Mathematics Commons^™

Apollonius Of Perga: Historical Background And Conic Sections, Ashley Allen

Senior Honors Theses

Apollonius of Perga greatly contributed to geometry, specifically in the area of conics. Through the study of the “Golden Age” of Greek mathematics from about 300 to 200 B.C., Apollonius, who lived from about 262 to 190 B.C., is seen as an innovative geometer whose theories were eventually proved factual. By investigating his theorems on how the different kinds of conic sections are formed, the standard equations for each conic will be better understood. The achievements that have been made as a result of conics include flashlights to whispering galleries.

The Multilevel Structures Of Nurbs And Nurblets On Intervals, Weiwei Zhu

Dissertations

This dissertation is concerned with the problem of constructing biorthogonal wavelets based on non-uniform rational cubic B-Splines on intervals. We call non-uniform rational B-Splines ``NURBs", and such biorthogonal wavelets ``NURBlets". Constructing NURBlets is useful in designing and representing an arbitrary shape of an object in the industry, especially when exactness of the shape is critical such as the shape of an aircraft. As we know presently most popular wavelet models in the industry are approximated at boundaries. In this dissertation a new model is presented that is well suited for generating arbitrary shapes in the industry with mathematical exactness throughout …

Geometric Characterization Of Digital Objects: Algorithms And Applications To Image Analysis., Arindam Biswas Dr.

Doctoral Theses

Several problems of characterizing a digital object, and particularly, those related to boundary description, have been studied in this thesis. New algorithms and their applications to various aspects of image analysis and retrieval have been reported. A combinatorial technique for constructing the outer and inner isothetic covers of a digital object has been developed. The resolution of the background 2D grid can be changed by varying the grid spacing, and this procedure can be used to extract shape and topological information about the object. Next, an algorithm has been designed for constructing the orthogonal (convex) hull of a digital object …

Super-Relaxed (Η)-Proximal Point Algorithms, Relaxed (Η)-Proximal Point Algorithms, Linear Convergence Analysis, And Nonlinear Variational Inclusions, Ram U. Verma, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed (η)-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal (η)-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the …

The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser

Mathematical Sciences Technical Reports (MSTR)

Consider a subgroup of an elliptic curve generated by a point P of order n. It is possible to match any point Q to an integer k (mod n) such that Q = kP using a brute force method. By observing patterns in the digraph of the squaring map on the integers modulo n it is possible to perform this matching. These techniques can be applied to solving the Elliptic Curve Discrete Log Problem given a complete graph of the square mapping k P -> k^2 P for the elliptic curve points.

Classes Of Commutative Clean Rings, Wolf Iberkleid, Warren William Mcgovern

Mathematics Faculty Articles

Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every element α∈A there is an idempotent e = e²∈A such that α−e is a unit and αe belongs to I. A filter of ideals, say F, of A is Noetherian if for each I∈F there is a finitely generated ideal J∈F such that J⊆I. We characterize I-clean rings for the ideals 0, n(A), J( …

A Preliminary Mathematical Model Of Skin Dendritic Cell Trafficking And Induction Of T Cell Immunity, Amy H. Lin Erickson, Alison Wise, Stephen Fleming, Margaret Baird, Zabeen Lateef, Annette Molinaro, Miranda Teboh-Ewungkem, Lisette G. De Pillis

All HMC Faculty Publications and Research

Chronic inflammation is a process where dendritic cells (DCs) are constantly sampling antigen in the skin and migrating to lymph nodes where they induce the activation and proliferation of T cells. The T cells then travel back to the skin where they release cytokines that induce/maintain the inflammatory condition. This process is cyclic and ongoing. We created a differential equations model to reflect the initial stages of the inflammatory process. In particular, we modeled antigen stimulation of DCs in the skin, movement of DCs from the skin to a lymph node, and the subsequent activation of T cells in the …

Update Sequence Stability In Graph Dynamical Systems, Matthew Macauley, Henning S. Mortveit

Publications

In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield vastly different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability.

A Critical Centre-Stable Manifold For Schroedinger's Equation In R^3, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

Consider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t \psi + \Delta \psi + | \psi |^2 \psi = 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conversely, in a general setting, any solution …

New Estimates For A Time-Dependent Schroedinger Equation, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem.

Continuous Trace C*-Algebras, Gauge Groups And Rationalization, John R. Klein, Claude Schochet, Samuel B. Smith

Mathematics Faculty Research Publications

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter Shiue

Scholarship

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

Reduced Order Models For Fluid-Structure Interaction Systems By Mixed Finite Element Formulation, Ye Yang

Dissertations

In this work, mixed finite element formulations are introduced for acoustoelastic fluid- structure interaction (FSI) systems. For acoustic fluid, in addition to displacement- pressure (u/p) mixed formulation, a three-field formulation, namely, displacement-pressure-vorticity moment formulation (u - p -Λ) is employed to eliminate some zero frequencies. This formulation is introduced in order to compute the coupled frequencies without the contamination of nonphysical spurious non-zero frequencies. Furthermore, gravitational forces are introduced to include the coupled sloshing mode. In addition, u/p mixed formulation is the first time employed in solid. The numerical examples will demonstrate that the mixed formulations are capable of predicted …

Operations Research Methods For Optimization In Radiation Oncology, M Ehrgott, Allen Holder

Mathematical Sciences Technical Reports (MSTR)

Operations Research has a successful tradition of applying mathematical analysis to a wide range of applications, with one of the burgeoning areas of growth being in medical physics. The original application was in the optimal design of the influence map for a radiotherapy treatment, a problem that has continued to receive attention. However, operations research has been applied to other clinical problems like patient scheduling, vault design, and image alignment. The overriding theme of this article is to present how techniques in operations research apply to clinical problems, which we accomplish in three parts. First, we present the perspective from …

Contact Process In A Wedge, J. Theodore Cox, Nevena Maric, Rinaldo Schinazi

Mathematics - All Scholarship

We prove that the supercritical one-dimensional contact process survives in certain wedge-like space-time regions, and that when it survives it couples with the unrestricted contact process started from its upper invariant measure. As an application we show that a type of weak coexistence is possible in the nearest-neighbor "grass-bushes-trees'' successional model introduced in Durrett and Swindle (1991).

Normal Surfaces And Heegaard Splittings Of 3-Manifolds., Tejas Kalelkar Dr.

Doctoral Theses

This thesis deals with various questions regarding normal surfaces and Heegaard splittings of 3-manifolds.Chapter 1The first chapter is divided into two parts. In the first, we give an outline of normal surface theory and mention some of its important applications. The second part gives an overview of the theory of Heegaard splitting surfaces and a few of its applications. None of the material covered in this chapter is original and it is meant solely as an exposition of known results.Chapter 2In this chapter, we give a lower bound on the Euler characteristic of a normal surface, a topological invariant, in …

Lifting Bailey Pairs To Wp-Bailey Pairs, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

A pair of sequences (α_n(a,k,q),β_n(a,k,q)) such that α₀(a,k,q)=1 and β_n(a,k,q)=∑ⁿ_j=0 (k/a;q)_n−j (k;q)_n+j / (q;q)_n−j (aq;q)_n+j α_j (a,k,q) is termed a WP-Bailey Pair. Upon setting k=0 in such a pair we obtain a Bailey pair.

In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some …

A Simple Model Of Cortical Dynamics Explains Variability And State Dependence Of Sensory Responses In Urethane-Anesthetized Auditory Cortex, Carina Curto, Shuzo Sakata, Stephan Marguet, Vladimir Itskov, Kenneth D. Harris

Department of Mathematics: Faculty Publications

The responses of neocortical cells to sensory stimuli are variable and state dependent. It has been hypothesized that intrinsic cortical dynamics play an important role in trial-to-trial variability; the precise nature of this dependence, however, is poorly understood. We show here that in auditory cortex of urethane-anesthetized rats, population responses to click stimuli can be quantitatively predicted on a trial-by-trial basis by a simple dynamical system model estimated from spontaneous activity immediately preceding stimulus presentation. Changes in cortical state correspond consistently to changes in model dynamics, reflecting a nonlinear, self-exciting system in synchronized states and an approximately linear system in …

Transition To Mixing And Oscillations In A Stokesian Viscoelastic Flow, Becca Thomases, Michael Shelley

Mathematics Sciences: Faculty Publications

In seeking to understand experiments on low-Reynolds-number mixing and flow transitions in viscoelastic fluids, we simulate the dynamics of the Oldroyd-B model, with a simple background force driving the flow. We find that at small Weissenberg number, flows are "slaved" to the extensional geometry imposed by forcing. For large Weissenberg number, such solutions become unstable and transit to a structurally dissimilar state dominated by a single large vortex. This new state can show persistent oscillatory behavior with the production and destruction of smaller-scale vortices that drive mixing.

Application Of Dual-Tree Complex Wavelet Transforms To Burst Detection And Rf Fingerprint Classification, Randall W. Klein

Theses and Dissertations

This work addresses various Open Systems Interconnection (OSI) Physical (PHY) layer mechanisms to extract and exploit RF waveform features (”fingerprints”) that are inherently unique to specific devices and that may be used to provide hardware specific identification (manufacturer, model, and/or serial number). This is addressed by applying a Dual-Tree Complex Wavelet Transform (DT-CWT) to improve burst detection and RF fingerprint classification. A ”Denoised VT” technique is introduced to improve performance at lower SNRs, with denoising implemented using a DT-CWT decomposition prior to Traditional VT processing. A newly developed Wavelet Domain (WD) fingerprinting technique is presented using statistical WD fingerprints with …

Human Capital And Economic Growth: Theory And Policy., Bidisha Chakraborty Dr.

Doctoral Theses

Growth theory is one of the most important branches of macroeconomics. Growth theory helps us to understand the intertemporal behaviour of a dynamic economy and to understand the properties of the long-run rate of economic growth. It identifies the factors causing the deviation of the actual rate of growth from the socially efficient rate of growth and analyses the effectiveness of various policies in removing this gap. It analyses the condition of stability of the long-run equilibrium and also attempts to establish links between the long run equilibrium and the persistence of underdevelopment.With the emergence of the ‘new’ growth theory, …

Independent Domination In Complementary Prisms., Joel Agustin Gongora

Electronic Theses and Dissertations

Let G be a graph and G̅ be the complement of G. The complementary prism GG̅ of G is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. For example, if G is a 5-cycle, then GG̅ is the Petersen graph. In this paper we investigate independent domination in complementary prisms.

The Last Of The Mixed Triple Systems., Ernest Jum

Electronic Theses and Dissertations

In this thesis, we consider the decomposition of the complete mixed graph on v vertices denoted M_v, into every possible mixed graph on three vertices which has (like M_v) twice as many arcs as edges. Direct constructions are given in most cases. Decompositions of theλ-fold complete mixed graph λM_v, are also studied.

Existence Of Pseudo Almost Automorphic Solutions For The Heat Equation With Sp-Pseudo Almost Automorphic Coefficients, Toka Diagana, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

We obtain the existence of pseudo almost automorphic solutions to the N-dimensional heat equation with Sp-pseudo almost automorphic coefficients.

Flattening A Cone, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

We want to manufacture a cut-off slanted cone from a flat sheet of metal. If the cone were a normal right cone we know that we would simply cut out a sector of a circle and roll it up. However the cone is slanted. We want to know what the flattened shape looks like so that we can cut it out and roll it up to closely approximate correct final shape. We also want to minimize the amount of wasted metal after the shape is cut out.

The problem, and it generalizations may be solved analytically but the analytical solution …

A Remark On The Topology Of (N,N) Springer Varieties, Stephan M. Wehrli

Mathematics - All Scholarship

We prove a conjecture of Khovanov [Kho04] which identifies the topological space underlying the Springer variety of complete flags in C²ⁿ stabilized by a fixed nilpotent operator with two Jordan blocks of size n.

Solving The P-Median Problem With Insights From Discrete Vector Quantization, Gino J. Lim, Allen Holder, Josh Reese

Mathematical Sciences Technical Reports (MSTR)

The goals of this paper are twofold. First, we formally equate the p-median problem from facility location to the optimal design of a vector quantizer. Second, we use the equivalence to show that the Maranzana Algorithm can be interpreted as a projected Lloyd Algorithm, a fact that improves complexity. Numerical results verify significant improvements in run-time.

A Decomposition Of The Pure Parsimony Problem, Allen Holder, Thomas M. Langley

Mathematical Sciences Technical Reports (MSTR)

We partially order a collection of genotypes so that we can represent the problem of inferring the least number of haplotypes in terms of substructures we call g-lattices. This representation allows us to prove that if the genotypes partition into chains with certain structure, then the NP-Hard problem can be solved efficiently. Even without the specified structure, the decomposition shows how to separate the underlying integer programming model into smaller models.

A Clustering Approach For Optimizing Beam Angles In Imrt Planning, Gino J. Lim, Allen Holder, Josh Reese

Mathematical Sciences Technical Reports (MSTR)

In this paper we introduce a p-median problem based clustering heuristic for selecting efficient beam angles for intensity-modulated radiation therapy. The essence of the method described here is the clustering of beam angles according to probability that an angle will be observed in the final solution and similarities among different angles and the selection of a representative angle from each of the p resulting cluster cells. We conduct experiments using several combinations of modeling parameters to find the conditions where the heuristic best performs. We found a combination of such parameters that outperformed all other parameters on three of the …

An Economical Model With Allee Effect, Rafael Luís, Saber Elaydi, Henrique Oliveira

Mathematics Faculty Research

The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hypothesis of Marx …