Mathematics Commons^™

Interlace Polynomials Of Friendship Graphs, Christina Eubanks-Turner, Aihua Li

Department of Mathematics Faculty Scholarship and Creative Works

In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdös, Rényi and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph.

Inverse Function: Pre-Service Teachers’ Techniques And Meanings, Teo Paoletti, Irma E. Stevens, Natalie L.F. Hobson, Kevin C. Moore, Kevin R. Laforest

Department of Mathematics Faculty Scholarship and Creative Works

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss …

A Covariational Understanding Of Function: Putting A Horse Before The Cart, Teo Paoletti, Kevin C. Moore

Department of Mathematics Faculty Scholarship and Creative Works

Supporting students developing understandings of function has been a notoriously elusive task in mathematics education. We present Thompson and Carlson’s (2017) description of a covariational meaning of function and provide an example of a student who maintains meanings compatible with this description. We use this student’s activity to illustrate nuances in a covariational meaning of function and to highlight how such meanings can be powerful for students. In doing so, we argue that a student who has develop meanings compatible with the covariational meaning of function presented by Thompson and Carlson has the foundational meanings needed to understand a formal …

The Rsa Cryptosystem, Rodrigo Iglesias

Williams Honors College, Honors Research Projects

This paper intends to present an overview of the RSA cryptosystem. Cryptosystems are mathematical algorithms that disguise information so that only the people for whom the information is intended can read it. The invention of the RSA cryptosystem in 1977 was a significant event in the history of cryptosystems. We will describe in detail how the RSA cryptosystem works and then illustrate the process with a realistic example using fictional characters. In addition, we will discuss how cryptosystems worked prior to the invention of RSA and the advantage of using RSA over any of the previous cryptosystems. This will help …

Generalized Characteristics Of A Generic Polytope, Tommy Naugle

Electronic Theses and Dissertations

For a smooth hypersurface S ⊂ R 2n given by the level set of a Hamiltonian function H, a symplectic form ω on R2n induces a vector field XH which flows tangent to S. By the nondegeneracy of ω, there exists a distinguished line bundle LS whose characteristics are the integral curves of XH. When S is the boundary of a smooth convex domain K˜ ⊂ R 2n, then the least action among closed characteristics of LS is equal to the Ekeland-Hofer-Zehnder capacity, a symplectic invariant. From a result due to Artstein-Avidan and Ostrover, there exists a continuous extension of …

A Two-Dimensional Finite Element Model Of The Grain Boundary Based On Thermo-Mechanical Strain Gradient Plasticity, Yooseob Song, George Z. Voyiadjis

Civil Engineering Faculty Publications

In this work, a two-dimensional finite element model for the grain boundary flow rule is developed based on the thermo-mechanical gradient-enhanced plasticity theory. The proposed model is temperature-dependent. A special attention is given to physical and micromechanical nature of dislocation interactions in combination with thermal activation on stored and dissipated energy. Thermodynamic conjugate microforces are decomposed into energetic and dissipative components. Correspondingly, two different grain boundary material length scales are present in the proposed model. Finally, numerical examples are solved in order to explore characteristics of the proposed grain boundary flow rule.

On The Girth And Diameter Of Generalized Johnson Graphs, Louis Anthony Agong, Carmen Amarra, John Caughman, Ari J. Herman, Taiyo S. Terada

Mathematics and Statistics Faculty Publications and Presentations

Let v > k > i be non-negative integers. The generalized Johnson graph, J(v,k,i), is the graph whose vertices are the k-subsets of a v-set, where vertices A and B are adjacent whenever |A∩B|= i. In this article, we derive general formulas for the girth and diameter of J(v,k,i). Additionally, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection.

High-Order Method For Evaluating Derivatives Of Harmonic Functions In Planar Domains, Jeffrey S. Ovall, Samuel E. Reynolds

Mathematics and Statistics Faculty Publications and Presentations

We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions in planar domains that are specified by their Dirichlet data.

Clustering And Multifacility Location With Constraints Via Distance Function Penalty Methods And Dc Programming, Mau Nam Nguyen, Thai An Nguyen, Sam Reynolds, Tuyen Tran

Mathematics and Statistics Faculty Publications and Presentations

This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide different numerical examples to test our method.

Results On The Gold Grabbing Game, Stephen Acampa

Online Theses and Dissertations

In this paper, we will contribute to research on a Graph Theory problem known as the Gold Grabbing Game. The game consists of two players and a tree in which each vertex has a positive integer value of gold. Players take turns removing leaves from the tree and deleting the associated edge until the graph is entirely empty. A winning condition is acquiring at least half of the total gold. Existing research shows that for a tree with an even number of vertices, Player 1 can always win.

It can also be shown via simple examples that for a tree …

Positive Symmetric Solutions Of A Boundary Value Problem With Dirichlet Boundary Conditions, Tek Nath Dhakal

Online Theses and Dissertations

We apply a recent extension of a compression-expansion fixed point theorem of function type to a second order boundary value problem with Dirichlet boundary conditions. We show the existence of positive symmetric solutions of this boundary value problem.

A Non-Commutative Julia Inequality, John E. Mccarthy, James E. Pascoe

Mathematics Faculty Research

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in C^d. We look at differentiability at a boundary point for functions that have a certain regularity there.

Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

Mathematics Faculty Research

In this paper, we provide a direct and constructive proof of weak factorization of h1 (ℝ×ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] in the sense of h1 (ℝ×ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ║fh1║(ℝ×ℝ) is given in terms of ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this directly implies a lower bound on the norm of …

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.