Mathematics Commons^™

Topological Structures In The Equities Market Network, Gregory Leibon, Scott Pauls, Daniel Rockmore, Robert Savell

Dartmouth Scholarship

We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on “partition decoupled null models,” a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an application, we analyze a correlation matrix derived from 4 years of close prices of equities in the New York Stock Exchange (NYSE) and National Association of Securities Dealers Automated Quotation (NASDAQ). In this example, we expose (i) a natural structure composed of 2 interacting partitions of …

Unfolding Convex Polyhedra Via Quasigeodesic Star Unfoldings, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.

Unfolding Manhattan Towers, Mirela Damian, Robin Flatland, Joseph O'Rourke

Computer Science: Faculty Publications

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4×5×1 refinement of the vertex grid.

Metrology And Proportion In The Ecclesiastical Architecture Of Medieval Ireland, Avril Behan, Rachel Moss

Conference Papers

The aim of this paper is to examine the extent to which detailed empirical analysis of the metrology and proportional systems used in the design of Irish ecclesiastical architecture can be analysed to provide historical information not otherwise available. Focussing on a relatively limited sample of window tracery designs as a case study, it will first set out to establish what, if any, systems were in use, and then what light these might shed on the background, training and work practices of the masons, and, by association, the patrons responsible for employing them.

Cauchy’S Arm Lemma On A Growing Sphere, Zachary Abel, David Charlton, Sébastien Collette, Erik D. Demaine, Martin L. Demaine, Stefan Langerman, Joseph O'Rourke, Val Pinciu, Godfried Toussaint

Computer Science: Faculty Publications

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.

Grid Vertex-Unfolding Orthogonal Polyhedra, Mirela Damian

Computer Science: Faculty Publications

No abstract provided.

A Class Of Convex Polyhedra With Few Edge Unfoldings, Alex Benton, Joseph O'Rourke

Computer Science: Faculty Publications

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.

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 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.

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 …

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.

Tessellations Of The Hyperbolic Plane, Roberto Carlos Soto

Theses Digitization Project

In this thesis, the two models of hyperbolic geometry, properties of hyperbolic geometry, fundamental regions created by Fuchsian groups, and the tessellations that arise from such groups are discussed.

Foundations Of Geometry, Lawrence Michael Clarke

Theses Digitization Project

In this paper, a brief introduction to the history, and development of Euclidean geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.

Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi

Pomona Faculty Publications and Research

We show that, given any n and alpha, any embedding of any sufficiently large complete graph in R³ contains an oriented link with components Q₁,...,Q_n such that for every i not equal to j, Ilk(Q_i,Q_j)I greater than or equal to alpha and la₂(Q_i)l greater than or equal to alpha, where a₂(Q_i) denotes the second coefficient of the Conway polynomial of Q_i.

Optimal Packings Of Up To 6 Equal Circles On A Triangular Flat Torus, Anna Castelaz, William Dickinson, Daniel Guillot, Sandi Xhumari

Student Summer Scholars Manuscripts

How do you optimally pack equal circles into the standard triangular torus? In this paper, we proved the optimal packings of 1 through 6 equal circles. In order to do this we used techniques from graph theory and also mathematical softwares like Maple and LaTeX. I only worked on proving the optimal packing of 6 equal circles on this special container called the standard triangular torus. In addition, I also contributed in improving the previous proofs for 5 or less equal circles. The purpose of my research was to prove the best packing of 6 equal circles into this at …

N-Linear Algebra Of Type Ii, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to …

Weight Systems For Milnor Invariants, Blake Mellor

Mathematics Faculty Works

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.

Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi

Mathematics Faculty Works

We show that, given any n and α, every embedding of any sufficiently large complete graph in R³ contains an oriented link with components Q₁, ..., Q_n such that for every i≠j, $|\lk(Q_i,Q_j)|\geq\alpha$ and |a₂(Qi)|≥α, where a₂(Q_i) denotes the second coefficient of the Conway polynomial of Q_i.

Musical Actions Of Dihedral Groups, Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra

Mathematics Faculty Works

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.