Mathematics Commons^™

Generalized Plithogenic Whole Hypersoft Set, Pfhss-Matrix, Operators And Applications As Covid-19 Data Structures, Shazia Rana, Muhammad Saeed, Madiha Qayyum, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This article is a preliminary draft for initiating and commencing a new pioneer dimension of expression. To deal with higher-dimensional data or information flowing in this modern era of information technology and artificial intelligence, some innovative super algebraic structures are essential to be formulated. In this paper, we have introduced such matrices that have multiple layers and clusters of layers to portray multi-dimensional data or massively dispersed information of the plithogenic universe made up of numerous subjects their attributes, and sub-attributes. For grasping that field of parallel information, events, and realities flowing from the micro to the macro level of …

Exploding Haystacks: Solutions For Fermi Questions, March 2023, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Shrub Sphericity: Solutions For Fermi Questions, January 2023, John Adam

Mathematics & Statistics Faculty Publications

Answers the questions: Question 1: a. Show that α = 4.836, and hence find the sphericity index for (i) a cube and (ii) two identical "kissing" spheres, i.e., spheres in tangential contact. b. Estimate your sphericity index.

Question 2: Estimate χ for the yucca plant in the picture. It is about 1 m in diameter.

Weierstrass Vertices On Finite Graphs, Abrianna L. Gill

Honors Undergraduate Theses

The intent of this thesis is to explore whether any patterns emerge among families or through graph operations regarding the appearance of Weierstrass vertices on graphs. Currently, patterns have been identified and proven on cycles, complete graphs, complete bipartite graphs, and the house and house-x graphs. A Python program developed as part of this thesis to perform the algorithms used in this analysis confirms these findings. This program also revealed a pattern: if v is a Weierstrass vertex, then the vertex v* added to the graph as a pendant vertex to v is also a Weierstrass vertex. The converse is …

Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, Carissa Slone

Theses and Dissertations--Mathematics

We provide the slice (co)towers of \(\Si{V} H_{C_2}\ul M\) for a variety of \(C_2\)-representations \(V\) and \(C_2\)-Mackey functors \(\ul M\). We also determine a characterization of all 2-slices of equivariant spectra over the Klein four-group \(C_2\times C_2\). We then describe all slices of integral suspensions of the equivariant Eilenberg-MacLane spectrum \(H\ulZ\) for the constant Mackey functor over \(C_2\times C_2\). Additionally, we compute the slices and slice spectral sequence of integral suspensions of $H\ulZ$ for the group of equivariance $Q_8$. Along the way, we compute the Mackey functors \(\mpi_{k\rho} H_{K_4}\ulZ\) and $\mpi_{k\rho} H_{Q_8}\ulZ$.

Normalization Techniques For Sequential And Graphical Data, Cole Pospisil

Theses and Dissertations--Mathematics

Normalization methods have proven to be an invaluable tool in the training of deep neural networks. In particular, Layer and Batch Normalization are commonly used to mitigate the risks of exploding and vanishing gradients. This work presents two methods which are related to these normalization techniques. The first method is Batch Normalized Preconditioning (BNP) for recurrent neural networks (RNN) and graph convolutional networks (GCN). BNP has been suggested as a technique for Fully Connected and Convolutional networks for achieving similar performance benefits to Batch Normalization by controlling the condition number of the Hessian through preconditioning on the gradients. We extend …

Lattice Minors And Eulerian Posets, William Gustafson

Theses and Dissertations--Mathematics

We study a partial ordering on pairings called the uncrossing poset, which first appeared in the literature in connection with a certain stratified space of planar electrical networks. We begin by examining some of the relationships between the uncrossing poset and Catalan combinatorics, and then proceed to study the structure of lower intervals. We characterize the lower intervals in the uncrossing poset that are isomorphic to the face lattice of a cube. Moving up in complexity certain lower intervals are isomorphic to the poset of simple vertex labeled minors of an associated graph.

Inspired by this structure, we define a …

Novel Architectures And Optimization Algorithms For Training Neural Networks And Applications, Vasily I. Zadorozhnyy

Theses and Dissertations--Mathematics

The two main areas of Deep Learning are Unsupervised and Supervised Learning. Unsupervised Learning studies a class of data processing problems in which only descriptions of objects are known, without label information. Generative Adversarial Networks (GANs) have become among the most widely used unsupervised neural net models. GAN combines two neural nets, generative and discriminative, that work simultaneously. We introduce a new family of discriminator loss functions that adopts a weighted sum of real and fake parts, which we call adaptive weighted loss functions. Using the gradient information, we can adaptively choose weights to train a discriminator in the direction …

Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

Theses and Dissertations--Mathematics

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …

Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson

Theses and Dissertations--Mathematics

Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map …

Toric Bundles As Mori Dream Spaces, Courtney George

Theses and Dissertations--Mathematics

A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream …

Geometry Of Pipe Dream Complexes, Benjamin Reese

Theses and Dissertations--Mathematics

In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of …

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, …