Mathematics Commons^™

Commutative Doubly-Idempotent Semirings Determined By Chains And By Preorder Forests, Natanael Alpay, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

A commutative doubly-idempotent semiring (cdi-semiring) (S,V,·,0,1) is a semilattice (S,V,0) with x V 0 = x and a semilattices (S,·,1) with identity 1 such that x0 = 0, and x(y V z) = xy V xz holds for all x, y, z ϵ S. Bounded distributive lattices are cdi-semirings that satisfy xy = x ^ y, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a V-semilattice of height …

Circada: Shiny Apps For Exploration Of Experimental And Synthetic Circadian Time Series With An Educational Emphasis, Lisa Cenek, Liubou Klindziuk, Cindy Lopez, Eleanor Mccartney, Blanca Martin Burgos, Selma Tir, Mary E. Harrington, Tanya L. Leise

Psychology: Faculty Publications

Circadian rhythms are daily oscillations in physiology and behavior that can be assessed by recording body temperature, locomotor activity, or bioluminescent reporters, among other measures. These different types of data can vary greatly in waveform, noise characteristics, typical sampling rate, and length of recording. We developed 2 Shiny apps for exploration of these data, enabling visualization and analysis of circadian parameters such as period and phase. Methods include the discrete wavelet transform, sine fitting, the Lomb-Scargle periodogram, autocorrelation, and maximum entropy spectral analysis, giving a sense of how well each method works on each type of data. The apps also …

Decompositions Of Complete Uniform Multipartite Hypergraphs, Patrick Ward

Honors Projects

In recent years, researchers have studied the existence of complete uniform hypergraphs into small-order hypergraphs. In particular, results on small 3-uniform graphs including loose 3, 4, and 5 cycles have been studied, as well as 4-uniform loose cycles of length 3. As part of these studies, decompositions of multipartite hypergraphs were constructed. In this paper, we extend this work to higher uniformity and order as well as expand the class of hypergraphs.

On The Mersenne Prime Numbers, Julia Vanlandingham

Undergraduate Honors Thesis Projects

The prime numbers have been an important field of research for thousands of years and are intertwined with most other fields of mathematics. One topic that has piqued the interest of mathematicians young and old is the Mersenne prime numbers, which have applications in many mathematics and computer science fields. The Mersenne primes get a lot of attention because there is not much known about them. However, we do have a very simple primality test for Mersenne numbers, which is why the largest currently known primes are Mersenne primes. These primes are also very closely related to another class of …

Mathematical And Physical Description Of Conical Intersections, Nicholas Dewey

Honors Projects

Conical intersections (CIs) are points of degeneracy between two or more potential energy surfaces. Due to the intersection point, it is much easier for molecules to transition their electronic, vibrational, and rotational energies between surfaces. Therefore, CIs are critical to the study of excited states, particularly in the context of photochemistry. However, they are difficult to study because they interfere with traditional thermodynamics and deviate from the Born-Oppenheimer approximation. The purpose of this review is to explain why that is the case by exploring the foundation for the mathematics and physics behind CIs. Also included is an application of how …

Alexander Polynomials Of 3-Braid Knots, Marwa Emad Alrefai

Mathematical Sciences Theses

A knot is an embedding of a circle S¹ into the three-dimensional sphere S³. A component link is an embedding of n disjoint circles Ⅱᵢ^ⁿ =₁ S¹ into S³. The main objective of knot theory is to classify knots and links up to natural deformations called isotopies. While there is no simple algorithm that helps decide whether two given knots (or links) are equivalent, various topological invariants have been developed to help distinguish between non-equivalent knots and links. The Alexander polynomial ∆_L(t) is one of the oldest such tools. It …

Ground States And Gibbs Measures Of Λ-Model On Cayley Tree Of Order Two, Rauda Humaid Mohammed Al Shamsi

Mathematical Sciences Theses

Abstract

Statistical mechanics deals with the average properties of a mechanical system. Some examples are; the water in a kettle, the atmosphere inside a room and the number of atoms in a magnet bar. These kinds of systems are made up of a large number of components, usually molecules. The observer has restricted power to consider all the components. All that can be done is to specify a few average quantities of the system such as its density, pressure or temperature. The main objective of statistical mechanics is to predict the relationship between the observable macroscopic properties of the system, …

Searching Games: A Bound For The Responder, Jose Garcia

Student Scholars Day Oral Presentations

A searching game with two unknowns and a lie involves two players, the responder and the questioner. Before the start of the game, the two parties predetermine an amount of numbers n to consider, and how many questions k the questioner can ask before the game ends with a victory (or loss) for the responder. The responder thinks of two secret numbers. The questioner asks questions of the form "How many of your two numbers are in the subset Q of the set {0,...,n-1}?", in an attempt to search and find what the two secret numbers are. If the questioner …

Automated Conjecture Making: Domination On Planar Graphs, Jose Garcia

Student Scholars Day Posters

A planar graph G = (V,E) is a graph that can be embedded in the plane, i.e. it can be drawn in the plane so that no edges intersect except at the vertices. A subset S of vertices in a graph G is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. The domination number of a graph G is the smallest cardinality of a dominating set; we denote the domination number as γ(G). Automated conjecture making is the process of having a computer generate …

Structure-Activity Relationship Of Novel Diphenyl Ureas Targeting Mycobacterium, Piper Burghduf

Student Scholars Day Posters

In 2017, the World Health Organization reported that 10 million people were infected with tuberculosis, 1.6 million of whom died. Tuberculosis is caused by a bacterium called Mycobacterium tuberculosis, which primarily infects an individual’s lungs. Unfortunately, failure to adhere to the long and arduous drug regimen has contributed to the emergence of antibiotic-resistant strains of M. tuberculosis. Therefore, the need for novel antibiotics is imperative to saving millions of lives. Our lab has recently developed a family of diphenyl ureas that exhibited increased antimicrobial activity toward Mycobacterium. Reported herein is the continuation of our previous research involving the synthesis of …

Optimal Control Applied To Cancer Vaccine Protocols, Brady Fritz

Student Scholars Day Posters

This research focused on a mathematical model for the administration of a cancer vaccine. This model involves time delays, making it a more difficult optimal control problem to solve. The mathematical model describes the administration of the vaccine along with certain groups of cells affected over time. The model was programmed into a specialized software called the Sparse Optimization Suite which could output a solution. This solution was then analyzed in order to properly describe it.

