Mathematics Commons^™

Labs For Calculus: Learning Through Collaborative Discovery, Ryan Mckinney

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

No abstract provided.

A Supply Chain Profile Of A School-Based Feeding Program Using The Centralized Kitchen Model, Eden Delight Miro, J. Lemuel Martin, Leslie Lopez, Joselito Secson, Carmela Oracion, Jhoel Loanzon

Mathematics Faculty Publications

There has recently been renewed interest and a growing demand for school feeding programs. In the Philippines, the government, through the Department of Education (DepEd), Department of Social Welfare and Development (DSWD), and non-government organizations such as the Ateneo Center for Educational Development (ACED), has initiated such programs to address the prevalence of malnutrition among Filipino school-age children. In 2011, ACED introduced the ACED Blueplate Centralized Kitchen (ABCK) model for large-scale school feeding. This study aims to provide a supply chain profile of the first and largest city-wide implementation of the ABCK model in the Philippines to date, which is …

String C-Groups Of Order 1024, Yasushi Gomi, Mark L. Loyola, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This paper determines the nondegenerate string C-groups of order 1024. For groups of rank 3, we use the technique of central extension of string C-groups of order 512. For groups of rank at least 4, we compute for quotients of universal string C-groups.

Deep Linear Networks With Arbitrary Loss: All Local Minima Are Global, Thomas Laurent

Mathematics, Statistics and Data Science Faculty Works

We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possible in the following sense: If the loss is convex and Lipschitz but not differentiable then deep linear networks can have sub-optimal local minima.

The Multilinear Structure Of Relu Networks, Thomas Laurent

Mathematics, Statistics and Data Science Faculty Works

We study the loss surface of neural networks equipped with a hinge loss criterion and ReLU or leaky ReLU nonlinearities. Any such network defines a piecewise multilinear form in parameter space. By appealing to harmonic analysis we show that all local minima of such network are non-differentiable, except for those minima that occur in a region of parameter space where the loss surface is perfectly flat. Non-differentiable minima are therefore not technicalities or pathologies; they are heart of the problem when investigating the loss of ReLU networks. As a consequence, we must employ techniques from nonsmooth analysis to study these …

Stochastic Maximum Principle For Partial Information Optimal Investment And Dividend Problem Of An Insurer, Yanping Ma

Mathematics, Statistics and Data Science Faculty Works

We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Lévy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions …

An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, Jacob Liddy

Williams Honors College, Honors Research Projects

An abundant number is said to be primitive if none of its proper divisors are abundant. Dickson proved that for an arbitrary positive integer d there exists only finitely many odd primitive abundant numbers having exactly d prime divisors. In this paper we describe a fast algorithm that finds all primitive odd numbers with d unique prime divisors. We use this algorithm to find all the number of odd primitive abundant numbers with 6 unique Divisors. We use this algorithm to prove that an odd weird number must have at least 6 prime divisors.

Group Rings, Christopher Wrenn

Masters Essays

No abstract provided.

Mathematical Curves In The High School Classroom, Magdalena Zook

Masters Essays

No abstract provided.

Advanced Enrichment Topics In An Honors Geometry Course, Kayla Woods

Masters Essays

No abstract provided.

A General Lower Bound On Gallai-Ramsey Numbers For Non-Bipartite Graphs, Colton Magnant

Theory & Applications of Graphs

Given a graph H and a positive integer k, the k-color Gallai-Ramsey number gr_k(K3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph K_n contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs H to a value that we conjecture to …

Oer Women In Math Course, Cynthia J. Huffman Ph.D.

Open Educational Resources - Math

This Women in Math course takes a look at 11 notable women mathematicians through readings, quizzes, discussion boards, and activities related to the mathematical contributions of the women. Several of the activities are original and can be found separately at Pittsburg State University Digital Commons (https://digitalcommons.pittstate.edu/ - search by author for Huffman, Cynthia).

Course URL

The OER Women in Math course can be accessed at https://pittstate.instructure.com/courses/1081791 .

Geometry Of Derived Categories On Noncommutative Projective Schemes, Blake Alexander Farman

Theses and Dissertations

Noncommutative Projective Schemes were introduced by Michael Artin and J.J. Zhang in their 1994 paper of the same name as a generalization of projective schemes to the setting of not necessarily commutative algebras over a commutative ring. In this work, we study the derived category of quasi-coherent sheaves associated to a noncommutative projective scheme with a primary emphasis on the triangulated equivalences between two such categories.

