Physical Sciences and Mathematics Commons^™

Course In Computational Number Theory, David Bressoud, Stan Wagon

Stan Wagon, Retired

No abstract provided.

How To Pick Out The Integers In The Rationals: An Application Of Number Theory To Logic, Daniel Flath, Stan Wagon

Stan Wagon, Retired

No abstract provided.

The 99th Fibonacci Identity, Arthur T. Benjamin, Alex K. Eustis '06, Sean S. Plott '08

All HMC Faculty Publications and Research

We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences G_n that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well.

[1] A. T. Benjamin and J. J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions, 27, Mathematical Association of America, Washington, DC, 2003

Geometric Primitives In Digital Images: Analyses And Applications Using Digital Geometry., Partha Bhowmick Dr.

Doctoral Theses

No abstract provided.

Local Cohomology And Support For Triangulated Categories, Dave Benson, Srikanth Iyengar, Henning Krause

Department of Mathematics: Faculty Publications

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the …

Percentage-Based Versus Power-Based Vote Tabulation Statistical Audits, John Mccarthy, Howard Stanislevic, Mark Lindeman, Arlene S. Ash, Vittorio Addona, Mary Batcher

Vittorio Addona

An audit model is presented to address inefficient percentage-based election audits. Presented to state and local election officials and legislators.

On Isoconcentration Surfaces Of Three Dimensional Turing Patterns, Tilmann Glimm, H. George E. Hentschel

Mathematics Faculty Publications

We consider three-dimensional Turing patterns and their isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of "high" and "low" concentrations in Turing patterns. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Remarkably, they are often very well approximated by certain minimal surfaces. We give a dynamical explanation for the emergence of Scherk's surface in certain cases, a structure …

The Morphostatic Limit For A Model Of Skeletal Pattern Formation In The Vertebrate Limb, Mark Alber, Tilmann Glimm, H. George E. Hentschel, Bogdan Kazmierczak, Yong-Tao Zhang, Jianfeng Zhu, Stuart (Stuart A.) Newman

Mathematics Faculty Publications

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722,2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally …

On The Stochastic Beverton-Holt Equation With Survival Rates, Paul H. Bezandry, Toka Diagana, Saber Elaydi

Mathematics Faculty Research

The paper studies a Beverton-Holt difference equation, in which both the recruitment function and the survival rate vary randomly. It is then shown that there is a unique invariant density, which is asymptotically stable. Moreover, a basic theory for random mean almost periodic sequence on Z₊ is given. Then, some suffcient conditions for the existence of a mean almost periodic solution to the stochastic Beverton-Holt equation are given.

Reorganizing Freshman Business Mathematics Ii: Authentic Assessment In Mathematics Through Professional Memos, Kris H. Green, W. Allen Emerson

Mathematical and Computing Sciences Faculty/Staff Publications

Part I of this paper described the development of a new Freshman Business Mathematics (FBM) course at our college. In this second part of the paper, we discuss our assessment tool, the business memo, as a venue for students to apply mathematical skills, via mathematical modeling, to realistic business problems. These memos have proven a crucial step in turning our FBM course around from a dreaded course with little connection to students’ intended careers into a course where students experience the power of mathematics for solving problems and informing decisions. Comments from students in the course throughout its six-year history …

Crafty Counting, Frank A. Farris

Mathematics and Computer Science

To count a set means to put it in one-to-one correspondence with a set of integers {1, 2, 3,...,n}. Direct counting is nice, but in complicated situations it pays to be more crafty. A problem with patterns of colored tiles gives us a chance to illustrate a popular counting principle known by various names. We'll call it the Burnside-Cauchy-Frobenius formula. It is also popularly called the Burnside Orbit-Counting Lemma, though wags refer to it as "not Burnside," because it was known long before Burnside was born. Later Pólya generalized the formula, so some readers may recognize this as Pólya Enumeration.

Paint It Black -- A Combinatorial Yawp, Arthur T. Benjamin, Jennifer J. Quinn, James A. Sellers, Mark A. Shattuck

All HMC Faculty Publications and Research

No abstract provided in this paper.

A Combinatorial Approach To Fibonomial Coefficients, Arthur T. Benjamin, Sean S. Plott '08

All HMC Faculty Publications and Research

A combinatorial argument is used to explain the integrality of Fibonomial coefficients and their generalizations. The numerator of the Fibonomial coeffcient counts tilings of staggered lengths, which can be decomposed into a sum of integers, such that each integer is a multiple of the denominator of the Fibonomial coeffcient. By colorizing this argument, we can extend this result from Fibonacci numbers to arbitrary Lucas sequences.

Emergent Decision-Making In Biological Signal Transduction Networks, Tomáš Helikar, Jack Heidel, Jim A. Rogers, Jimmy Rogers

Mathematics Faculty Publications

The complexity of biochemical intracellular signal transduction networks has led to speculation that the high degree of interconnectivity that exists in these networks transforms them into an information processing network. To test this hypothesis directly, a large scale model was created with the logical mechanism of each node described completely to allow simulation and dynamical analysis. Exposing the network to tens of thousands of random combinations of inputs and analyzing the combined dynamics of multiple outputs revealed a robust system capable of clustering widely varying input combinations into equivalence classes of biologically relevant cellular responses. This capability was nontrivial in …

Fourier Series Of Orthogonal Polynomials, Nataniel Greene

Publications and Research

Explicit formulas for the Fourier coefficients of the Legendre polynomials can be found in the Bateman Manuscript Project. However, similar formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials.

