Mathematics Commons^™

Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox

Faculty Publications

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …

Normalized Ground State Of A Mixed Dispersion Nonlinear Schrodinger Equation With Combined Power-Type Nonlinearities, Zhouji Ma, Xiaojun Chang, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.

The Inextricability Of Students’ Mathematical And Physical Reasoning In Quantum Mechanics Problems, Kaitlyn Stephens Serbin, Megan Wawro

School of Mathematical & Statistical Sciences Faculty Publications

Reasoning with mathematics plays an important role in university students’ learning throughout their courses in the scientific disciplines, such as physics. In addition to understanding mathematical concepts and procedures, physics students often must mathematize physical constructs in terms of their associated mathematical structures and interpret mathematical entities in terms of the physical context. In this study, we investigate physics students’ reasoning about mathematics in relation to physics content addressed in two quantum mechanics problems. Through qualitative analysis of interview data from twelve students, results show that 1) students use intricate, nonuniform problem-solving methods with reasoning that moves fluidly between structural …

Visualizing The Standard Deviation Via Revolution Using R/Rstudio, Hieu Nguyen '25, Trung Pham '25, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

The standard deviation is a commonly used statistical measure to quantify the level of variation present in a set of numbers or in a random variable. Sarkar and Rashid (2016) introduced an interpretation of the population standard deviation as the radius of a cylinder with a volume equivalent to that of the solid of revolution when the 2-D graph of the empirical cumulative distribution function is revolved about the vertical line through the mean. This article demonstrates step-by-step how to use the RevSD package in R/RStudio to visualize the standard deviation of data using this innovative technique. The RevSD package …

The Eliahou-Kervaire Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Alexis Hardesty

School of Mathematical & Statistical Sciences Faculty Publications

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.

Faithful Representation Of Free Groups And Congruent Subgroups Of Sl3(Z), Julius Kurian

Theses

This thesis is concerned with the matrix representation of a free nonabelian group by matrices of size ≥ 3. We proceed from defining an equivalence class and then transitioning to free groups.We discuss in details the group G_n(k) which is the group generated by the matrices filled with first, (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The filled places are occupied by the same parameter k. An alternative proof for the known fact that G_n(3) is not …

Pricing Asian Options During Crises With The Impact Of Exogenous Event, Moath Ali Bakour

Theses

Asian options are financial derivatives products whose value depends on the value of underlying asset prices. These options are called path-dependent since their payoff is built on the prices of the underlying asset over some time. As in the case of the European options, the Asian option pricing problem is primarily subject to the prediction model for asset prices. The pioneer Black-Scholes model in the paper [2] suggests a GBM-Geometric Brownian motion. The Black-Scholes formula has several shortcomings. For instance, the Geometric Brownian motion does not take into consideration crises. Another problem in the Black- Scholes model is that it …

On The Projections And Unitary Groups Of Unital C*-Algebras, Fouzia Shaheen

Dissertations

H. Dye proved that the unitary group in a factor determines the algebraic type of that factor. Al-Rawashdeh, Booth and Giordano established that, for a large class of simple unital C*-algebras, an isomorphism between the unitary groups induces an isomorphism between their K₀-ordered group and 1-groups. Then using the results of Dadarlat-Elliot-Gong and Kirchberg-Phillips, the C*-algebras are isomorphic. Dye introduced special projections P_i,j (a) of the matrix algebra M_n(A), and he used it as a main tool to establish his results in the case of von Neumann factors. Precisely, in case of von Neumann algebra, …

Analyzing The Influence Of Design And Operating Conditions On Combustion And Emissions In Premixed Turbulent Flames: A Comprehensive Review, Medhat Elkelawy Prof. Dr. Eng., E. A. El Shenawy Prof. Dr., Hagar Alm-Eldin Bastawissi, Ibrahim Ali Mousa Eng., Mohamed M. Abdel-Raouf Ibrahim Dr. Eng.

