Mathematics Commons^™

Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d-dimensional torus ^d = ℝ^d/ℤ^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^−n/2, where C₁ = C(n, d) is …

Heights And Diophantine Problems, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at Rice University, September 2004.

Putnam, Pizza & Problem Solving, Andrew J. Bernoff, Francis E. Su

All HMC Faculty Publications and Research

Ok, here's a difficult question for you.. How can you get roughly 10% of the student body at your college to get up early on a Saturday and spend six hours working on an incredibly difficult exam for which many will get a score of zero?

A Liouville-Gelfand Equation For K-Hessian Operators, Jon T. Jacobsen

All HMC Faculty Publications and Research

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand problem from combustible gas dynamics and prescribed curvature problems.

An Experimental Study Of Micron-Scale Droplet Aerosols Produced Via Ultrasonic Atomization, Thomas D. Donnelly, J. Hogan '03, A. Mugler '04, N. Schommer '04, M. Schubmehl '02, Andrew J. Bernoff, B. Forrest '02

All HMC Faculty Publications and Research

In the last 10 years, laser-driven fusion experiments performed on atomic clusters of deuterium have shown a surprisingly high neutron yield per joule of input laser energy. Results indicate that the optimal cluster size for maximizing fusion events should be in the 0.01–μm diameter range, but an appropriate source of droplets of this size does not exist. In an attempt to meet this need, we use ultrasonic atomization to generate micron-scale droplet aerosols of high average density, and we have developed and refined a reliable droplet sizing technique based on Mie scattering. Harmonic excitation of the fluid in …

Magical Miscellany, Francis Su

All HMC Faculty Publications and Research

What is a Math Fun Fact, you ask? A Math Fun Fact is any mathematical tidbit that can be presented or grasped quickly, is surprising or captivating, can be generally enjoyed by friends of mathematics, and is hopefully fun! Of course, part of the fun is thinking about why the Fun Fact is true--so we won't spoil the fun. Though, we may give you some hints and references

However, since there are infinitely many Math Fun Facts (prove this), we can only bring you a few each time... here are a few whose conclusions might be considered "magical".

Using Humanistic Content And Teaching Methods To Motivate Students And Counteract Negative Perceptions Of Mathematics, Roger Haglund

Humanistic Mathematics Network Journal

This paper examines the following questions: How is math commonly taught, why is it taught this way, and what are the outcomes? Who are some of the voices calling for change and what are they saying? Can a humanistic approach produce positive results in students who have learned to dislike math and have not been successful in a traditional classroom?

Man's Cards And God's Dice: A Conceptual Analysis Of Probability For The Advanced Student, Elie Feder

Humanistic Mathematics Network Journal

No abstract provided.

What Are Mathematical Problems?, Emam Hoosain

Humanistic Mathematics Network Journal

No abstract provided.

A Linear Perspective To Art, Sarah Littler

Humanistic Mathematics Network Journal

No abstract provided.

Humanistic Mathematics As Mathematics For All, Michael N. Fried

Humanistic Mathematics Network Journal

No abstract provided.

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

Humanistic Mathematics Network Journal

No abstract provided.

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 …

Taxicab Geometry As A Vehicle For The Journey Toward Enlightenment, Neil Greenspan

Humanistic Mathematics Network Journal

No abstract provided.

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.

Mathematical Magic, Arthur T. Benjamin

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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.

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

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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 …

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

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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.

Is Mathematics Education Taking A Step Backward?, Frances Kuwahara Chinn

Humanistic Mathematics Network Journal

This paper considers the recent history of mathematics teaching.

Tesselland: A Mathematical Oddment, Martin Glover

Humanistic Mathematics Network Journal

No abstract provided.

Bridging To Infinity, Mike Pinter

Humanistic Mathematics Network Journal

The author's own experiences as a mathematics student and teacher have influenced how he thinks about the infinite. Author Madeleine L'Engle has also shaped his thinking with her writing. The author offers some thoughts that connect some of L'Engle's writing with his experience.

Mathematics, The Liberal Arts, And Slavish Devotions, J. D. Phillips

Humanistic Mathematics Network Journal

No abstract provided.

Humanistic Mathematics: Personal Evaluation And Excavations, Stephen I. Brown

Humanistic Mathematics Network Journal

No abstract provided.

A Brief Look At Mathematics And Theology, Philip J. Davis

Humanistic Mathematics Network Journal

No abstract provided.

Innumeracy And Its Perils, Numeracy And Its Promises, Ramakrishnan Menon

Humanistic Mathematics Network Journal

No abstract provided.

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

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.

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 …

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

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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.