Physical Sciences and Mathematics Commons^™

Teaching Mathematics With Poetry: Some Activities, Alexis E. Langellier

Journal of Humanistic Mathematics

During the summer of 2021, I experimented with a new way of getting children excited about mathematics: math poetry. Math can be a trigger word for some children and many adults. I wanted to find a way to make learning math fun—without the students knowing they’re doing math. In this paper I describe some activities I used with students ranging from grades K-12 to the college level and share several poem examples, from students in grades two to eight.

Matrix Completion Problems For The Positiveness And Contraction Through Graphs, Louis C. Christopher

Theses and Dissertations

In this work, we study contractive and positive real matrix completion problems which are motivated in part by studies on sparce (or dense) matrices for weighted sparse recovery problems and rating matrices with rating density in recommender systems. Matrix completions problems also have many applications in probability and statistics, chemistry, numerical analysis (e.g. optimization), electrical engineering, and geophysics. In this paper we seek to connect the contractive and positive completion property to a graph theoretic property. We then answer whether the graphs of real symmetric matrices having loops at every vertex have the contractive completion property if and only if …

Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roma Sverdlov

Mathematics & Statistics ETDs

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points …

From Mirrors To Wallpapers: A Virtual Math Circle Module On Symmetry, Nicole A. Sullivant, Christina L. Duron, Douglas T. Pfeffer

Journal of Math Circles

Symmetry is a natural property that children see in their everyday lives; it also has deep mathematical connections to areas like tiling and objects like wallpaper groups. The Tucson Math Circle (TMC) presents a 7-part module on symmetry that starts with reflective symmetry and culminates in the deconstruction of wallpapers into their ‘generating tiles’. This module utilizes a scaffolded, hands-on approach to cover old and new mathematical topics with various interactive activities; all activities are made available through free web-based platforms. In this paper, we provide lesson plans for the various activities used, and discuss their online implementation with Zoom, …

When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, deep neural networks work better than the traditional "shallow" ones, however, in some cases, the shallow neural networks lead to better results. At present, deciding which type of neural networks will work better is mostly done by trial and error. It is therefore desirable to come up with some criterion of when deep learning is better and when shallow is better. In this paper, we argue that this depends on whether the corresponding situation has natural symmetries: if it does, we expect deep learning to work better, otherwise we expect shallow learning to be more effective. …

Making Upper-Level Math Accessible To A Younger Audience, Allyson Roller

WWU Honors College Senior Projects

Symmetry is all around us. It appears on fabrics and on the buildings that surround us. Believe it or not, there is actually quite a bit of math that goes into generating these patterns, which are known as the seven frieze patterns. In my work, I explain how each unique pattern is generated using different types of symmetries. I also created a PDF of a children’s book about frieze patterns to ensure that people of all ages have the opportunity to learn about seemingly complex patterns.

On The Quadruple Sequence Spaces Of Fuzzy Complex Numbers, Aqeel Mohammed Hussein

Al-Qadisiyah Journal of Pure Science

In this paper, the quadruple sequence spaces of fuzzy complex numbers are shown, and several features such as solidity, symmetry, monotonicity, and convergence-free are discussed.

Many Known Quantum Algorithms Are Optimal: Symmetry-Based Proofs, Vladik Kreinovich, Oscar Galindo, Olga Kosheleva

Departmental Technical Reports (CS)

Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal -- in some reasonable sense -- or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries).

Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.

Doctoral Theses

No abstract provided.

Higher Cohomologies For Presheaves Of Commutative Monoids, Pilar Carrasco, Antonio M. Cegarra

Turkish Journal of Mathematics

We present an extension of the classical Eilenberg-MacLane higher order cohomology theories of abelian groups to presheaves of commutative monoids (and of abelian groups, then) over an arbitrary small category. These high-level cohomologies enjoy many desirable properties and the paper aims to explore them. The results apply directly in several settings such as presheaves of commutative monoids on a topological space, simplicial commutative monoids, presheaves of simplicial commutative monoids on a topological space, commutative monoids or simplicial commutative monoids on which a fixed monoid or group acts, and so forth. As a main application, we state and prove a precise …

A New Approach To The Q-Conjugacy Character Tables Of Finite Groups, Ali Moghani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study the Q-conjugacy character table of an arbitrary finite group and introduce a general relation between the degrees of Q-conjugacy characters with their corresponding reductions. This could be accomplished by using the Hermitian symmetric form. We provide a useful technique to calculate the character table of a finite group when its corresponding Qconjugacy character table is given. Then, we evaluate our results in some useful examples. Finally, by using GAP (Groups, Algorithms and Programming) package, we calculate all the dominant classes of the sporadic Conway group Co₂ enabling us to find all possible the integer-valued …

