Mathematics Commons^™

Selected Problems Of Inference On Branching Processes And Poisson Shock Model, Satrajit Roychoudhury

Dissertations

This dissertation explores the development of statistical methodology for some problems of branching processes and poisson shock model.

Branching process methods have become extremely popular in recent days. This dissertation mainly explores two fundamental inference problems of Galton-Watson processes. The first problem is concerned with statistical inference regarding the nature of the process. Two methodologies have been developed to develop a statistical test for the null hypothesis that the process is supercritical versus an alternative hypothesis that the process is non-supercritical. Another problem we investigate involves the estimation of the 'age' of a Galton-Watson Process. Three different methods are discussed …

Some Heuristics About Elliptic Curves, Mark Watkins

Mathematics - All Scholarship

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E,1)=0. We find that we expect there to be about c₁X^19/24(log X)^3/8 curves with |Delta|< X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX^5/6 total curves, this implies that asymptotically almost all even parity curves have rank 0. We …

Essays On Individual And Collective Powers In A Voting Body., Sonali Roy Dr.

Doctoral Theses

MotivationThe issue of measurement of voting power is a very important topic of discussion in social science these days. The concept of voting power concerns any collective decision making body (or, equivalently, a collectivity) which makes ‘yes’ or ‘no’ decisions on any issue, by the process of voting. Examples of such bodies abound in today’s world. The United Nations Security Council, The Council of Ministers in the European Union, the Parliament of the republic of India, the board room of any corporate house etc., are all examples of such decision making bodies.The voting process of each of these bodies is …

Preprojective Representations Of Valued Quivers And Reduced Words In The Weyl Group Of A Kac-Moody Algebra, Mark Kleiner, Allen Pelley

Mathematics - All Scholarship

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced …

Categorification Of The Colored Jones Polynomial And Rasmussen Invariant Of Links, Anna Beliakova, Stephan Wehrli

Mathematics - All Scholarship

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that, for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples, where this invariant is a …

A Complete Characterization Of Maximal Symmetric Difference-Free Families On {1,…N}., Travis Gerarde Buck

Electronic Theses and Dissertations

Prior work in the field of set theory has looked at the properties of union-free families. This thesis investigates families based on a different set operation, the symmetricc difference. It provides a complete characterization of maximal symmetric differencefree families of subsets of {1, . . . n}

Movie Math: Mathematical Talent = Mental Illness, Christopher D. Goff

College of the Pacific Faculty Presentations

No abstract provided.

Sequences Of Reflection Functors And The Preprojective Component Of A Valued Quiver, Mark Kleiner, Helene R. Tyler

Mathematics - All Scholarship

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated to an indecomposable symmetrizable generalized Cartan …

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang

Mathematics Research Reports

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.

Nash Equilibria In Cauchy-Random Zero-Sum And Coordination Matrix Games, David P. Roberts

Mathematics Publications

We consider zero-sum games (A, − A) and coordination games (A,A), where A is an m-by-n matrix with entries chosen independently with respect to the Cauchy distribution. In each case, we give an exact formula for the expected number of Nash equilibria with a given support size and payoffs in a given range, and also asymptotic simplications for matrices of a fixed shape and increasing size. We carefully compare our results with recent results of McLennan and Berg on Gaussian random bimatrix games (A,B), and describe how …

Teaching Time Savers: Style Points, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

When I began as an assistant professor, I had a pretty good sense of how much time it would take for me to prepare for each class. After a few conversations with my new colleagues, I even had a good sense of how much time I should devote to tasks like office hours and committee work. Somewhere in the middle of grading my first exam, though, it became painfully clear that I had underestimated the amount of time I would need to grade exams!

Exponents For B-Stable Ideals, Eric Sommers, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1, m 2,. . .,m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n, B n, C n and some other types. When I = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the …

Chainability And Hemmingsen's Theorem, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

On the surface, the definitions of chainability and Lebesgue covering dimension ⩽1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and AA is a lattice base for a nondegenerate continuum, then AA is elementarily equivalent to a lattice base for a continuum Y …

The Effect Of Using A Video-Case Curriculum To Promote Preservice Teachers’ Development Of A Reflective Stance Towards Mathematics Teaching, Shari L. Stockero

Dissertations

This study investigates the effects of using a coherent video-case curriculum in a university methods course. In particular, three issues are addressed: (1) howthe use of a video-case curriculum affects the reflective stance of preservice teachers; (2) the extent to which a reflective stance developed while reflecting on other teachers' practice transfers to reflecting on one's own practice; and (3) how preservice teachers' reflective stance that is developed via sustained and focused reflection using a video-case curriculum compares to the reflective stance of peers who engaged in less sustained and focused reflection. Althoughvideo cases are increasingly being used in teacher …

Formant Analysis For Kiswahili Vowels, Y. Y. Sungita, E. E. Mhamilawa

Tanzania Journal of Science

Vowels' spectral characteristics in a language have been studied for suitability in speech recognition by using formants analysis technique. Other techniques do mostly require large computer memories for speech processing and analysis. In this paper, the formant analysis for Kiswahili vowels has been presented. The spectrographs for each vowel and their respective average formant frequencies are tabled. The distribution of formants for the vowels modelled in the form of an articulatory model is shown. The results show that there is a big separation of formant frequencies among the Kiswahili vowels that signify the suitability for automatic speech recognition.

Uncertainty Principles On Nilpotent Lie Groups., Sanjay Parui Dr.

Doctoral Theses

No abstract provided.

Markov Chain Monte Carlo Methods For Regression Splines With A Penalized Acceptance Ratio, David Keith Stamps

Dissertations

An increasingly popular method for fitting complex models, particularly with a hierchical structure involvese the use of Markov Chain Monte Carlo simulation. Within a Bayesian framework, two major strategies are Gibbs sampling and Metropolis-Hastings methods. Recent research in the area of MCMC methods has witnessed the emergence of modeling efforts which permit the movement of the chain across models of varying dimensions. When properly constructed, such Markov chains converge to the joint posterior distribution of the parameters to be estimated, making Bayesian averaging an attractive option after convergence has occurred. With this transdimensional methodology, the Bayesian averaging process takes place …

Image Segmentation By Energy And Related Functional Minimization Methods, Eric Hudson Mason

Dissertations

Effective and efficient methods for partitioning a digital image into image segments, called ¿image segmentation,¿ have a wide range of applications that include pattern recognition, classification, editing, rendering, and compressed data for image search. In general, image segments are described by their geometry and similarity measures that identify them. For example, the well-known optimization model proposed and studied in depth by David Mumford and Jayant Shah is based on an L₂ total energy functional that consists of three terms that govern the geometry of the image segments, the image fidelity (or closeness to the observed image), and the prior …

On The N-Body Problem, Zhifu Xie

Theses and Dissertations

In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses …

Knots Not For Naught, Sharleen Adrienne Roberts

Theses and Dissertations

The goal of this paper is to find the Homfly polynomial for each knot in a specific family of knots. This family of knots is generated from placing the Whitehead link into a solid torus, slicing the torus at a spot where the Whitehead has no crossings and then twisting the torus 360 degrees in either direction an integral number of times. Let L(n) denote the knot obtained by twisting the torus 360 degrees, n times. Note that n is an integer. Let the twists be towards the center of the torus for positive n and away from the center …

Conjugacy Classes Of The Piecewise Linear Group, Matthew L. Housley

Theses and Dissertations

The piecewise linear group is the set of all piecewise linear orientation preserving homeomorphisms from the interval to itself under the operation of composition. We present here a complete set of invariants to classify the conjugacy classes of this group. Our approach to this problem relies on the factorization of elements into elements having only a single breakpoint.

Topics On The Spectral Theory Of Automorphic Forms, Dustin David Belt

Theses and Dissertations

We study the analytic properties of the Eisenstein Series of $frac {1}{2}$-integral weight associated with the Hecke congruence subgroup $Gamma_0(4)$. Using these properties we obtain asymptotics for sums of certain Dirichlet $L$-series. We also obtain a formula reducing the study of Selberg's Eigenvalue Conjecture to the study of the nonvanishing of the Eisenstein Series $E(z,s)$ for Hecke congruence subgroups $Gamma_0(N)$ at $s=frac {1+i}{2}$.

Proven Cases Of A Generalization Of Serre's Conjecture, Jonathan H. Blackhurst

Theses and Dissertations

In the 1970's Serre conjectured a correspondence between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the first examples of proven cases of this generalized conjecture.

On The Combinatorics Of Certain Garside Semigroups, Christopher R. Cornwell

Theses and Dissertations

In his dissertation, F.A. Garside provided a solution to the word and conjugacy problems in the braid group on n-strands, using a particular element that he called the fundamental word. Others have since defined fundamental words in the generalized setting of Artin groups, and even more recently in Garside groups. We consider the problem of finding the number of representations of a power of the fundamental word in these settings. In the process, we find a Pascal-like identity that is satisfied in a certain class of Garside groups.

Erratum: The Emergence Of A Large-Scale Coherent Structure Under Small-Scale Random Bombardments (Communications On Pure And Applied Mathematics (2006) 59:4 (467-500)), Andrew Majda, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

Large Deviations For Stochastic Systems With Memory, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

We establish a large deviations principle for stochastic delay equations driven by small multiplicative white noise. Both upper and lower large deviations estimates are obtained.

Ricci Curvature Rigidity For Weakly Asymptotically Hyperbolic Manifolds, Vincent Bonini, Pengzi Miao, Jie Qing

Mathematics

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasilocal mass characterization of Euclidean balls from [9] [14] and eigenfunction compactification ideas from [12].

An In-Depth Investigation Of The Divine Ratio, Birch Fett

The Mathematics Enthusiast

The interesting thing about mathematical concepts is that we can trace their development or discoveries throughout history. Most cultures of the ancient world had some form of mathematics, and these basic skills developed into what we now call modern mathematics. The divine ratio is similar in that it was used in many different sections of history. The divine ratio, sometimes called the golden ratio or golden section, has been found in very diverse areas. The mathematical concepts of the golden ration have been found throughout nature, in architecture, music as well as in art. Phi is an astonishing number because …

The Effects Of Blended E-Learning On Mathematics And Computer Attitudes In Pre-Calculus Algebra, Balarabe Yushau

The Mathematics Enthusiast

This study examines the influence of blended e-learning on students' attitude towards mathematics and computers. A random sample of 70 students of the preparatory year program of King Fahd University of Petroleum & Minerals (KFUPM), Dhahran served as the sample of this study. Data were collected at the beginning (pre-program) and the end (post-program) of the semester using Aiken Mathematics Attitude Scale and Greessen and Loyd Computer Attitude Scale. The result indicates that the subjects have positive attitude towards mathematics and computer. However, analysis of variance shows no statistically significant change in students’ attitudes towards mathematics and computer except for …

World-Class High Quality Mathematics Education For All K-12 American Students, Om P. Ahuja

The Mathematics Enthusiast

In September 1989, the United States’ Governors Conference in Charlottesville, Virginia set an ambitious goal by declaring that “By the year 2000, United States students will be first in the world in mathematics and science achievements”. However, recent results of the ‘Programme for International Student Achievement’ and ‘Trends in International Mathematics and Science Study’ indicate that the United States students’ achievements in mathematics are far below world class standards. This paper seeks to discuss issues in an international context related to the goal of creating world-class high quality mathematics education for all K-12 American students. In particular, the author also …