Physical Sciences and Mathematics Commons^™

Book Review: Fermat's Enigma By Simon Singh, Matthew Becker

Humanistic Mathematics Network Journal

No abstract provided.

Base And Subbase In A Number System, Walter S. Sizer

Humanistic Mathematics Network Journal

No abstract provided.

Are You A Quantitative Or Qualitative Runner?: 5.13 Miles And Rosemary-Lilac Shampoo, Shelly Sheats Harkness

Humanistic Mathematics Network Journal

No abstract provided.

Teaching For Change: The Leadership In Environmental Education Partnership, Paul Faulstich

Pitzer Faculty Publications and Research

Humans are transforming earth's landscape from a natural matrix with pockets of civilization to just the opposite. Most of us realize that this pattern is not sustainable. I live and work in Claremont, California, a charming college town in the midst of suburban sprawl. The town has a central village of terminally tasteful, overpriced bungalows nestled in the shade of tall, largely exotic trees. Indeed, most of the landscape of this "city of trees and Ph.D.s" has been imported; only a remnant parcel of coastal sage scrub that the Claremont Colleges have reluctantly preserved remains.

Radon Transforms And The Finite General Linear Groups, Michael E. Orrison

All HMC Faculty Publications and Research

Using a class sum and a collection of related Radon transforms, we present a proof G. James’s Kernel Intersection Theorem for the complex unipotent representations of the finite general linear groups. The approachis analogous to that used by F. Scarabotti for a proof of James’s Kernel Intersection Theorem for the symmetric group. In the process, we also show that a single class sum may be used to distinguish between distinct irreducible unipotent representations.

Random Walks With Badly Approximable Numbers, Doug Hensley, Francis Su

All HMC Faculty Publications and Research

Using the discrepancy metric, we analyze the rate of convergence of a random walk on the circle generated by d rotations, and establish sharp rates that show that badly approximable d-tuples in Rd give rise to walks with the fastest convergence.

Mathematical Magic, Arthur T. Benjamin

All HMC Faculty Publications and Research

In this paper, we present simple strategies for performing mathematical calculations that appear magical to most audiences. Specifically, we explain how to square large numbers, memorize pi to 100 places and determine the day of the week of any given date.

Blowup And Dissipation In A Critical-Case Unstable Thin Film Equation, Thomas P. Witelski, Andrew J. Bernoff, Andrea L. Bertozzi

All HMC Faculty Publications and Research

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

Semilinear Equations With Discrete Spectrum, Alfonso Castro

All HMC Faculty Publications and Research

This is an overview of the solvability of semilinear equations where the linear part has discrete spectrum. Semilinear elliptic and hyperbolic equations, as well as Hammerstein integral equations, are used as motivating examples. The presentation is intended to be accessible to non experts.

An Existence Result For A Class Of Sublinear Semipositone Systems, Alfonso Castro, C. Maya, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the existence of positive solutions for the system

-Δu_i = λ[f_i(u₁,u₂,...,u_m) - h_i]; Ω

u_i = 0; ∂Ω

where λ > 0 is a parameter, Δ is the Laplacian operator, Ω is a bounded domain in Rⁿ; n ≥ 1 with a smooth boundary ∂Ω, f_i are C¹ functions satisfying f₁(0,0,...,0) = 0, lim z→∞ f_i(z,z,...,z) = ∞ and lim z→∞ f_i(z,z,...,z)/z = 0, and h_i are nonnegative continuous functions in Ω for i = 1,2,...,m. …

Examples Of Cayley 4-Manifolds, Weiqing Gu, Christopher Pries '03

All HMC Faculty Publications and Research

We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.

We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We …

Essential P-Spaces: A Generalization Of Door Spaces, Emad Abu Osba, Melvin Henriksen

All HMC Faculty Publications and Research

An element f of a commutative ring A with identity element is called a von Neumann regular element if there is a g in A such that f²g=f. A point p of a (Tychonoff) space X is called a P-point if each f in the ring C(X) of continuous real-valued functions is constant on a neighborhood of p. It is well-known that the ring C(X) is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case X is called a P-space. If all but at most one point of X …

Small Zeros Of Quadratic Forms With Linear Conditions, Lenny Fukshansky

CMC Faculty Publications and Research

Given a quadratic form and M linear forms in N + 1 variables with coefficients in a number field K, suppose that there exists a point in KN+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.