Strategies And Algorithms Of Sudoku, Callie Weaver

Mathematics Senior Capstone Papers

This paper discusses different strategies for the game of Sudoku and how those strategies relate to other problem solving techniques while also attempting to use those other techniques in a way that improves the strategies for Sudoku. This includes a thorough analysis of the general algorithm and an algorithm that is formed by the Occupancy Theorem and Preemptive Sets. This paper also compares these algorithms that directly relate to Sudoku with algorithms to similar combinatorial problems such as the Traveling Salesman problem and more. With the study of game theory becoming more popular, these strategies have also been shown to …

The Mystery Of Frobenius Symmetry, Maciej Piwowarczyk

DePaul Discoveries

In this project we studied the mathematical concept of the Frobenius number and some curious patterns that come with it. One common example of the Frobenius number is the Coin Problem: If handed two denominations of coins, say 4¢ and 5¢, and asked to create all possible values, we will eventually find ourselves in a position where we can make any value. With 4¢ and 5¢ coins, we can create any value above 11¢, but not 11¢ itself. So, that makes 11 the Frobenius number of 4 and 5. What we explore in this paper is a pattern we call …

Patterns, Symmetries, And Mathematical Structures In The Arts, Sarah C. Deloach

Honors College Theses

Mathematics is a discipline of academia that can be found everywhere in the world around us. Mathematicians and scientists are not the only people who need to be proficient in numbers. Those involved in social sciences and even the arts can benefit from a background in math. In fact, connections between mathematics and various forms of art have been discovered since as early as the fourth century BC. In this thesis we will study such connections and related concepts in mathematics, dances, and music.

Infinite Mode Quantum Gaussian States., Tiju Cherian John Dr.

Doctoral Theses

No abstract provided.

Symmetry And Measuring: Ways To Teach The Foundations Of Mathematics Inspired By Yupiaq Elders, Jerry Lipka, Barbara Adams, Monica Wong, David Koester, Karen Francois

Journal of Humanistic Mathematics

Evident in human prehistory and across immense cultural variation in human activities, symmetry has been perceived and utilized as an integrative and guiding principle. In our long-term collaborative work with Indigenous Knowledge holders, particularly Yupiaq Eskimos of Alaska and Carolinian Islanders in Micronesia, we were struck by the centrality of symmetry and measuring as a comparison-of-quantities, and the practical and conceptual role of qukaq [center] and ayagneq [a place to begin]. They applied fundamental mathematical principles associated with symmetry and measuring in their everyday activities and in making artifacts. Inspired by their example, this paper explores the question: Could symmetry …

Fast Adjustable Npn Classification Using Generalized Symmetries, Xuegong Zhou, Lingli Wang, Peiyi Zhao, Alan Mishchenko

Mathematics, Physics, and Computer Science Faculty Articles and Research

NPN classification of Boolean functions is a powerful technique used in many logic synthesis and technology mapping tools in FPGA design flows. Computing the canonical form of a function is the most common approach of Boolean function classification. In this paper, a novel algorithm for computing NPN canonical form is proposed. By exploiting symmetries under different phase assignments and higher-order symmetries of Boolean functions, the search space of NPN canonical form computation is pruned and the runtime is dramatically reduced. The algorithm can be adjusted to be a slow exact algorithm or a fast heuristic algorithm with lower quality. For …

Crystallographic Patterns In Philippine Indigenous Textiles, Ma. Louise Antonette N. De Las Peñas, Agnes Garciano, Debbie Marie Verzosa, Eduard C. Taganap

Mathematics Faculty Publications

The aim of this study was to analyze a representative sample of Philippine indigenous textiles in order to capture the range of symmetries and color symmetries present. This paper examines the existence of symmetries in finite designs, and classifies the plane-group and frieze-group symmetry types of the repeated patterns in woven textiles. The tendency of a particular symmetry to be more or less common than another can indicate relationships between the symmetries and the weaving technique or the culture that produced them. This paper will also examine designs and patterns with color symmetry found in these textiles. The sample consisted …

Symmetry Of Numerical Range And Semigroup Generation Of Infinite Dimensional Hamiltonian Operators, Junjie Huang, Jie Liu, Alatancang Chen

Turkish Journal of Mathematics

This paper deals with the infinite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating $C_0$ semigroups is further given.

Geometric Serendipity, Dakota Becker

AUCTUS: The Journal of Undergraduate Research and Creative Scholarship

The central focus of my practice is the serendipitous exploration into geometry, symmetry, design, and color. I have found more and more that the affinity I have for hard-edge geometric abstraction is a deeper reflection of the way in which I process my thoughts and surroundings. In the past year, I have sought to challenge myself by questioning the core of my practice and pushing it to go beyond its individual elements. In this way, I seek to create work that is more than its parts. As a result, I have become more purposeful with my designs and push both …

The Roots Of Early Group Theory In The Works Of Lagrange, Janet Heine Barnett

Abstract Algebra

No abstract provided.

Essays On Economic Behaviour And Regulation., Subrato Banerjee Dr.

