Mathematics Commons^™

Interpolation And Complex Symmetry, Stephan Ramon Garcia, Mihai Putinar

Pomona Faculty Publications and Research

In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.

A Model For Rolling Swarms Of Locusts, Chad M. Topaz, Andrew J. Bernoff, Sheldon Logan '06, Wyatt Toolson '07

All HMC Faculty Publications and Research

We construct an individual-based kinematic model of rolling migratory locust swarms. The model incorporates social interactions, gravity, wind, and the effect of the impenetrable boundary formed by the ground. We study the model using numerical simulations and tools from statistical mechanics, namely the notion of H-stability. For a free-space swarm (no wind and gravity), as the number of locusts increases, the group approaches a crystalline lattice of fixed density if it is H-stable, and in contrast becomes ever denser if it is catastrophic. Numerical simulations suggest that whether or not a swarm rolls depends on the statistical mechanical properties of …

Symmetric Representation Of The Elements Of Finite Groups, Barbara Hope Gwinn-Edwards

Theses Digitization Project

The main purpose of this thesis is to construct finite groups as homomorphic images of infinite semi-direct products.

Thermal Roots Of Correlation-Based Complexity, Philip Fraundorf

Physics Faculty Works

Bayesian maxent lets one integrate thermal physics and information theory points of view in the quantitative study of complex systems. Since net surprisal (a free energy analog for measuring “departures from expected”) allows one to place second law constraints on mutual information (a multimoment measure of correlations), it makes a quantitative case for the role of reversible thermalization in the natural history of invention, and suggests multiscale strategies to monitor standing crop as well. It prompts one to track evolved complexity starting from live astrophysically observed processes, rather than only from evidence of past events. Various gradients and boundaries that …

The Philosophy Of Mathematics, Erin Wilding-Martin

Sabbaticals

The philosophy of mathematics considers what is behind the math that we do. What is mathematics? Is it some cosmic truth we discover, or is it created by humans? Do mathematical objects such as numbers and functions really exist, or are they just symbols we have invented? Two of the great debates in the history of mathematical philosophy center around ontology and epistemology. Where did mathematics come from? How do we know that it is true?

Where did mathematics come from? Is it discovered or created? Ontological questions are concerned with the nature and status of mathematical objects. Some people …

Mirror Principle For Flag Manifolds, Vehbi Emrah Paksoy

Mathematics Faculty Articles

In this paper, using mirror principle developped by Lian, Liu and Yau [8, 9, 10, 11, 12, 13] we obtained the A and B series for the equivariant tangent bundles over homogenous spaces using Chern polynomial. This is necessary to obtain related cohomology valued series for given arbitrary vector bundle and multiplicative characteristic class. Moreover, this can be used as a valuable testing ground for the theories which associates quantum cohomologies and J functions of non-abelian quotient to abelian quotients via quantization

Review: On Rank-One Perturbations Of Normal Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: Shift-Type Invariant Subspaces Of Contractions, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: Hypercyclic Pairs Of Coanalytic Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Historical Tidbits, The Shoah, And The Teaching Of Mathematics, Wolff-Michael Roth

The Mathematics Enthusiast

In this extended essay, I use cultural-historical activity theory to look at the questions Theodore Eisenberg raises about the inclusion of historical facts, both historical tidbits and ethically questionable tendencies and horrific actions (the Shoah), in the teaching of mathematics. I conclude by suggesting that the ultimate answer has to be one that involves a decision, which means that an answer cannot be provided a priori or be determined by any antecedent. Deciding to include this or that in a mathematical curriculum is an ethical act.

Mathematically Gifted Elementary Students' Revisiting Of Euler's Polyhedron Theorem, Jaehoon Yim, Sanghun Song, Jiwon Kim

The Mathematics Enthusiast

This paper explores how the constructions of mathematically gifted fifth and sixth grade students using Euler’s polyhedron theorem compare to those of mathematicians as discussed by Lakatos (1976). Eleven mathematically gifted elementary school students were asked to justify the theorem, find counterexamples, and resolve conflicts between the theorem and counterexamples. The students provided two types of justification of the theorem. The solid figures suggested as counterexamples were categorized as 1) solids with curved surfaces, 2) solids made of multiple polyhedra sharing points, lines, or faces, 3) polyhedra with holes, and 4) polyhedra containing polyhedra. In addition to using the monster-barring …

Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky

CMC Faculty Publications and Research

Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 <= L <= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over (Q) over bar of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.

