Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Education (4)
- Mathematics (4)
- Combinatorics (3)
- Heights (3)
- Pedagogy (3)
-
- Lattice theory (2)
- Linear forms (2)
- Siegel formula (2)
- Adequate ring (1)
- Algebraically real closed field (1)
- Analytic entire function (1)
- Arbitrary 3-manifolds (1)
- Audience (1)
- B'ezout domain (1)
- Bases (1)
- Blocking Probability (1)
- Calabi–Yau threefolds (1)
- Cartan–Dieudonné theorem (1)
- Cellular adhesion (1)
- Cellular automata (1)
- Clark operators (1)
- Clark perturbations (1)
- Communiation Networks (1)
- Commutative rings (1)
- Complex symmetric operators (1)
- Conjugation (1)
- Counting (1)
- Cozerosets (1)
- Crossbar Switching Networks (1)
- Cubes (1)
Articles 1 - 30 of 31
Full-Text Articles in Physical Sciences and Mathematics
Teaching Time Savers: Some Advice On Giving Advice, Michael E. Orrison Jr.
Teaching Time Savers: Some Advice On Giving Advice, Michael E. Orrison Jr.
All HMC Faculty Publications and Research
There are always a lot of questions that need to be answered at the beginning of a course. When are office hours? What are the grading policies? How many exams will there be? Will late homework be accepted? We have all seen the answers to these sorts of questions form the bulk of a standard course syllabus, and most of us feel an obligation (and rightly so) to provide such information.
Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz
Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz
All HMC Faculty Publications and Research
No abstract provided in this article.
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
All HMC Faculty Publications and Research
By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice.
Teaching Time Savers: Style Points, Michael E. Orrison Jr.
Teaching Time Savers: Style Points, Michael E. Orrison Jr.
All HMC Faculty Publications and Research
When I began as an assistant professor, I had a pretty good sense of how much time it would take for me to prepare for each class. After a few conversations with my new colleagues, I even had a good sense of how much time I should devote to tasks like office hours and committee work. Somewhere in the middle of grading my first exam, though, it became painfully clear that I had underestimated the amount of time I would need to grade exams!
Quadratic Forms And Height Functions, Lenny Fukshansky
Quadratic Forms And Height Functions, Lenny Fukshansky
CMC Faculty Publications and Research
The effective study of quadratic forms originated with a paper of Cassels in 1955, in which he proved that if an integral quadratic form is isotropic, then it has non-trivial zeros of bounded height. Here height stands for a certain measure of arithmetic complexity, which we will make precise. This theorem has since been generalized and extended in a number of different ways. We will discuss some of such generalizations for quadratic spaces over a fixed number field as well as over the field of algebraic numbers. Specifically, let K be either a number field or its algebraic closure, and …
Hole Dynamics In Polymer Langmuir Films, James C. Alexander, Andrew J. Bernoff, Elizabeth K. Mann, J. Adin Mann Jr., Lu Zou
Hole Dynamics In Polymer Langmuir Films, James C. Alexander, Andrew J. Bernoff, Elizabeth K. Mann, J. Adin Mann Jr., Lu Zou
All HMC Faculty Publications and Research
This article develops a model for the closing of a gaseous hole in a liquid domain within a two-dimensional fluid layer coupled to a Stokesian subfluid substrate, and compares this model to experiments following hole dynamics in a polymer Langmuir monolayer. Closure of such a hole in a fluid layer is driven by the line tension at the hole boundary and the difference in surface pressure within the hole and far outside it. The observed rate of hole closing is close to that predicted by our model using estimates of the line tension obtained by other means, assuming that the …
Teaching Time Savers: A Recommendation For Recommendations, Michael E. Orrison Jr.
Teaching Time Savers: A Recommendation For Recommendations, Michael E. Orrison Jr.
All HMC Faculty Publications and Research
I admit it — I enjoy writing recommendation letters for my students. I like
learning about their hopes and dreams, where they have been and where they want to go. A recommendation letter is an opportunity to remind myself how much my students can grow while they are in college, and how much I have grown as an instructor, advisor, and mentor.
The Local Gromov–Witten Invariants Of Configurations Of Rational Curves, Dagan Karp, Chiu-Chu Melissa Liu, Marcos Mariño
The Local Gromov–Witten Invariants Of Configurations Of Rational Curves, Dagan Karp, Chiu-Chu Melissa Liu, Marcos Mariño
All HMC Faculty Publications and Research
We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of ℙ1’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of ℙ3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the …
The Linear Complexity Of A Graph, David L. Neel, Michael E. Orrison Jr.
The Linear Complexity Of A Graph, David L. Neel, Michael E. Orrison Jr.
All HMC Faculty Publications and Research
The linear complexity of a matrix is a measure of the number of additions, subtractions, and scalar multiplications required to multiply that matrix and an arbitrary vector. In this paper, we define the linear complexity of a graph to be the linear complexity of any one of its associated adjacency matrices. We then compute or give upper bounds for the linear complexity of several classes of graphs.
