Mathematics Commons^™

Large-Time Behavior Of Non-Symmetric Fokker-Planck Type Equations, Anton Arnold, Eric Carlen, Qiangchang Ju

Communications on Stochastic Analysis

No abstract provided.

The Role Of Prediction In The Teaching And Learning Of Algebra, Lisa Kasmer

Dissertations

Research has shown that including prediction questions within reading and science instruction has been advantageous for students, yet minimal research existed regarding the use of such questions within mathematics instruction. In order to extend and build on our knowledge about the effects of prediction in mathematics instruction, this study explored the impact of this paradigm in the teaching and learning of algebra. Specifically, this study probed whether utilizing prediction questions provided students opportunities for engaging in mathematical thinking, retrieving prior knowledge, and discussing related mathematical ideas, could increase such students' conceptual understanding and mathematical reasoning in the content area of …

Bioinspired Broadband Antireflection Coatings On Gasb, Wei-Lun Min, Amaury P. Betancourt, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report an inexpensive yet scalable templating technique for fabricating moth-eye antireflection gratings on gallium antimonide substrates. Non-close-packed colloidal monolayers are utilized as etching masks to pattern subwavelength-structured nipple arrays on GaSb. The resulting gratings exhibit superior broadband antireflection properties and thermal stability than conventional multilayer dielectric coatings. The specular reflection of the templated nipple arrays match with the theoretical predictions using a rigorous coupled-wave analysis model. The effect of the nipple shape and size on the antireflection properties has also been investigated by the same model. These biomimetic coatings are of great technological importance in developing efficient thermophotovoltaic cells.

Analysis And Application Of Perfectly Matched Layer Absorbing Boundary Conditions For Computational Aeroacoustics, Sarah Anne Parrish

Mathematics & Statistics Theses & Dissertations

The Perfectly Matched Layer (PML) was originally proposed by Berenger as an absorbing boundary condition for Maxwell's equations in 1994 and is still used extensively in the field of electromagnetics. The idea was extended to Computational Aeroacoustics in 1996, when Hu applied the method to Euler's equations. Since that time much of the work done on PML in the field of acoustics has been specific to the case where mean flow is perpendicular to a boundary, with an emphasis on Cartesian coordinates. The goal of this work is to further extend the PML methodology in a two-fold manner: First, to …

A Closer Look At The Crease Length Problem, Sean F. Ellermeyer

Faculty Articles

An optimization problem that appears as an exercise in most modern calculus textbooks is the crease length problem. Here, Ellermeyer provides a solution of the general crease length problem in which all possible foldings of a corner to the opposite edge are taken into account. One of his findings will be that the minimum crease length is never produced by a Case 2 fold and hence that the general crease length problem always yields a different minimum than the constrained problem that is treated in the textbooks. He discovers a criterion that determines which foldings must be performed in order …

A Wavelet-Based Method For Overcoming The Gibbs Phenomenon, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.

The Mixed Problem For Harmonic Functions In Polyhedra, Moises Venouziou, Gregory C. Verchota

Mathematics - All Scholarship

R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses.

Failure Of Metric Regularity For Major Classes Of Variational Systems, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to the study of metric regularity, which is a remarkable property of set-valued mappings playing an important role in many aspects of nonlinear analysis and its applications. We pay the main attention to metric regularity of the so- called parametric variational systems that contain, in particular, various classes of parameterized/perturbed variational and hemivariational inequalities, complementarity systems, sets of optimal solutions and corresponding Lagrange multipliers in problems of parametric optimization and equilibria, etc. Based on the advanced machinery of generalized differentiation1 we surprisingly reveal that metric regularity fails for certain major classes of parametric variational systems, which …

Hua's Matrix Equality And Schur Complements, Chris Paige, George P. H. Styan, Bo-Ying Wang, Fuzhen Zhang

Mathematics Faculty Articles

The purpose of this paper is to revisit Hua's matrix equality (and inequality) through the Schur complement. We present Hua's original proof and two new proofs with some extensions of Hua's matrix equality and inequalities. The new proofs use a result concerning Shur complements and a generalization of Sylvester's law of inertia, each of which is useful in its own right.

Preconditioned Conjugate Gradient Algorithm, Elena Caraba

Honors Capstones

No abstract provided.

Beam Selection In Radiotherapy Design, M Ehrgott, Allen G. Holder, Josh Reese

Mathematics Faculty Research

The optimal design of a radiotherapy treatment depends on the collection of directions from which radiation is focused on the patient. These directions are manually selected by a physician and are typically based on the physician's previous experiences. Once the angles are chosen, there are numerous optimization models that decide a fluency pattern (exposure times) that best treats a patient. So, while optimization techniques are often used to decide the length of time a patient is exposed to a high-energy particle beam, the directions themselves are not optimized. The problem with optimally selecting directions is that the underlying mixed integer …

An Overview Of Conditionals And Biconditionals In Probability, Nataniel Greene

Publications and Research

Conditional and biconditional statements are a standard part of symbolic logic but they have only recently begun to be explored in probability for applications in artificial intelligence. Here we give a brief overview of the major theorems involved and illustrate them using two standard model problems from conditional probability.

Grid Vertex-Unfolding Orthogonal Polyhedra, Mirela Damian

Computer Science: Faculty Publications

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 …

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.

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 …

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.

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.

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.

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 …

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 …

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 …

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.

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.