Journal of Engineering Research

Recently, premixed combustion has dominated the field of combustion research worldwide. The current work is a review that addresses the effects of design and operating regimes on the combustion and emission characteristics of premixed turbulent flames. The study accounts for recent developments aimed at overcoming combustor operability issues that influence emissions and flame stability. Various experimental setups have been utilized in investigations, with results pertaining to performance and emissions concerning premixed turbulent flames. Thus, the objective of this paper is to provide a comprehensive review of the effects of swirl vane angles and equivalence fuel-air ratios for tests conducted both …

Error Analysis Of Learners’ Problem-Solving Abilities In Mathematics Courses: A Newman Error Analysis (Nea) Approach, Aprilita Ekasari, Don Jaya Putra

Jurnal Pendidikan Sains

This study endeavors to systematically discern the spectrum of learner errors manifesting in problem-solving processes within the domain of mathematics, specifically focusing on the intricacies of integration, utilizing Newman Error Analysis (NEA) as the guiding framework. The cohort under investigation comprises 14 first-year students hailing from the Department of Physics Education at Musamus University, Merauke. Employing a descriptive methodology underscored by a qualitative orientation, this research incorporates tests and interviews as primary instruments for data collection. The tests are designed to meticulously evaluate the array of errors committed by learners, as per the tenets of Newman’s model. Additionally, a subset …

Can Argument-Driven Inquiry-Assisted Virtual Laboratory Improve Pre-Service Physics Teachers’ Mastery Concept Of Simple Harmonic Oscillation?, Yolanda Dwi Vivian Karawahenni, Sutopo Sutopo, Arif Hidayat, Maya Mustika

Jurnal Pendidikan Sains

This study endeavors to ascertain the efficacy of the Argument-Driven Inquiry (ADI)- assisted virtual laboratory learning model in enhancing the comprehension of simple harmonic oscillation concepts among pre-service physics teachers. Employing a mixed- method approach with an embedded experimental design, the investigation employs a reasoned multiple-choice instrument comprising 15 questions. Analysis of the results reveals a noteworthy 48.38% increase in the average mastery of concepts between the pre-test and post-test, yielding an N-gain value of 0.63. Statistical scrutiny through paired sample t-tests corroborates significant disparities between pre-test and post-test scores, evidenced by a tcount (16.28) exceeding the critical ttable value …

Fostering Engagement And Learning Outcomes: A Comparative Analysis Of Ethnochemical And Stem-Based Pedagogies For Chemistry Learning In Vocational High Schools, Maulidya Husnul Khatimah, Lismi Animatul Chisbiyah

Jurnal Pendidikan Sains

In contemporary education, the perceived difficulty of learning chemistry, attributed to its abstract nature, remains a challenge for many students. This difficulty is compounded by a notable lack of alignment between chemistry concepts and students’ daily experiences, despite the inherent relevance of chemistry to their everyday phenomena. Building on previous research affirming the efficacy of ethnochemical modules in enhancing student learning and achievement, this study investigates the suitability of a STEM-based approach for chemistry education, particularly in vocational high school. The research methodology employs a literature study method, utilizing the PICO(T) framework for keyword selection. A rigorous review and analysis …

Differences In Biology Students’ Self-Regulation During Online And Offline Learning And Its Relationship To Learning Outcomes, Adi Kus Rochman Jk, Dwi Listyorini, Fatchur Rohman, Yessi Affriyenni

Jurnal Pendidikan Sains

The cultivation of self-regulatory skills holds paramount importance for individuals’ future success, particularly in the context of evolving educational modalities prompted by the Covid-19 pandemic. As learning transitions between online during the pandemic and offline post-pandemic, understanding the nuances of self-regulation becomes imperative. Thus, this study sought to achieve three objectives: (1) analyzing disparities in student self-regulation between online and offline settings, (2) examining the interplay between online and offline self-regulation, and (3) scrutinizing the relationship between self-regulation and learning outcomes. Employing a methodological approach involving questionnaire administration, interviews, and collecting academic scores data, data analysis was conducted utilizing the …

An Exploration Of Secondary School Science Teacher’S Epistemological Belief In Malang, Anindyta Nur Rizkyana Safitri, Endang Purwaningsih, Nuril Munfaridah, Alif Darmawan