Modeling Algae Self-Replenishment, V. S. Manoranjan, Miguel A. Olmos Gomez, R. Corban Harwood

Faculty Publications - Department of Mathematics

This paper presents a sunlight-dependent algae growth model. Driven by the circumstances surrounding Lake Chapala, Mexico, this theoretical model is an endeavor to understand the resilient sustainability of algae that threatens the area’s ecosystem. In this paper, free-floating algae (phytoplankton) are treated as two distinct populations according to their location in the body of water: the vibrant sunlit upper region and the stagnate lower region where photosynthesis is not possible. The numerical solution for the model is analyzed and results are discussed in light of previous studies and the state of Lake Chapala.

Scale-Distortion Inequalities For Mantissas Of Finite Data Sets, Arno Berger, Theodore P. Hill, Kent E. Morrison

Research Scholars in Residence

In scientific computations using floating point arithmetic, rescaling a data set multiplicatively (e.g., corresponding to a conversion from dollars to euros) changes the distribution of the mantissas, or fraction parts, of the data. A scale-distortion factor for probability distributions is defined, based on the Kantorovich distance between distributions. Sharp lower bounds are found for the scale-distortion of n-point data sets, and the unique data set of size n with the least scale-distortion is identified for each positive integer n. A sequence of real numbers is shown to follow Benford’s Law (base b) if and only if the scale-distortion (base b) …

Results And Examples Regarding Bifurcation With A Two-Dimensional Kernal, Scott R. Kaschner

Scholarship and Professional Work - LAS

Many problems in pure and applied mathematics entail studying the structure of solutions to F(x; y) = 0, where F is a nonlinear operator between Banach spaces and y is a real parameter. A parameter value where the structure of solutions of F changes is called a bifurcation point. The particular method of analysis for bifurcation depends on the dimension of the kernel of D_xF(0,λ), the linearization of F.

The purpose of our study was to examine some consequences of a recent theorem on bifurcations with 2-dimensional kernels. This resent theorem was compared to previous methods. Also, some …

Bailey's Hexameters, Jeremiah Farrell, Al Shapiro

Scholarship and Professional Work - LAS

Nat. Bailey was the most important English lexicographer before Samuel Johnson. Our interest in this essay is to report on some of the 1730 dictionary's entries for the "Entertainmen of the Curious".

Cube Mentalism, Jeremiah Farrell, Ivan Moscovich

Scholarship and Professional Work - LAS

Our tour of multidimensional cubes begins with the marking of the eight comers of a 3-cube with the eight words HOT, POT, POD, HOD, HAD, HAT, PAT, and PAD. The figure below shows how these eight inherit the labels of the HOT-PAD).die where the letter H is opposite P, the letter O is opposite T and the letter T is opposite the letter D...

The Eight Monarchs (Some Mathematical Magic), Jeremiah Farrell, Eric Nelson

Scholarship and Professional Work - LAS

The eight Monarchs are the four Kings and four Queens of an ordinary deck of cards. We can perform our magic without a deck by using the grid below with the K-Q token ( a coin can be used instead if one wishes).

The Effect: The magician's back will be turned while Mark, the subject, places the token on one of the suit nodes. Mark is to remember this starting position. Then Mark makes a sequence of moves; a move being one of four possibilities: a horizontal move, a vertical move, or a diagonal move to a new node or …

The First Gathering, Jeremiah Farrell

Scholarship and Professional Work - LAS

The following is a photocopy of a letter I sent to my good friend James P. Fink shortly after the first Gathering in 1993. It is not reqritten or edited in any way in hopes of conveying to you some sense of my awe at how very special this event really way. My son David, then a student in Boston, was also privileged to be invited by Tom Rodgers and David and I still feel overwhelmed by the experience.

There are so many memories- and so many friends. After you have read the letter, I will remark on some of …

The Magic Octagon, Jeremiah Farrell, Tom Rodgers

Scholarship and Professional Work - LAS

The black nodes mark the corners of an octagon and each of these nodes in connected to four others by lines. The (rather hard) puzzle is to assign the sixteen numbers 0 through 15 to each of the sixteen lines so that each black node has a sum of 30 when the line numbers leading into it are added.

The word version of the puzzle was described in the article "Most-Perfect Word Magic", Oscar Thumpbindle, Word Ways Vol. 40(4). Nov. 2007.