Doctoral Theses

To sum up, this thesis looks at agent behaviour in the laboratory, in the field, and in the market. Firstly, we impose a requirement in the laboratory (Chapter 2) that mimics a regulatory environment (similar to the introduction of a maximum retail price, or a legal fare subject to which an economic transaction must take place), and study individual behaviour subject to our (imposed) requirements. We then study the effect of real-life regulation on the behaviour of economic agents in the field. While the effect of regulation is seen in the field (that is, we see that many auto drivers …

On Commensurability And Symmetry, David Pierce

Journal of Humanistic Mathematics

Commensurability and symmetry have diverged from a common Greek origin. We review the history of this divergence. In mathematics, symmetry is now a kind of measure that is different from size, though analogous to it. Size being given by numbers, the concept of numbers and their equality comes into play. For Euclid, two magnitudes were symmetric when they had a common measure; also, numbers were magnitudes, commonly represented as bounded straight lines, for which equality was congruence. When Billingsley translated Euclid into English in the sixteenth century, he used the word "commensurable" for Euclid's symmetric magnitudes; but the word had …

Polyhedral Painting With Group Averaging, Frank A. Farris, Ryan Tsao

Mathematics and Computer Science

The technique of group-averaging produces colorings of a sphere that have the symmetries of various polyhedra. The concepts are accessible at the undergraduate level, without being well-known in typical courses on algebra or geometry. The material makes an excellent discovery project, especially for students with some background in computer science; indeed, this is where the authors first worked through the material, as teacher and student, producing a previously unseen type of artistic image. The process uses a photograph as a palette, whose colors and textures appear in kaleidoscopic form on the surface of a sphere. We depict tetrahedral, octahedral, and …

Kaleidoscopes, Chessboards, And Symmetry, Tricia M. Brown

Journal of Humanistic Mathematics

This paper describes the n-queens problem on an n by n chessboard. We discuss the possible symmetries of n-queens solutions and show how solutions to this classical chess question can be used to create examples of colorful artwork.

Solving Ordinary Differential Equations Using Differential Forms And Lie Groups, Richard M. Shumate

Senior Honors Theses

Differential equations have bearing on practically every scientific field. Though they are prevalent in nature, they can be challenging to solve. Most of the work done in differential equations is dependent on the use of many methods to solve particular types of equations. Sophus Lie proposed a modern method of solving ordinary differential equations in the 19th century along with a coordinate free variation of finding the infinitesimal generator by combining the influential work of Élie Cartan among others in the field of differential geometry. The driving idea behind using symmetries to solve differential equations is that there exists a …

Determination Of Lie Superalgebras Of Supersymmetries Of Super Differential Equations, Xuan Liu

Electronic Thesis and Dissertation Repository

Superspaces are an extension of classical spaces that include certain (non-commutative) supervariables. Super differential equations are differential equations defined on superspaces, which arise in certain popular mathematical physics models. Supersymmetries of such models are superspace transformations which leave their sets of solutions invariant. They are important generalization of classical Lie symmetry groups of differential equations.

In this thesis, we consider finite-dimensional Lie supersymmetry groups of super differential equations. Such supergroups are locally uniquely determined by their associated Lie superalgebras, and in particular by the structure constants of those algebras. The main work of this thesis is providing an algorithmic method …

On The Representation Of Inverse Semigroups By Difunctional Relations, Nathan Bloomfield

Graduate Theses and Dissertations

A semigroup S is called inverse if for each s in S, there exists a unique t in S such that sts = s and tst = t. A relation σ contained in X x Y is called full if for all x in X and y in Y there exist x' in X and y' in Y such that (x, y') and (x', y) are in σ, and is called difunctional if σ satisfies the equation σ σ^-1 σ = σ. Inverse semigroups were introduced by Wagner and Preston in 1952 and 1954, respectively, and difunctional relations were …

Logarithmic Spirals And Projective Geometry In M.C. Escher's "Path Of Life Iii", Heidi Burgiel, Matthew Salomone

Journal of Humanistic Mathematics

M.C. Escher's use of dilation symmetry in Path of Life III gives rise to a pattern of logarithmic spirals and an oddly ambiguous sense of depth.

Bi-Integrable And Tri-Integrable Couplings And Their Hamiltonian Structures, Jinghan Meng

USF Tampa Graduate Theses and Dissertations

An investigation into structures of bi-integrable and tri-integrable couplings is undertaken. Our study is based on semi-direct sums of matrix Lie algebras. By introducing new classes of matrix loop Lie algebras, we form new Lax pairs and generate several new bi-integrable and tri-integrable couplings of soliton hierarchies through zero curvature equations. Moreover, we discuss properties of the resulting bi-integrable couplings, including infinitely many commuting symmetries and conserved densities. Their Hamiltonian structures are furnished by applying the variational identities associated with the presented matrix loop Lie algebras.

The goal of this dissertation is to demonstrate the efficiency of our approach and …