Physical Sciences and Mathematics Commons^™

Newton, Maclaurin, And The Authority Of Mathematics, Judith V. Grabiner

Pitzer Faculty Publications and Research

Sir Isaac Newton revolutionized physics and astronomy in his Principia. How did he do it? Would his method work on any area of inquiry, not only in science, but also about society and religion? We look at how some Newtonians, most notably Colin Maclaurin, combined sophisticated mathematical modeling and empirical data in what has come to be called the "Newtonian Style." We argue that this style was responsible not only for Maclaurin’s scientific success but for his ability to solve problems ranging from taxation to insurance to theology. We show how Maclaurin’s work strengthened the prestige of Newtonianism and …

A (Not So) Complex Solution To A² + B² = Cⁿ, Arnold M. Adelberg, Arthur T. Benjamin, David I. Rudel '99

All HMC Faculty Publications and Research

No abstract provided in this article.

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 …

On The Polarization Of Closed Strings By Ramond-Ramond Fluxes, Vatche Sahakian

All HMC Faculty Publications and Research

In the Green-Schwarz formalism, the closed string worldsheet of the IIB theory couples to Ramond-Ramond (RR) fluxes through spinor bilinears. We study the effect of such fluxes by analyzing the supersymmetry transformation of the worldsheet in general backgrounds. We show that, in the presence RR fields, the closed string can get `polarized', as the spinors acquire non-zero vevs in directions correlating with the orientation of close-by D-branes. Reversing the argument, this may allow for worldsheet configurations—with non-trivial spinor structure—that source RR moments.

Heights And Diophantine Problems, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at Rice University, September 2004.

Immunogold Labeling To Enhance Contrast In Optical Coherence Microscopy Of Tissue Engineered Corneal Constructs, Chris B. Raub, Elizabeth J. Orwin, Richard C. Haskell

All HMC Faculty Publications and Research

Our lab has used an optical coherence microscope (OCM) to assess both the structure of tissue-engineered corneal constructs and their transparency. Currently, we are not able to resolve cells versus collagen matrix material in the images produced. We would like to distinguish cells in order to determine if they are viable while growing in culture and also if they are significantly contributing to the light scattering in the tissue. In order to do this, we are currently investigating the use of immunogold labeling. Gold nanoparticles are high scatterers and can create contrast in images. We have conjugated gold nanoparticles to …

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?

Visualizing Early Frog Development With Motion-Sensitive 3-D Optical Coherence Microscopy, Richard C. Haskell, Mary E. Williams, Daniel C. Petersen, Barbara M. Hoeling, Andrew J. Schile, J. D. Pennington, M. G. Seetin, J. M. Castelaz, Scott E. Fraser, Cyrus Papan, Hongwu Ren, Johannes F. De Boer, Zhongping Chen

All HMC Faculty Publications and Research

A motion-sensitive en-face-scanning 3-D optical coherence microscope (OCM) has been designed and constructed to study critical events in the early development of plants and animals. We describe the OCM instrument and present time-lapse movies of frog gastrulation, an early developmental event in which three distinct tissue layers are established that later give rise to all major organ systems. OCM images constructed with fringe-amplitude data show the mesendoderm migrating up along the blastocoel roof, thus forming the inner two tissue layers. Motion-sigma data, measuring the random motion of scatterers, is used to construct complementary images that indicate the presence of Brownian …

Femtosecond Spectrotemporal Magneto-Optics, J.-Y. Bigot, L. Guidoni, E. Beaurepaire, Peter N. Saeta

All HMC Faculty Publications and Research

A new method to measure and analyze the time and spectrally resolved polarimetric response of magnetic materials is presented. It allows us to study the ultrafast magnetization dynamics of a CoPt₃ ferromagnetic film. The analysis of the pump-induced rotation and ellipticity detected by a broad spectrum probe beam shows that magneto-optical signals predominantly reflect the spin dynamics in ferromagnets.

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 …

Patterning Polymer Thin Films: Lithographically Induced Self Assembly And Spinodal Dewetting, Regina C. Carns

Pomona Senior Theses

In an age in which the microchip is ubiquitous, the rewards for novel methods of microfabrification are great, and the vast possibilities of nanotechnology lie just a little ahead. Various methods of microlithography offer differing benefits, and even as older techniques such as optical lithography are being refined beyond what were once considered their upper limits of resolution, new techniques show great promise for going even further once they reach their technological maturity. Recent developments in optical lithography may allow it to break the 100-nm limit even without resorting to x-rays.

Behavior Of Coupled Automata, Reuben Gann, Jessica Venable, Eric J. Friedman, Adam S. Landsberg

WM Keck Science Faculty Papers

We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of "synchronization" between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles.

Closed Strings In Ramond-Ramond Backgrounds, Vatche Sahakian

All HMC Faculty Publications and Research

We write the IIB Green-Schwarz action in certain general classes of curved backgrounds threaded with Ramond-Ramond fluxes. The fixing of the kappa symmetry in the light-cone gauge and the use of supergravity Bianchi identities simplify the task. We find an expression that truncates to quartic order in the spacetime spinors and relays interesting information about the vacuum structure of the worldsheet theory. The results are particularly useful in exploring integrable string dynamics in the context of the holographic duality.

Double Excitations Within Time-Dependent Density Functional Theory Linear Response, Neepa T. Maitra, Fan Zhang, Robert J. Cave, Kieron Burke

All HMC Faculty Publications and Research

Within the adiabatic approximation, time-dependent density functional theory yields only single excitations. Near states of double excitation character, the exact exchange–correlation kernel has a strong dependence on frequency. We derive the exact frequency-dependent kernel when a double excitation mixes with a single excitation, well separated from the other excitations, in the limit that the electron–electron interaction is weak. Building on this, we construct a nonempirical approximation for the general case, and illustrate our results on a simple model.

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".

Limits To Performance Improvement Provided By Balanced Interferometers And Balanced Detection In Oct/Ocm Instruments, David Liao, Adam E. Pivonka, Brendan R. Haberle, Daniel C. Petersen, Barbara M. Hoeling, Richard C. Haskell

All HMC Faculty Publications and Research

We compare the dynamic range of OCT/OCM instruments configured with unbalanced interferometers, e.g., Michelson interferometers, with that of instruments utilizing balanced interferometers and balanced photodetection. We define the dynamic range (DR) as the ratio of the maximum fringe amplitude achieved with a highly reflecting surface to the root-mean-square (rms) noise. Balanced systems achieve a dynamic range 2.5 times higher than that of a Michelson interferometer, enabling an image acquisition speed roughly 6 times faster. This maximum improvement occurs at light source powers of a few milliwatts. At light source powers higher than 30 mW, the advantage in acquisition speed of …

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.

An Unusual Lava Cave From Ol Doinyo Lengai, Tanzania, Donald A. Mcfarlane, Joyce Lundberg, Frederick Belton

WM Keck Science Faculty Papers

A new type of lava cave is described from the summit crater of Ol Doinyo Lengai, a unique active carbonatite volcano in Tanzania. This and other similar caves on Ol Doinyo Lengai are formed by thermal erosion and aqueous dissolution of otherwise solid spatter cones. Meteoritic water and endogenous condensates act to form speleothems of complex mineralogy up to 3 m in length. We propose the new classification of “polygenetic spatter cone cave”.

Guano, Donald A. Mcfarlane

WM Keck Science Faculty Papers

This is an encyclopedia article.

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

Humanistic Mathematics Network Journal

This paper considers the recent history of mathematics teaching.

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?

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

Humanistic Mathematics Network Journal

No abstract provided.

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.

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.

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

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.