ < 0.5. The intermediate and short waves received a lot less attention, and one reason is perhaps the fact that the corresponding approximations lead to more complicated, nonlinear and nonlocal equations. In [3] an integral equation for surface waves has been proposed for arbitrary wavelengths and finite depth. The problem has been studied in [4] and model equations both for long and short waves are derived from the governing equations as well. The short-wave effects usually compete with the capillarity effects and then resonances can be observed — these have been studied quite a lot, see for example [5–12]. For the intermediate long waves or for waves on deep water the so-called Benjamin–Ono (BO) [13–15] and the Intermediate Long Wave Equation (ILWE) [16–18] are derived for the internal waves below a flat surface, which leads to some simplifications and these models are in fact integrable.

The Lagrangian Formulation For Wave Motion With A Shear Current And Surface Tension, Conor Curtin, Rossen Ivanov

Articles

The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler-Lagrange equations we proceed to derive some …

Boundedness Of Bergman Projections Acting On Weighted Mixed Norm Spaces, Ivana Savkovic

Turkish Journal of Mathematics

We prove that Bergman projections on weighted mixed norm spaces on smoothly bounded domains in $\mathbb{R}^n$ are bounded for a certain range of parameters of such spaces and assuming certain conditions on weights. The proof relies on estimates of integral means of $M_p(P_\gamma f, r)$ in terms of integral means of $f$. This result complements earlier result on boundedness of $P_\gamma$ on a closely related space $L^{p,q}_\alpha(\Omega).$

Between Graphical Zonotope And Graph-Associahedron, Marko Pesovic, Tanja Stojadinovic

Turkish Journal of Mathematics

This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph, and the last is the graph-associahedron associated to it. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs. In the last section, we study some properties related to $\mathcal{H}$-polytopes.

New Insights Into Binocular Rivalry From The Reconstruction Of Evolving Percepts Using Model Network Dynamics, Kenneth Barkdoll , '24, Yuhua Lu , '24, Victor J. Barranca

Mathematics & Statistics Faculty Works

When the two eyes are presented with highly distinct stimuli, the resulting visual percept generally switches every few seconds between the two monocular images in an irregular fashion, giving rise to a phenomenon known as binocular rivalry. While a host of theoretical studies have explored potential mechanisms for binocular rivalry in the context of evoked model dynamics in response to simple stimuli, here we investigate binocular rivalry directly through complex stimulus reconstructions based on the activity of a two-layer neuronal network model with competing downstream pools driven by disparate monocular stimuli composed of image pixels. To estimate the dynamic percept, …

Quantum Hooks And Mirror Symmetry For Flag Varieties, Linda Chen, E. Kalashnikov

Mathematics & Statistics Faculty Works

Given a flag variety F1 (n; r₁,...,r_p), there is natural ring morphism from the symmetric polynomial ring in r₁ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions λ, the image of s_λ under the ring homomorphism is a Schubert class which is described by partitioning λ into a quantum hook (or q-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new …

Structure And Dynamics That Specialize Neurons For High-Frequency Coincidence Detection In The Barn Owl Nucleus Laminaris, Ben Drucker , '22, Joshua H. Goldwyn

Mathematics & Statistics Faculty Works

A principal cue for sound source localization is the difference in arrival times of sounds at an animal’s two ears (interaural time difference, ITD). Neurons that process ITDs are specialized to compare the timing of inputs with submillisecond precision. In the barn owl, ITD processing begins in the nucleus laminaris (NL) region of the auditory brain stem. Remarkably, NL neurons are sensitive to ITDs in high-frequency sounds (kilohertz-range). This contrasts with ITD-based sound localization in analogous regions in mammals where ITD sensitivity is typically restricted to lower-frequency sounds. Guided by previous experiments and modeling studies of tone-evoked responses of NL …

Generalized Sasakian Structures From A Poisson Geometry Viewpoint, Janet Talvacchia

Mathematics & Statistics Faculty Works

In this paper we define a canonical Poisson structure on a normal generalized contact metric space and use this structure to define a generalized Sasakian structure. We show also that this canonical Poisson structure enables us to distinguish generalized Sasakian structures from generalized coKähler structures.

The Square Roots Of Some Classical Operators, Javad Mashreghi, Marek Ptak, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.