We adapt Artin and Zhang’s noncommutative projective schemes for the language of differential graded categories and work in Ho (dgcatk), the homotopy category of differential graded categories, making extensive use of Bertrand Toën’s Derived …

Quick Trips: On The Oriented Diameter Of Graphs, Garner Paul Cochran

Theses and Dissertations

In this dissertation, I will discuss two results on the oriented diameter of graphs with certain properties. In the first problem, I studied the oriented diameter of a graph G. Erdos et al. in 1989 showed that for any graph with |V | = n and δ(G) = δ the maximum the diameter could possibly be was 3 n/ δ+1. I considered whether there exists an orientation on a given graph with |G| = n and δ(G) = δ that has a small diameter. Bau and Dankelmann (2015) showed that there is an orientation of diameter 11 n/ δ+1 + …

Local Rings And Golod Homomorphisms, Thomas Schnibben

Theses and Dissertations

The Poincaré series of a local ring is the generating function of the Betti numbers for the residue field. The question of when this series represents a rational function is a classical problem in commutative algebra. Golod rings were introduced by Golod in 1962 and are one example of a class of rings that have rational Poincaré series. The idea was generalized to Golod homomorphisms by Levin in 1975.

In this paper we prove two homomorphisms are Golod. The first is a class of ideals such that the natural projection to the quotient ring is a Golod homomorphism. The second …

Special Fiber Rings Of Certain Height Four Gorenstein Ideals, Jaree Hudson

Theses and Dissertations

Let S be a set of four variables, k a field of characteristic not equal to two such that k contains all square roots, and I a height four Gorenstein ideal of k[S] generated by nine quadratics so that I has a Gorenstein-linear resolution. We define a complex X• so that each module of X• is the tensor product of a certain polynomial ring Q in nine variables and a direct sum of indecomposable k[Sym(S)]-modules and the differential maps are Q- and k[Sym(S)]-module homomorphisms. Work with the Macaulay2 software suggests that H0(X•) is the special fiber ring of I and …

Turán Problems And Spectral Theory On Hypergraphs And Tensors, Shuliang Bai

Theses and Dissertations

Turán problems on uniform hypergraphs have been actively studied for many decades. However, on non-uniform hypergraphs, these problems are rarely considered. We refer a non-uniform hypergraph as an R-hypergraph where R is the set of cardinalities of all edges. An R-graph H is called degenerate if it has the smallest Turán density |R(H)|−1. What do the degenerate R-graphs look like? For the special case R = {r}, the answer to this question is simple: they are r-partite r-uniform hypergraphs. However, it is more intrigue for the other cases. A degenerate hypergraph is called trivial if it is contained in the …

A Quest For Positive Definite Matrices Over Finite Fields, Erin Patricia Hanna

Theses and Dissertations

Positive definite matrices make up an interesting and extremely useful subset of Hermitian matrices. They are particularly useful in exploring convex functions and finding minima for functions in multiple variables. These matrices admit a plethora of equivalent statements and properties, one of which is an existence of a unique Cholesky decomposition. Positive definite matrices are not usually considered over finite fields as some of the definitions and equivalences are quickly seen to no longer hold. Motivated by a result from the theory of pressing sequences, which almost mirrors an equivalent statement for positive definite Hermitian matrices, we consider whether any …

States And The Numerical Range In The Regular Algebra, James Patrick Sweeney

Theses and Dissertations

In this dissertation we investigate the algebra numerical range defined by the Banach algebra of regular operators on a Dedekind complete complex Banach lattice, i.e., V (Lr(E), T) = {Φ(T) : Φ ∈ Lr(E)∗, ||Φ|| = 1 = Φ(I)}. For T in the center Z(E) of E we prove that V (Lr(E), T) = co(σ(T)). For T ⊥ I we prove that V (Lr(E), T) is a disk centered at the origin. We then consider the part of V (Lr(E), T) obtained by restricting ourselves to positive states Φ ∈ Lr(E)∗. In this case we show that we get a …

Optimal Supply Delivery Under Military Specific Constraints, Talena Fletcher

College of Graduate Studies: Theses & Dissertations

Through-out military history, the need to safely and effectively allocate resources to various military operations was a task of extreme importance. Satisfying the needs of multiple consumers by optimally pairing with appropriate suppliers falls into the category of vehicle routing problems (VRP), which has been intensively studied over the years. In general, finding the optimal solution to VRP is known to be NP-hard. The proposed solutions rely on mathematical programming and the size of the problems that can be optimally solved is typically limited. In military settings, balancing the needs of multiple consumers with the current operational environment has always …

Old English Character Recognition Using Neural Networks, Sattajit Sutradhar

College of Graduate Studies: Theses & Dissertations

Character recognition has been capturing the interest of researchers since the beginning of the twentieth century. While the Optical Character Recognition for printed material is very robust and widespread nowadays, the recognition of handwritten materials lags behind. In our digital era more and more historical, handwritten documents are digitized and made available to the general public. However, these digital copies of handwritten materials lack the automatic content recognition feature of their printed materials counterparts. We are proposing a practical, accurate, and computationally efficient method for Old English character recognition from manuscript images. Our method relies on a modern machine learning …