Spectral Analysis Of The Supreme Court, Brian L. Lawson, Michael E. Orrison, David T. Uminsky
Spectral Analysis Of The Supreme Court, Brian L. Lawson, Michael E. Orrison, David T. Uminsky
All HMC Faculty Publications and Research
The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small voting bodies like the United States Supreme Court. Our goal is to show how simple ideas from linear algebra can come together to say something interesting about voting. And what could be more simple than where our story begins— with counting.
A Framework For Inclusive Teaching In Stem Disciplines, Lois Reddick, Wayne Jacobson, Angela Linse, Darryl Yong
A Framework For Inclusive Teaching In Stem Disciplines, Lois Reddick, Wayne Jacobson, Angela Linse, Darryl Yong
All HMC Faculty Publications and Research
A wide body of literature exists recounting the ways in which inclusive teaching practices and principles benefit students and positively impact learning, student retention, and professional development across disciplines. However, STEM faculty do not readily accept the traditional approach of examining course content from multiple perspectives as relevant to their course content or useful in their teaching. In this chapter, we propose a Framework for Inclusive Teaching in STEM Disciplines that reflects the contexts of teaching in these disciplines, and extends James Banks’ Five Dimensions of Multicultural Education to the distinct needs of STEM faculty in their classes. We also …
Double Birthday Magic Square, Arthur T. Benjamin
Double Birthday Magic Square, Arthur T. Benjamin
All HMC Faculty Publications and Research
No abstract provided.
Combinatorial Interpretations Of Spanning Tree Identities, Arthur T. Benjamin, Carl R. Yerger
Combinatorial Interpretations Of Spanning Tree Identities, Arthur T. Benjamin, Carl R. Yerger
All HMC Faculty Publications and Research
We present a combinatorial proof that the wheel graph Wn has L2n − 2 spanning trees, where Ln is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci number. Our proofs avoid the use of induction, determinants, or the matrix tree theorem.
Some Promising Approaches To Tumor-Immune Modeling, Lisette G. De Pillis, Ami E. Radunskaya
Some Promising Approaches To Tumor-Immune Modeling, Lisette G. De Pillis, Ami E. Radunskaya
All HMC Faculty Publications and Research
Mathematical models of tumor-immune interactions provide an analytical framework in which to address specific questions regarding tumor-immune dynamics. We present a brief summary of several approaches we are currently exploring to model tumor growth, tumor-immune interactions, and treatments. Results to date have shown that simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge, are able to reproduce data from published studies. We additionally present some of our current efforts in the investigation of optimal control to aid in determining improved treatment strategies.
Spatial Tumor-Immune Modeling, Lisette G. De Pillis, D G. Mallet, Ami E. Radunskaya
Spatial Tumor-Immune Modeling, Lisette G. De Pillis, D G. Mallet, Ami E. Radunskaya
All HMC Faculty Publications and Research
In this paper, we carry out an examination of four mechanisms that can potentially lead to changing morphologies in a growing tumor: variations in nutrient consumption rates, cellular adhesion, excessive consumption of nutrients by tumor cells and immune cell interactions with the tumor. We present numerical simulations using a hybrid PDE-cellular automata (CA) model demonstrating the effects of each mechanism before discussing hypotheses about the contribution of each mechanism to morphology change.
The Linking Probability Of Deep Spider-Web Networks, Nicholas Pippenger
The Linking Probability Of Deep Spider-Web Networks, Nicholas Pippenger
All HMC Faculty Publications and Research
We consider crossbar switching networks with base b (that is, constructed from b x b crossbar switches), scale k (that is, with bk inputs, bk outputs, and bk links between each consecutive pair of stages), and depth l (that is, with l stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least k stages. We assume that each vertex is independently idle with probability q, the vacancy probability. We assume that b ≥ 2 and the vacancy probability q are fixed, and that k …
Communicating Applied Mathematics: Four Examples, Daniel E. Finkel, Christopher Kuster, Matthew Lasater, Rachel Levy, Jill P. Reese, Ilse C. F. Ipsen
Communicating Applied Mathematics: Four Examples, Daniel E. Finkel, Christopher Kuster, Matthew Lasater, Rachel Levy, Jill P. Reese, Ilse C. F. Ipsen
All HMC Faculty Publications and Research
Communicating Applied Mathematics is a writing- and speaking-intensive graduate course at North Carolina State University. The purpose of this article is to provide a brief description of the course objectives and the assignments. Parts A–D of of this article represent the class projects and illustrate the outcome of the course:
• The Evolution of an Optimization Test Problem: From Motivation to Implementation, by Daniel E. Finkel and Jill P. Reese
• Finding the Volume of a Powder from a Single Surface Height Measurement, by Christopher Kuster
• Finding Oscillations in Resonant Tunneling Diodes, by Matthew Lasater
• …
Optimal Therapy Regimens For Treatment-Resistant Mutations Of Hiv, Weiqing Gu, Helen Moore
Optimal Therapy Regimens For Treatment-Resistant Mutations Of Hiv, Weiqing Gu, Helen Moore
All HMC Faculty Publications and Research