Klein-Four Covers Of The Projective Line In Characteristic Two, Darren B. Glass

Math Faculty Publications

In this paper we examine curves defined over a field of characteristic 2 which are (Z/2Z)2-covers of the projective line. In particular, we prove which 2-ranks occur for such curves of a given genus and where possible we give explicit equations for such curves.

Equivalences On Acyclic Orientations, Matthew Macauley, Henning S. Mortveit

Publications

The cyclic and dihedral groups can be made to act on the set Acyc(Y ) of acyclic orientations of an undirected graph Y , and this gives rise to the equivalence relations ∼κ and ∼δ, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers.

In this paper we construct the graphs C(Y ) and D(Y ) with vertex sets Acyc(Y ) and whose connected components encode the equivalence classes. The number of connected components …

From College Prep High School Courses To College Remedial Courses : Bridging The Gap, Tracy K. Abar

Theses, Dissertations and Culminating Projects

This study investigated students’ use of, and access to, the calculator in high school mathematics courses and compared it to the accessibility of a calculator during college placement tests. In spring 1999, at the request of the College Board, the Educational Testing Service (ETS) conducted a survey on calculator use in the nation’s schools. Ninety-nine point nine percent of the schools surveyed indicated they either required or allowed calculators for part of their college preparatory mathematics sequence. Accompanying the increased role of calculators in mathematics learning and the use of technology in the classroom, significant changes were introduced into many …

Multisymplectic Theory Of Balance Systems, I, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation C of a balance system B_C is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle π : Y ͢ X and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms …

Templated Biomimetic Multifunctional Coatings, Chih-Hung Sun, Adriel Gonzalez, Nicholas C. Linn, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report a bioinspired templating technique for fabricating multifunctional optical coatings that mimic both unique functionalities of antireflective moth eyes and superhydrophobic cicada wings. Subwavelength-structured fluoropolymer nipple arrays are created by a soft-lithography-like process. The utilization of fluoropolymers simultaneously enhances the antireflective performance and the hydrophobicity of the replicated films. The specular reflectivity matches the optical simulation using a thin-film multilayer model. The dependence of the size and the crystalline ordering of the replicated nipples on the resulting antireflective properties have also been investigated by experiment and modeling. These biomimetic materials may find important technological application in self-cleaning antireflection coatings.

Broadband Moth-Eye Antireflection Coatings On Silicon, Chih-Hung Sun, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report a bioinspired templating technique for fabricating broadband antireflection coatings that mimic antireflective moth eyes. Wafer-scale, subwavelength-structured nipple arrays are directly patterned on silicon using spin-coated silica colloidal monolayers as etching masks. The templated gratings exhibit excellent broadband antireflection properties and the normal-incidence specular reflection matches with the theoretical prediction using a rigorous coupled-wave analysis (RCWA) model. We further demonstrate that two common simulation methods, RCWA and thin-film multilayer models, generate almost identical prediction for the templated nipple arrays. This simple bottom-up technique is compatible with standard microfabrication, promising for reducing the manufacturing cost of crystalline silicon solar cells.

A Class Of Convex Polyhedra With Few Edge Unfoldings, Alex Benton, Joseph O'Rourke

Computer Science: Faculty Publications

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.

Frobenius Number, Covering Radius, And Well-Rounded Lattices, Lenny Fukshansky, Sinai Robins

CMC Faculty Publications and Research

Lecture given at the Joint Mathematics Meeting in San Diego, January 2008.

How To Cleverly Count Pattern-Avoiding Words, Lara Pudwell

Lara K. Pudwell

No abstract provided.

Honey, I Shrunk The Dollar, Lisa Yocco, Ronald Harshbarger

Lisa S. Yocco

No abstract provided.

Characterization Of Digraphs With Equal Domination Graphs And Underlying Graphs, Kim A. S. Factor, Larry L. Langley

College of the Pacific Faculty Articles

A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u,v}∈E[dom(D)] whenever (u,z)∈A(D) or (v,z)∈A(D) for every other vertex z∈V(D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)=dom(D). The maximum and minimum number of single arcs in these digraphs, and their characteristics, is given.

Illustrating The Use Of The Nine Chapters In The Classroom, Joel K. Haack

Faculty Publications

No abstract provided.

Numerical Analysis Of Elliptic Inverse Problems With Interior Data, Mark Gockenbach

Department of Mathematical Sciences Publications

A number of algorithms have been proposed and analyzed for estimating a coefficient in an elliptic boundary value problem when interior data is available. Most of the analysis has been done for the simple scalar BVP

-Δ a Δ u = f in Ω,

a (∂ u / ∂ n) g on ∂ Ω

However, some methods and the associated analysis have been extended to the problem of estimating the Lamé moduli in the system of linear, isotropic elasticity. Under certain idealized conditions, convergence of estimates to the exact Lame moduli has been proved for two techniques, the output …

Algebraic Properties Of Edge Ideals, Rachelle R. Bouchat

University of Kentucky Doctoral Dissertations

Given a simple graph G, the corresponding edge ideal I_G is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for …

Solitary Wave Families In Two Non-Integrable Models Using Reversible Systems Theory, Jonathan Leto

Electronic Theses and Dissertations

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal …