Mathematics Commons^™

Benson's Theorem For Partial Geometries, Ellen J. Kamischke

Dissertations, Master's Theses and Master's Reports - Open

In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values …

Computer Programming To Advance Gravitational Lensing, Alex Roche

Undergraduate Review

The purpose of this research was to create a computer code that would numerically test a Poisson equation relating the mass distribution of a lens galaxy cluster to weak gravitational shear. Einstein’s theory of general relativity predicts that space-time is bent by massive objects, and in weak gravitational lensing, galaxy clusters act as lenses. The observable result is that galaxies far behind the gravitational lens will appear slightly more elliptical than they actually are. The ellipticity of the background galaxies is quantifiable and is directly related to the weak gravitational shear, and the shear is used to determine the mass …

A Note On Chaos In Product Maps, Risong Li, Xiaoliang Zhou

Turkish Journal of Mathematics

In this paper, we mainly discuss how chaos conditions on semi-flows carry over to their products. We show that if two semi-flows (or even one of them) are sensitive, so does their product. On the other side, the product of two topologically transitive semi-flows need not be topologically transitive. We then provide several sufficient conditions under which the product of two chaotic semi-flows is chaotic in the sense of Devaney. Also, stronger forms of sensitivity and transitivity for product systems are studied. In particular, we introduce the notion of ergodic sensitivity and prove that for any given two (not-necessarily continuous) …

Total Acquisition In Graphs, Timothy D. Lesaulnier, Noah Prince, Paul S. Wenger, Douglas B. West, Pratik Worah

Faculty Publications & Research

Let G be a weighted graph in which each vertex initially has weight 1. A total acquisition move transfers all the weight from a vertex u to a neighboring vertex v, under the condition that before the move the weight on v is at least as large as the weight on u. The (total) acquisition number of G, written a_t(G), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves. Among connected n-vertex graphs, a_t(G) is maximized by trees. …

An Approach To Discrete Quantum Gravity, S. Gudder

Mathematics Preprint Series

This article presents a simplified version of the author’s previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we …

A Dynamics For Discrete Quantum Gravity, Stan Gudder

Mathematics Preprint Series

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the “completed” universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρn on H that satisfy normalization and consistency conditions. …

Visualizing Chaos, Margaret Peterson

Honors Theses, 1963-2015

Chaos is typically visualized on an infinite 2D plane. By using stereographic projection, my colleague Preston Hardy and I utilized a third dimension to plot basin maps of iterative root finding methods on a subset of the complex plane onto a sphere. These spheres are then shaded in accordance to the speed in which the particular initial point converges, creating images that can be used to visualize all basins of attraction on the complex plane on a finite 3D surface. The resulting images are used to explore efficiency of root finding methods as well as evaluating the choice of addition …

The Growth In Normal Subgroups Under Direct Products, Whitney Radil

Honors Theses, 1963-2015

This paper will consider the growth of the number of normal subgroups in a nonabelian group under the direct product operation. The groups considered in this paper are dihedral groups and semidirect products with a similar structure. Many proofs will rely on the use of a key property of normal subgroups, namely closure under conjugation, to predict this growth.

Nim On Groups, Marie Meyer

Honors Theses, 1963-2015

The traditional game of Nim comprises of two players who take turns removing objects from distinct piles. The player who takes the last object is the winner. We consider the game Nim on Cayley graphs of finite groups, where the piles are located on the vertices and the number of objects in each pile is denoted as the weight of the vertex. In this version of the game, a player wins by trapping the opponent on a vertex with weight zero so he or she is unable to further reduce the weight of that vertex. We examine winning strategies for …

With One Breath, Jessica Solfest

Honors Theses, 1963-2015

Biologists know that the cardiovascular and respiratory systems interact and coordinate via the baroreflex and other pathways, but there is still much to discover regarding the dynamical aspects of their communication. Synchronization patterns between the respiratory and cardiovascular systems evince their interaction. This thesis focuses on the synchronization of cardiovascular and respiratory activity. I analyze the frequency of this synchrony to determine patterns and characterize the systems’ adaptability to changes in the environment. I applied time series analysis and Markov processes to dynamical systems in an attempt to unearth meaning behind cardiorespiratory communicative mysteries. Through this research, I found that …

Coding Theory-Based Cryptopraphy: Mceliece Cryptosystems In Sage, Christopher Roering

Honors Theses, 1963-2015

Unlike RSA encryption, McEliece cryptosystems are considered secure in the presence of quantum computers. McEliece cryptosystems leverage error-correcting codes as a mechanism for encryption. The open-source math software Sage provides a suitable environment for implementing and exploring McEliece cryptosystems for undergraduate research. Using our Sage implementation, we explored Goppa codes, McEliece cryptosystems, and Stern’s attack against a McEliece cryptosystem.

Ioana's Superrigidity Theorem And Orbit Equivalence Relations, Samuel Coskey

Mathematics Faculty Publications and Presentations

We give a survey of Adrian Ioana’s cocycle superrigidity theoremfor profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana’s theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas’s theorem that the classification of torsion-free abelian groups of …

Navigating The Seas Of Mathematics Education: New Waves In Research To Improve Student Learning, Jinfa Cai, Sasha Wang

Mathematics Faculty Publications and Presentations

This issue focuses on research in the domain of mathematics education. Although mathematics has been a subject of study for many centuries, mathematics education is a relatively new field of scholarly inquiry, having been established as an independent field of research only in the early twentieth century. The most significant milestone was the establishment of the International Commission on Mathematical Instruction (ICMI) in 1908. Since 1969, ICMI has organized the International Congress on Mathematical Education, a quadrennial international meeting whose aim is to present the current states and trends in mathematics education research and in the practice of mathematics teaching …

The Archimedean - January 2013, Department Of Mathematics, Syracuse University

Newsletters from the Department of Mathematics - The Archimedean

No abstract provided.

New Bounds And Computations On Prime-Indexed Primes, Jonathan Bayless, Dominic Klyve, Tomas Oliveira E Silva

Mathematics Faculty Scholarship

In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime.

We give explicit upper and lower bounds for π²(x), the number of prime-indexed primes up to …

Methods In Visual Mathematics: Reductionism In Researching Mathematical Principles In Art, Lauren N. Colie

AUCTUS: The Journal of Undergraduate Research and Creative Scholarship

People traditionally rely on visual arts as an effective communication tool and medium of self-expression for when words fail to convey abstract concepts. Thera Mjaaland, anthropologist and professional photographer, writes, “Art is capable of negotiating conceptual gaps caused by a dichotomized epistemology” (393). In essence, Mjaaland asserts that art helps relate different modes of thinking by illustrating the abstract and difficult to grasp—privileging the communicative value of an image over that of text. Within this method of communication is a collection of works acknowledged by public consensus to be of an elevated status or value. The art world is deeply …

Intersections Of Loops And The Andersen–Mattes–Reshetikhin Algebra, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

Given two free homotopy classes α₁,α₂ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points # (α₁,α₂) of loops in these two classes.

We show that, for α₁≠α₂, the number of terms in the Andersen–Mattes–Reshetikhin Poisson bracket of α₁ and α₂ is equal to # (α₁,α₂). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of α₁ and α₂.

The …

A New Recursion For Three-Column Combinatorial Macdonald Polynomials, Elizabeth Niese

Mathematics Faculty Research

The Hilbert series of the Garsia–Haiman module M_μ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n . Since there are n ! fillings that generate , it is desirable to find recursions to reduce the number of fillings that need to be considered when computing combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!ⁿn!).

Rational Map Of Cp^2 With No Invariant Foliation, Scott R. Kaschner, Rodrigo A. Perez, Roland K.W. Roeder

Scholarship and Professional Work - LAS

Conference Poster presented at: Midwest Dynamical Systems Conference, Champaign/Urbana, IL November 1-3, 2013.

Chasing Squares [Pamphlet], Jeremiah Farrell

Scholarship and Professional Work - LAS

Pamphlet accompanying Jeremiah Farrell's mass-produced "Chasing Squares" puzzle.