Bifurcation Levels Of The Integral Manifolds Of The Newtonian N-Body Problem, Hannah G. Havel

CURE Proceedings

The N-body problem, first proposed by Isaac Newton, is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It has significance to many areas of science, including physics and computer science, and is crucial in understanding how the universe works. In fact, it was a primary motivation for Newton's development of calculus. An important application of the N-body problem is within celestial mechanics and involves how planets and other celestial bodies move with mutual gravitational attraction. It is important in developing how satellites behave in space using complicated …

Complete Study Of An Original Power-Exponential Transformation Approach For Generalizing Probability Distributions, Mustafa S. Shama, Farid El Ktaibi, Jamal N. Al Abbasi, Christophe Chesneau, Ahmed Z. Afify

All Works

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are …

Odd Solutions To Systems Of Inequalities Coming From Regular Chain Groups, Daniel Slilaty

Mathematics and Statistics Faculty Publications

Hoffman’s theorem on feasible circulations and Ghouila-Houry’s theorem on feasible tensions are classical results of graph theory. Camion generalized these results to systems of inequalities over regular chain groups. An analogue of Camion’s result is proved in which solutions can be forced to be odd valued. The obtained result also generalizes the results of Pretzel and Youngs as well as Slilaty. It is also shown how Ghouila-Houry’s result can be used to give a new proof of the graph- coloring theorem of Minty and Vitaver.

Peer-To-Peer Energy Trading In Smart Residential Environment With User Behavioral Modeling, Ashutosh Timilsina

Theses and Dissertations--Computer Science

Electric power systems are transforming from a centralized unidirectional market to a decentralized open market. With this shift, the end-users have the possibility to actively participate in local energy exchanges, with or without the involvement of the main grid. Rapidly reducing prices for Renewable Energy Technologies (RETs), supported by their ease of installation and operation, with the facilitation of Electric Vehicles (EV) and Smart Grid (SG) technologies to make bidirectional flow of energy possible, has contributed to this changing landscape in the distribution side of the traditional power grid.

Trading energy among users in a decentralized fashion has been referred …

Graphs, Adjacency Matrices, And Corresponding Functions, Yang Hong

Honors Theses

Stable polynomials, in the context of this research, are two-variable polynomials like $p(z_1,z_2) = 2 - z_1 - z_2$ that are guaranteed to be non-zero if both input variables have an absolute value less than one in the complex plane. Stable polynomials are used in a variety of mathematical fields, thus finding ways to construct stable polynomials is valuable. An important property of these polynomials is whether they have boundary zeros, which are points in the complex plane where the polynomial equals zero and both variables have an absolute value of 1. Overall, it is challenging to find stable polynomials …

On Prime Labelings Of Uniform Cycle Snake Graphs, M. A. Ollis

Emerson Authors, Researchers, & Creators

A reseaech paper in graph theory, a subfield of math. At the time of the research Agam Bedi and Samiksha Ramesh were undergraduate students at Emerson College and the work was completed as part of the SOC320 Research Co-Curricular in the summer of 2022. The work has a Creative Commons BY-NC licence.

Some Estimates On The Exponential Stability Of Solutions Of Nonlinear Neutral Type Systems With Periodic Coefficients, Yener Altun

Turkish Journal of Mathematics

In this present study, we pay attention to a class of nonlinear neutral type systems (NNSs) with periodic coefficients and construct some assumptions guaranteeing the exponential stability (ES) of the trivial solutions of the system considered. To get specific conditions guaranteeing the ES, we use a modified Lyapunov functional. In conclusion, we get some estimates for the exponential decay of the solutions at infinity with the constructed sufficient conditions. We give two examples to demonstrate the applicability of the results obtained with the constructed assumptions.

Object Detection And Image Categorization By Transferring Commonsense Knowledge With Premises And Quantifiers, Irina Chernyavsky

Theses, Dissertations and Culminating Projects

Domestic, or household robots, are autonomous robots designed to make our home-life easier by performing chores and mundane tasks such as cleaning, or cooking. Currently domestic robots are specialized to complete a specific task and, therefore, are confined by factors such as mobility, size, and complexity. With the fast development of computer vision and robotics, the need for more compact, advanced and multi-task robots has emerged. Therefore, the robot needs to be multi-functional, able to discern the environment and the tasks. The aim of this paper is to categorize images in domestic robots as relevant to the culinary, laundry, vacuum …