A Syntactic Parser With Semantic Filtering For Biomedical Text, Gondy Leroy, Thomas C. Rindflesch, Bisharah Libbus, Halil Kilicoglu, Hsinchun Chen

CGU Faculty Publications and Research

No abstract provided.

Using Symbolic Knowledge In The Umls To Disambiguate Words In Small Datasets With A Naive Bayes Classifier, Gondy Leroy, Thomas C. Rindflesch

CGU Faculty Publications and Research

Current approaches to word sense disambiguation use and combine various machine-learning techniques. Most refer to characteristics of the ambiguous word and surrounding words and are based on hundreds of examples. Unfortunately, developing large training sets is time-consuming. We investigate the use of symbolic knowledge to augment machine-learning techniques for small datasets. UMLS semantic types assigned to concepts found in the sentence and relationships between these semantic types form the knowledge base. A naïve Bayes classifier was trained for 15 words with 100 examples for each. The most frequent sense of a word served as the baseline. The effect of increasingly …

Eminent Domain: B.L. Wiggins, Forestry, And The New South At Sewanee, Char Miller

Pomona Faculty Publications and Research

The history of the University of the South and of its forest is intertwined. The health of the forest—and of the university—hung in the balance when Benjamin Wiggins took charge of both in 1893.

Evaluation Of Multiple Models To Distinguish Closely Related Forms Of Disease Using Dna Microarray Data: An Application To Multiple Myeloma, Johanna S. Hardin, Michael Waddell, C. David Page, Fenghuang Zhan, Bart Barlogie, John Shaughnessy, John J. Crowley

Pomona Faculty Publications and Research

Motivation: Standard laboratory classification of the plasma cell dyscrasia monoclonal gammopathy of undetermined significance (MGUS) and the overt plasma cell neoplasm multiple myeloma (MM) is quite accurate, yet, for the most part, biologically uninformative. Most, if not all, cancers are caused by inherited or acquired genetic mutations that manifest themselves in altered gene expression patterns in the clonally related cancer cells. Microarray technology allows for qualitative and quantitative measurements of the expression levels of thousands of genes simultaneously, and it has now been used both to classify cancers that are morphologically indistinguishable and to predict response to therapy. It is …

Enabling Computer Decisions Based On Eeg Input, Benjamin J. Culpepper, Robert M. Keller

All HMC Faculty Publications and Research

Multilayer neural networks were successfully trained to classify segments of 12-channel electroencephalogram (EEG) data into one of five classes corresponding to five cognitive tasks performed by a subject. Independent component analysis (ICA) was used to segregate obvious artifact EEG components from other sources, and a frequency-band representation was used to represent the sources computed by ICA. Examples of results include an 85% accuracy rate on differentiation between two tasks, using a segment of EEG only 0.05 s long and a 95% accuracy rate using a 0.5-s-long segment.

A Combinatorial Approach To Hyperharmonic Numbers, Arthur T. Benjamin, David Gaebler '04, Robert Gaebler '04

All HMC Faculty Publications and Research

Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many interesting identities.

A Probabilistic View Of Certain Weighted Fibonacci Sums, Arthur T. Benjamin, Judson D. Neer, Daniel T. Otero, James A. Sellers

All HMC Faculty Publications and Research

In this article, we pursue the reverse strategy of using probability to derive a_n and develop an exponential generating function for a_n in Section 3. In Section 4, we present a method for finding an exact, non-recursive, formula for a_n.

Solitary Waves In Layered Nonlinear Media, Randall J. Leveque, Darryl H. Yong

All HMC Faculty Publications and Research

We study longitudinal elastic strain waves in a one-dimensional periodically layered medium, alternating between two materials with different densities and stress-strain relations. If the impedances are different, dispersive effects are seen due to reflection at the interfaces. When the stress-strain relations are nonlinear, the combination of dispersion and nonlinearity leads to the appearance of solitary waves that interact like solitons. We study the scaling properties of these solitary waves and derive a homogenized system of equations that includes dispersive terms. We show that pseudospectral solutions to these equations agree well with direct solutions of the hyperbolic conservation laws in the …

The Fibonacci Numbers -- Exposed More Discretely, Arthur T. Benjamin, Jennifer J. Quinn

All HMC Faculty Publications and Research

No abstract provided in this article.