Jurnal Pendidikan Sains

The objective of this study is to delve into the epistemological convictions of secondary school science teachers in Malang across various tenures, namely senior teachers, junior teachers, and pre-service teachers. Employing an explanatory sequential design model, the research employs a mixed-method approach. Quantitative data, gathered through the administration of the Scientific Epistemology Beliefs Questionnaire (n = 30), are complemented by qualitative insights drawn from interviews with teachers (n = 6) concerning their epistemological orientations. Descriptive statistics and qualitative analyses, utilizing an a priori coding technique aligned with the study’s conceptual framework, were conducted on the interview data. Quantitative analysis reveals …

Separation And Relative Quasiconvexity Criteria For Relatively Geometric Actions, Eduard Einstein

Mathematics & Statistics Faculty Works

Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite subgraphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that any group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube …

On The Subelliptic And Subparabolic Infinity Laplacian In Grushin-Type Spaces, Zachary Forrest

USF Tampa Graduate Theses and Dissertations

This thesis poses the ∞-Laplace equation in Grushin-type spaces. Grushin-type spaces G are defined by the vector fields which serve as a basis for their tangent spaces; by weighting the canonical (Euclidean) directional vectors {∂/∂x_i}ⁿ_i=1 by functions ρ_i that obey certain technical assumptions, we produce a class of metric spaces in which certain directions may not be accessible at all points in the space. We prove the existence and uniqueness of viscosity solutions to both Dirichlet problems and Cauchy-Dirichlet problems involving the∞-Laplacian over bounded Grushin-type domains. The main tool in proving uniqueness of …

The Invariantring Package For Macaulay2, Luigi Ferraro, Federico Galetto, Francesca Gandini, Hang Huang, Matthew Mastroeni, Xianglong Ni

School of Mathematical & Statistical Sciences Faculty Publications

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups, as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

A Cohomological Perspective To Nonlocal Operators, Nicholas White

Honors Program: Senior Projects (Public)

Nonlocal models have experienced a large period of growth in recent years. In particular, nonlocal models centered around a finite horizon have been the subject of many novel results. In this work we consider three nonlocal operators defined via a finite horizon: a weighted averaging operator in one dimension, an averaging differential operator, and the truncated Riesz fractional gradient. We primarily explore the kernel of each of these operators when we restrict to open sets. We discuss how the topological structure of the domain can give insight into the behavior of these operators, and more specifically the structure of their …

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Quasistationary Distribution For The Invasion Model On A Complete Bipartite Graph, Clayton Allard, Iddo Ben-Ari, Shrikant Chand, Van Hovenga, Edith Lee, Julia Shapiro

Journal of Stochastic Analysis

No abstract provided.

The Homotopy Cardinality Of The Representation Category, Justin Murray

LSU Doctoral Dissertations

Given a Legendrian knot in (R^3, ker(dz − ydx)) one can assign a combinatorial invariants called ruling polynomials. These invariants have been shown to recover not only a (normalized) count of augmentations but are also closely related to a categorical count of augmentations in the form of the homotopy cardinality of the augmentation category. In this article, we prove that that the homotopy cardinality of the n-dimensional represen- tation category is a multiple of the n-colored ruling polynomial. Along the way, we establish that two n-dimensional representations are equivalent in the representation category if they are “conjugate homotopic”. We also …

The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain

Rose-Hulman Undergraduate Mathematics Journal

We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.

Understanding Waveguides In Resonance, Pieter Johannes Daniel Vandenberge

Dissertations and Theses

Several important classes of modern optical waveguides, including anti-resonant reflecting and photonic bandgap fibers, make use of geometries that guide energy in low refractive index material, a property that makes them of significant interest in numerous applications, notably including high-power delivery and guidance. These waveguides frequently exhibit resonance phenomena, in which their ability to propagate an input signal is sharply curtailed at particular operating frequencies. In this work we detail new advances in understanding these resonance effects and their implications for numerical modeling of these structures.