Survey Of Results On The Schrodinger Operator With Inverse Square Potential, Richardson Saint Bonheur

College of Graduate Studies: Theses & Dissertations

In this paper we present a survey of results on the Schrodinger operator with Inverse ¨ Square potential, L_a= −∆ + a/|x|^2 , a ≥ −( d−2/2 )^2. We briefly discuss the long-time behavior of solutions to the inter-critical focusing NLS with an inverse square potential(proof not provided). Later we present spectral multiplier theorems for the operator. For the case when a ≥ 0, we present the multiplier theorem from Hebisch [12]. The case when 0 > a ≥ −( d−2/2 )^2 was explored in [1], and their proof will be presented for completeness. No improvements on the sharpness …

Classical And Quantum Kac’S Chaos, Rade Musulin

Theses and Dissertations

In 1956 Kac studied the Boltzmann equation, an integro-differential equation which describes the density function of the distribution of the velocities of the molecules of dilute monoatomic gases under the assumption that the energy is only transferred via collisions between the molecules. In an attempt at a solution to the Boltzmann equation, Kac introduced a property of the density function that he called the “Boltzmann property" which describes the behavior of the density function at a given fixed time as the number of particles tends to infinity. The Boltzmann property has been studied extensively since then, and has been abstracted …

Theory, Computation, And Modeling Of Cancerous Systems, Sameed Ahmed

Theses and Dissertations

This dissertation focuses on three projects. In Chapter 1, we derive and implement the compact implicit integration factor method for numerically solving partial differential equations. In Chapters 2 and 3, we generalize and analyze a mathematical model for the nonlinear growth kinetics of breast cancer stem cells. And in Chapter 4, we develop a novel mathematical model for the HER2 signaling pathway to understand and predict breast cancer treatment. Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential …

Active Calculus Multivariable: 2018 Edition, Steven Schlicker, David Austin, Matthew Boelkins

Open Textbooks

Active Calculus Multivariable is the continuation of Active Calculus to multivariable functions. The Active Calculus texts are different from most existing calculus texts in at least the following ways: the texts are freely readable online in HTML format (new in this version of Active Calculus Multivariable) and are also available for in PDF; in the electronic format, graphics are in full color; the texts are open source, and interested instructors can gain access to the original source files on GitHub; the style of the texts requires students to be active learners — there are very few worked examples in the …

Preceding-Jump Simplex Method, Natdanai Kafakthong

Chulalongkorn University Theses and Dissertations (Chula ETD)

A practical linear programming solver uses the simplex method to identify the optimal solution by iteratively searching among adjacent feasible vertices. It may visit numerous vertices before finding the optimal one for a large linear programming problem taking a long time to succeed. This research proposes a heuristic method to alleviate this issue by jumping to a new vertex close to the optimal one before performing the simplex method. The new vertex is identified by first jumping from the initial vertex along the gradient vector of the objective function or alternative direction if the objective direction points out of the …

Closed-Form Formula For Pricing Discretely-Sampled Moment Swaps On One-Dimensional Ito Process, Kittisak Chumpong

Chulalongkorn University Theses and Dissertations (Chula ETD)

Moment swaps are essentially forward contracts on realized higher moments of log-returns of a specified underlying asset, which play an important role in protection against different kinds of market shocks. Variance, skewness, and kurtosis swaps are examples of moment swaps currently traded in markets. To facilitate market practitioners, this work provides a simple and easy-to-use pricing formula of moment swaps on discrete sampling under the Ito process, extended Black-Scholes model for stock prices and Schwartz model for commodity prices. Furthermore, the interesting topics of the fair prices are also investigated. Finally, Monte Carlo simulations are performed to support the accuracy …

Covering Of Objects Related To Rupert Property, Pongbunthit Tonpho

Chulalongkorn University Theses and Dissertations (Chula ETD)

One problem related to covering problem is whether one object has Rupert property. An object K in R³ has the Rupert property if a hole could be cut through one copy of K with the same size to permit another copy to pass through it. There are objects in R³ which have Rupert property, such as a cube. But not all of objects in R³ has Rupert property, such as a sphere. It is interesting to check that a given object has Rupert property. In this work, we construct useful lemmas for studying this problem. Furthermore, we apply these lemmas …

Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu

EKU Faculty and Staff Scholarship

We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. …

Quasi-Gorenstein Modules, Alexander York

Electronic Theses and Dissertations

This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An R-module M of grade g will be quasi-Gorenstein if ExtiR(M, R) = 0 for i 6= g and there is an isomorphism M ∼= ExtgR(M, R). Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their …