Steps Toward The Creation Of A Carbon Nanotube Single Electron Transistor, R. Matthew Ferguson

Pomona Senior Theses

This report details work toward the fabrication of a single-electron transistor created from a single-walled carbon nanotube (SWNT). Specifically discussed is a method for growing carbon nanotubes (CNTs) via carbon vapor deposition (CVD). The growth is catalyzed by a solution of 0.02g Fe(NO3)3·9H2O, 0.005g MoO2(acac)2, and 0.015g of alumina particles in 15mL methanol. SWNT diameter ranges from 0.6 to 3.0 nm. Also discussed is a method to control nanotube growth location by patterning samples with small islands of catalyst. A novel “maskless” photolithographic process is used to focus light from a lightweight commercial digital projector through a microscope. Catalyst islands …

Maskless Projection Lithography, J. David Musgraves

Pomona Senior Theses

Photolithography is a key element of the modem integrated circuit process. It is photolithography, combined with metal deposition, that allows a three dimensional circuit to be built up on a two dimensional surface. Since it is such an important part of the semiconductor manufacturing industry, a massive base of research in this area already exists. The problem with this pre-existing research is that it is geared solely toward industrial purposes, as opposed to more academic research areas. The goal of my research is to move this industrial process into the academic setting of Pomom College.

Steps Toward The Creation Of A Carbon Nanotube Single Electron Transistor, R. Matthew Ferguson

Pomona Senior Theses

This report details work toward the fabrication of a single-electron transistor created from a single-walled carbon nanotube (SWNT). Specifically discussed is a method for growing carbon nanotubes (CNTs) via carbon vapor deposition (CVD). The growth is catalyzed by a solution of 0.02g Fe(NO3)3·9H2O, 0.005g MoO2(acac)2, and 0.015g of alumina particles in 15mL methanol. SWNT diameter ranges from 0.6 to 3.0 nm. Also discussed is a method to control nanotube growth location by patterning samples with small islands of catalyst. A novel “maskless” photolithographic process is used to focus light from a lightweight commercial digital projector through a microscope. Catalyst islands …

Genescene: Biomedical Text And Data Mining, Gondy Leroy, Hsinchun Chen, Jesse D. Martinez, Shauna Eggers, Ryan R. Falsey, Kerri L. Kislin, Zan Huang, Jiexun Li, Jie Xu, Daniel M. Mcdonald, Gavin Ng

CGU Faculty Publications and Research

To access the content of digital texts efficiently, it is necessary to provide more sophisticated access than keyword based searching. GeneScene provides biomedical researchers with research findings and background relations automatically extracted from text and experimental data. These provide a more detailed overview of the information available. The extracted relations were evaluated by qualified researchers and are precise. A qualitative ongoing evaluation of the current online interface indicates that this method to search the literature is more useful and efficient than keyword based searching.

A Sign-Changing Solution For A Superlinear Dirichlet Problem, Ii, Alfonso Castro, Pavel Drabek, John M. Neuberger

All HMC Faculty Publications and Research

In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of -Δ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the k-th and k+1-st eigenvalues there still exists a solution which changes sign at most k times. In particular, when k=1 the sign-changing exactly-once solution persists although one-sign solutions no …

Consensus-Halving Via Theorems Of Borsuk-Ulam And Tucker, Forrest W. Simmons, Francis E. Su

All HMC Faculty Publications and Research

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

The Computational Complexity Of Motion Planning, Jeff R.K. Hartline '01, Ran Libeskind-Hadas

All HMC Faculty Publications and Research

In this paper we show that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable. In particular, we show that the problem is PSPACE-complete. We begin with a review of NP-completeness and polynomial-time reductions, introduce the class PSPACE, and motivate the significance of PSPACE-complete problems. Afterwards, we prove that determining whether a given instance of a generalized Lunar Lockout puzzle is solvable is PSPACE-complete.

A Mathematical Model Of Immune Response To Tumor Invasion, Lisette De Pillis, Ami Radunskaya

All HMC Faculty Publications and Research

Recent experimental studies by Diefenbach et al. [1] have brought to light new information about how the immune system of the mouse responds to the presence of a tumor. In the Diefenbach studies, tumor cells are modified to express higher levels of immune stimulating NKG2D ligands. Experimental results show that sufficiently high levels of ligand expression create a significant barrier to tumor establishment in the mouse. Additionally, ligand transduced tumor cells stimulate protective immunity to tumor rechallenge. Based on the results of the Diefenbach experiments, we have developed a mathematical model of tumor growth to address some of the questions …