Part 1 focuses on the fields of slab waveguides, relatively simple structures for which …

Asymptotics Of The Rate Function In The Large Deviation Principle For Sums Of Independent Identically Distributed Random Variables, Iosif Pinelis

Michigan Tech Publications

Let Λ∗be the rate function in the large deviation principle for the sums X1 + · · · + Xn of independent identically distributed random variables X1, X2, …. It is shown that Λ∗(x) ∼ − ln P(X1 ≥ x) (as x → ∞) if and only if ln P(X1 ≥ x) ∼ L0(x) for some concave function L0. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in t ≥ 0 of the Bernstein–Chernoff upper bound e−txEetX1 on P(X1 ≥ x), which is amenable to analysis and, at the same time, is close …

Regularization By Deep Learning In Signal Processing, Carlos Ramirez Villamarin, Erwin Suazo, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we explore a new idea of using deep learning representations as a principle for regularization in inverse problems for digital signal processing. Specifically, we consider the standard variational formulation, where a composite function encodes a fidelity term that quantifies the proximity of the candidate solution to the observations (under a physical process), and a second regularization term that constrains the space of solutions according to some prior knowledge. In this work, we investigate deep learning representations as a means of fulfilling the role of this second (regularization) term. Several numerical examples are presented for signal restoration under …

Wang Tilings In Arbitrary Dimensions, Ian Tassin

Rose-Hulman Undergraduate Mathematics Journal

This paper makes a new observation about arbitrary dimensional Wang Tilings,
demonstrating that any d -dimensional tile set that can tile periodically along d − 1 axes must be able to tile periodically along all axes.
This work also summarizes work on Wang Tiles up to the present day, including
definitions for various aspects of Wang Tilings such as periodicity and the validity of a tiling. Additionally, we extend the familiar 2D definitions for Wang Tiles and associated properties into arbitrary dimensional spaces. While there has been previous discussion of arbitrary dimensional Wang Tiles in other works, it has been …

Algorithmic Design And Computational Modeling Using Dynamic Spectrum Allocation Techniques To Optimize Bandwidth Management In Wireless Communication Systems, Ankit Walishetti

Distinguished Student Work

This study aims to address the pressing need for efficient spectrum management methodologies in wireless communication systems by developing innovative sorting and allocation algorithms. Leveraging Dynamic Spectrum Allocation (DSA) techniques, this research devises strategies to optimize the utilization of bandwidth within existing spectrum space, ultimately reducing the need for network infrastructure expansion.

Ensuring thorough coverage of DSA techniques, 5 distinct transmitter sorting algorithms were programmed and tested across 8 performance metrics designed to measure specific capabilities. For consistency, a single bandwidth allocation program was designed to ‘pack’ transmitters starting from the left endpoint of the spectrum space. Progressively varying the …

Fusion In Supersolvable Hall Subgroups, Muhammet Yasi̇r Kizmaz

Turkish Journal of Mathematics

Let H be a supersolvable Hall π -subgroup of a finite group G. We prove that G has a normal π -complement if and only if H controls G-fusion in H.

Twisted Sasaki Metric On The Unit Tangent Bundle And Harmonicity, Liana Lotarets

Turkish Journal of Mathematics

The paper deals with the twisted Sasaki metric on the unit tangent bundle of n–dimensional Riemannian manifold Mn . The main purpose of the research is to find deformations that preserve the existence harmonic left-invariant unit vector fields on 3-dimensional unimodular Lie groups G with the left invariant metric and harmonic maps G → T1G in case of twisted Sasaki metric on the unit tangent bundle. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map Mn → T1Mn are obtained. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map M2 …

Langevin Delayed Equations With Prabhakar Derivatives Involving Two Generalized Fractional Distinct Orders, Mustafa Aydin

Turkish Journal of Mathematics

This paper is devoted to defining the delayed analogue of the Mittag-Leffler type function with three parameters and investigating a representation of a solution to Langevin delayed equations with Prabhakar derivatives involving two generalized fractional distinct orders, which are first introduced and investigated, by means of the Laplace integral transform. It is verified by showing the solution satisfies the introduced system. Special cases which are also novel are presented as examples. The findings are illustrated with the help of the RLC circuits.