Non-Attacking Queen And Rook Placements, Nicholas Layman

Student Scholars Day Posters

In 1848, Max Bezel introduced the problem of placing 8 queens on an 8 × 8 chess board so that none of the queens could attack each other. One generalization of this — the placement of n non-attacking queens on an n × n chess board — is the famous n-queens problem. A different but similar problem is that of placing non-attacking rooks on a generalized chess board which has connections to restricted permutations and has more general solutions known as compared to its queen counterpart. In this presentation, we investigate the intersection of these two problems — placing n …

Chebyshev Type Inequalities Involving The Fractional Integral Operator Containing Multi-Index Mittag-Leffler Function In The Kernel, S. D. Purohit, N. Jolly, M. K. Bansal, Jagdev Singh, Devendra Kumar

Applications and Applied Mathematics: An International Journal (AAM)

Recently, several authors have investigated Chebyshev type inequalities for numerous fractional integral operators. Being motivated by the work done by earlier researchers and their numerous applications in probability, transform theory, numerical quadrature, statistical problems and its significance in fractional boundary value problems. We aim to evaluate Chebyshev type inequalities involving fractional integral operator containing multi-index Mittag-Leffler function in the kernel. Admissible connections of the results mentioned in this article to those associated with previously established familiar fractional integral operators have been pointed out.

Ruscheweyh-Goyal Derivative Of Fractional Order, Its Properties Pertaining To Pre-Starlike Type Functions And Applications, Ritu Agarwal, G. S. Paliwal

Applications and Applied Mathematics: An International Journal (AAM)

The study of the operators possessing convolution form and their properties is considered advantageous in geometric function theory. In 1975 Ruscheweyh defined operator for analytic functions using the technique of convolution. In 2005, Goyal and Goyal generalized the Ruscheweyh operator to fractional order (which we call here Ruscheweyh-Goyal differential operator) using Srivastava-Saigo fractional differential operator involving hypergeometric function. Inspired by these earlier efforts, we discuss the properties of the Ruscheweyh-Goyal derivative of arbitrary order. We define a class of pre-starlike type functions involving the Ruscheweyh-Goyal fractional derivative and obtain the inclusion relation. Further, we prove that Ruscheweyh-Goyal derivative operator preserve …

Class Of Integrals Involving Generalized Hypergeometric Function, D. L. Suthar, Teklay Hailay, Hafte Amsalu, Jagdev Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we establish some definite integrals involving generalized hypergeometric function, product of algebraic functions, Jacobi function, Legendre function and general class of polynomials. Certain special cases of the main results are also pointed out.

Math 120bc – Precalculus, Virginia Thompson

Open Educational Resources

Students will explore advanced topics in algebra, functions and graphs, inverse functions, composite functions, polynomial and rational functions, trigonometry, exponential and logarithmic functions.

Local-Global Principles For Diophantine Equations, Benjamin Barham

Senior Honors Theses

The real number field, denoted ℝ, is the most well-known extension field of ℚ, the field of rational numbers, but it is not the only one. For each prime p, there exists an extension field ℚ_p of ℚ, and these fields, known as the p-adic fields, have some properties substantially different from ℝ. In this paper, we construct the p-adic numbers from the ground up and discuss the local-global principle, which concerns connections between solutions of equations found in ℚ and in ℚ_p. We state the Hasse-Minkowski theorem, which addresses a type …

Diameter Of 3-Colorable Graphs And Some Remarks On The Midrange Crossing Constant, Inne Singgih

Theses and Dissertations

The first part of this dissertation discussing the problem of bounding the diameter of a graph in terms of its order and minimum degree. The initial problem was solved independently by several authors between 1965 − 1989. They proved that for fixed δ ≥ 2 and large n, diam(G) ≤ 3n+ O(1). In 1989, Erdős, Pach, Pollack, and Tuza conjectured that the upper bound on the diameter can be improved if G does not contain a large complete subgraph K_k.

Let r, δ ≥ 2 be fixed integers and let G be a connected graph with n vertices …

Counting Number Fields By Discriminant, Harsh Mehta

Theses and Dissertations

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G.

We establish two main results in this work. In the first result we establish upper bounds for N_|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N _|F| (k,G;X) where G is a Frobenius …

Rationality Questions And The Derived Category, Alicia Lamarche

Theses and Dissertations

This document is roughly divided into four chapters. The first outlines basic preliminary material, definitions, and foundational theorems required throughout the text. The second chapter, which is joint work with Dr. Matthew Ballard, gives an example of a family of Fano arithmetic toric varieties in which the derived category is able to detect the existence of k-rational points. More succinctly, we show that if X is a generalized del Pezzo variety defined over a field k, then X contains a k-rational point (and is in fact k-rational, that is, birational to Pnk ) if and only if Db(X) admits a …