The Stenographic Affine Plane, Oscar Thumpbindle, Jeremiah Farrell

Scholarship and Professional Work - LAS

This square, composed of familiar words, is semimagic because any row or column (the rook sweeps) anagrams into STENOGRAPHIC. "Semi" means the diagonals don't. However, there are four very special diagonals; those that have words with one of the four vowels of STENOGRAPHIC in common. These four bishop sweeps will be important later. Keeping with the chess piece theme, the reader will take notice of the knight 4-tours using the consonants P, H, Rand C. For example one of these is the P-tour SAP, PEG, PIN and TOP. Another set of knight tours traces the letters S, T, G and …

Prague Six, Jeremiah Farrell

Scholarship and Professional Work - LAS

Draft of the 'Solution Page' for Jeremiah's puzzle "Prague Six", which was exchanged at the 2008 Prague International Puzzle Party. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exhange them. This puzzle is also mass produced by Kadon Enterprises as "Chasing Squares".

The Magic Octahedron, Jeremiah Farrell

Scholarship and Professional Work - LAS

An octahedral die has several advantages over its cubic cousin, not the least of which is its ability to magically model a four dimensional tesseract. We will use a four coloring of the die to illustrate the magic.

Octahedral Dice, Todd Estroff, Jeremiah Farrell

Scholarship and Professional Work - LAS

All five Platonic solids have been used as random number generators in games involving chance with the cube being the most popular. Martin Gardenr, in his article on dice (MG 1977) remarks: "Why cubical?... It is the easiest to make, its six sides accomodate a set of numbers neither too large nor too small, and it rolls easily enough but not too easily."

Gardner adds that the octahedron has been the next most popular as a randomizer. We offer here several problems and games using octahedral dice. The first two are extensions from Gardner's article. All answers will be given …

A Study On The B Family Of Shallow Water Wave Equations, Snehanhsu Saha

Mathematics Dissertations - Archive

In this dissertation we study b family of shallow water wave equations which include classical Korteweg-de Vries, Camassa-Holm and Degasperis-Procesi equations. We first establish the models of the Camassa-Holm and Degasperis-Procesi equations, deriving them from the shallow waterwave argument and then compare a large class of properties relating to the two equations. Then we consider the b family equation as the parent equation and derive the above mentioned two equations as special cases of the b-family as well as the classical KdV equation.Next we establish results of local well-posedness using Kato's semigroup theory,global existence and blow up solutions under certain …

The Art Of Teaching Mathematics, Garikai Campbell, Jon T. Jacobsen, Aimee S A Johnson, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity?

Final Comments On Eisenberg's Paper, Brian Greer

The Mathematics Enthusiast

I must begin by expressing appreciation for the spirit in which Ted accepted my comments as intended to be intellectually and not personally provocative. Let me also assure him that I was not at all offended by the paper – it would be more accurate to say that I was stimulated by it to voice some strong disagreements. Beyond the particular aspects on which we have different perspectives, I applaud his decision to raise ethical issues, too often ignored in writing about mathematics education.

Flaws And Idiosyncrasies In Mathematicians: Food For The Classroom, Ted Eisenberg

The Mathematics Enthusiast

This paper raises an ethical question: should aspects of a mathematician’s personality, political beliefs, physical handicaps, and the ironies surrounding their life be mentioned parenthetically or otherwise in our lessons? What about the political and social norms of the times in the countries in which they lived? There are no hard and fast guidelines on this other than to use good taste; but what is in good taste to one is often in bad taste to another. At the very least this paper presents tidbits of information and innuendo about mathematicians the reader might not know. But hopefully this paper …

Reflections Upon Teaching A Poorly-Conceived Lesson, Yuichi Handa

The Mathematics Enthusiast

Using a “failed” mathematics lesson as a mini-case study, I explore some of the challenges inherent in teaching an inquiry-style mathematics lesson. The ensuing discussion centers around two issues: the preparation and subsequent scaffolding involved in guiding students to desired understandings, along with the tension inherent in “telling” and “not telling” in an inquiry-style pedagogy. I resort to personal reflection, as well as to Wenger’s (1998) theoretical construct of participation and reification in revealing the underlying link between the manner of lesson preparation and consequent engagement of students during the enactment of the lesson.

Tme Volume 5, Number 1

The Mathematics Enthusiast

No abstract provided.