In this paper, we use control theory to determine optimal treatment regimens for HIV patients, taking into account treatment-resistant mutations of the virus. We perform optimal control analysis on a model developed previously for the dynamics of HIV with strains of various resistance to treatment (Moore and Gu, 2005). This model incorporates three types of resistance to treatments: strains that are not responsive to protease inhibitors, strains not responsive to reverse transcriptase inhibitors, and strains not responsive to either of these treatments. We solve for the optimal treatment regimens analytically and numerically. We find parameter regimes for which optimal dosing …
The Maximal Regular Ideal Of Some Commutative Rings, Emad Abu Osba, Melvin Henriksen, Osama Alkam, Frank A. Smith
The Maximal Regular Ideal Of Some Commutative Rings, Emad Abu Osba, Melvin Henriksen, Osama Alkam, Frank A. Smith
All HMC Faculty Publications and Research
In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring R has an ideal M (R) consisting of elements a for which there is an x such that axa=a, and maximal with respect to this property. Considering only the case when R is commutative and has an identity element, it is often not easy to determine when M(R) is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of a or 1-a has a von Neumann inverse, …
Residue Class Rings Of Real-Analytic And Entire Functions, Marek Golasiński, Melvin Henriksen
Residue Class Rings Of Real-Analytic And Entire Functions, Marek Golasiński, Melvin Henriksen
All HMC Faculty Publications and Research
Let A(ℝ) and E(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if m is a maximal ideal of A(ℝ), then A(ℝ)/m is isomorphic either to the reals or a real closed field that is an η1-set, while if m is a maximal ideal of E(ℝ), then E(ℝ)/m is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of a classical characterization of algebraically closed …
Reflections Acting Efficiently On A Building, Michael E. Orrison
Reflections Acting Efficiently On A Building, Michael E. Orrison
All HMC Faculty Publications and Research
We show how Radon transforms may be used to apply efficiently the class sum of reflections in the finite general linear group GLn(Fq) to vectorsin permutation modules arising from the action of GLn(Fq) on the building oftype An−1(Fq).
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
CMC Faculty Publications and Research
Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.
Super Solutions Of The Dynamical Yang-Baxter Equation, Gizem Karaali
Super Solutions Of The Dynamical Yang-Baxter Equation, Gizem Karaali
Pomona Faculty Publications and Research
Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r−matrices. A super dynamical r−matrix r satisfies the zero weight condition if
[h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ ɧ, λ ∈ ɧ ∗ .
In this paper we classify super dynamical r−matrices with zero weight.
Complex Symmetric Operators And Applications, Stephan Ramon Garcia, Mihai Putinar
Complex Symmetric Operators And Applications, Stephan Ramon Garcia, Mihai Putinar
Pomona Faculty Publications and Research
We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.
Review: Stability Of Bases And Frames Of Reproducing Kernels In Model Spaces, Stephan Ramon Garcia
Review: Stability Of Bases And Frames Of Reproducing Kernels In Model Spaces, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Analyzing Dna Microarrays With Undergraduate Statisticians, Johanna S. Hardin, Laura Hoopes, Ryan Murphy '06
Analyzing Dna Microarrays With Undergraduate Statisticians, Johanna S. Hardin, Laura Hoopes, Ryan Murphy '06
Pomona Faculty Publications and Research
With advances in technology, biologists have been saddled with high dimensional data that need modern statistical methodology for analysis. DNA microarrays are able to simultaneously measure thousands of genes (and the activity of those genes) in a single sample. Biologists use microarrays to trace connections between pathways or to identify all genes that respond to a signal. The statistical tools we usually teach our undergraduates are inadequate for analyzing thousands of measurements on tens of samples. The project materials include readings on microarrays as well as computer lab activities. The topics covered include image analysis, filtering and normalization techniques, and …
Conjugation And Clark Operators, Stephan Ramon Garcia
Conjugation And Clark Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Yeast Through The Ages: A Statistical Analysis Of Genetic Changes In Aging Yeast, Alison Wise '05, Johanna S. Hardin, Laura Hoopes
Yeast Through The Ages: A Statistical Analysis Of Genetic Changes In Aging Yeast, Alison Wise '05, Johanna S. Hardin, Laura Hoopes
Pomona Faculty Publications and Research
Microarray technology allows for the expression levels of thousands of genes in a cell to be measured simultaneously. The technology provides great potential in the fields of biology and medicine, as the analysis of data obtained from microarray experiments gives insight into the roles of specific genes and the associated changes across experimental conditions (e.g., aging, mutation, radiation therapy, drug dosage). The application of statistical tools to microarray data can help make sense of the experiment and thereby advance genetic, biological, and medical research. Likewise, microarrays provide an exciting means through which to explore statistical techniques.
Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor
Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor
Pomona Faculty Publications and Research
We prove that a graph is intrinsically linked in an arbitrary 3–manifold M if and only if it is intrinsically linked in S3. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S3.