2024-03-28T08:47:09Zhttps://thescholarship.ecu.edu/oai/requestoai:TheScholarship.intra.ecu.edu:10342/47032021-03-03T20:56:51Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
AN EXPOSITION OF THE RIEMANN ZETA FUNCTION
Molokach, John
Fucci, Guglielmo
This thesis is an exposition of the Riemann zeta function. Included are techniques of  analytic continuation and relationships to special functions. Some generalizations of  the Riemann zeta function are outlined, as well as the calculation of zeta constants  and the development of some identities. Additionally, one of the great unsolved  problems of mathematics, the Riemann hypothesis, is discussed. Â
East Carolina University
2014
Master's Thesis
http://hdl.handle.net/10342/4703
https://thescholarship.ecu.edu/bitstream/10342/4703/1/Molokach_ecu_0600O_11345.pdf
a43306810a865b7558cc6603d3b77c9a
https://thescholarship.ecu.edu/bitstream/10342/4703/2/Molokach_NEDL.pdf
215afd74386ea4af0e77503f52736065
https://thescholarship.ecu.edu/bitstream/10342/4703/3/Molokach_ecu_0600O_11345.pdf.txt
1c17969ae7a10fbab53b984ef2c7a58e
Mathematics
Complex analysis
Number theory
Zeta function
Series, Infinite
Analytic continuation
Riemann hypothesis
oai:TheScholarship.intra.ecu.edu:10342/44052022-12-09T15:26:37Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Economic Design of CUSUM Control Charts
Croskery, Thomas
Carolan, Christopher (Christopher A.)
In statistical process control, control charts are one tool for monitoring the control status of a process. One such type of chart is the cumulative sum (CUSUM) chart which has advantages over other styles of control chart. A study of the economic design of CUSUM control charts is undertaken via a comparative study of long-run hourly cost (LRHC) and a computational search algorithm is used to minimize LRHC for a CUSUM chart using nine parameters confined to their respective feasible parameter spaces as defined by the chart designer. Savings over similarly designed two stage Xbar charts are discovered and presented. Â
East Carolina University
2014
Master's Thesis
http://hdl.handle.net/10342/4405
https://thescholarship.ecu.edu/bitstream/10342/4405/3/Croskery.pdf
0d88fa93a3520fea97ac53e0212e4655
https://thescholarship.ecu.edu/bitstream/10342/4405/1/Croskery_ecu_0600O_11165.pdf
302e00234539135983e13adadfb1233a
https://thescholarship.ecu.edu/bitstream/10342/4405/2/Croskery_ecu_0600O_11165.pdf.txt
15a343ce3c531fdd403d46160152f5b4
Mathematics
Statistics
Control charts
CUSUM
Economic design
Long run hourly cost
Markov model
Search algorithm
Mathematical statistics--Research
Process control--Statistical methods--Research
CUSUM technique
oai:TheScholarship.intra.ecu.edu:10342/17632021-03-03T20:52:59Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Improved Tsunami Modeling Via q-Advanced Special Functions
Olivo, James M.
Spurr, Michael J.
This thesis studies q-advanced functions that are used as forcing terms in the forced wave equation and the Korteweg-de Vries equation in modeling tsunamis. The model improves existing tsunami models and is compared to data collected from the 2011 Japanese tsunami. The main results show that one can modify existing q-advanced functions to obtain more accuracy in the forcing of a tsunami, and, in turn, gain a more accurate model of a tsunami propagation and its approach to land. Â
East Carolina University
2013
Master's Thesis
http://hdl.handle.net/10342/1763
https://thescholarship.ecu.edu/bitstream/10342/1763/1/OlivoJr_ecu_0600M_10913.pdf
bf5f569d625e8623d03a4bc2680c76e2
https://thescholarship.ecu.edu/bitstream/10342/1763/2/OlivoJr_ecu_0600M_10913.pdf.txt
dc2afa957355120b7a03dcf5b4c87f02
Mathematics
Physics
Geology
Forced wave equation
KdV
Multiplicatively advanced differentiable equations
Q-advanced models
Tsunamis
Wavelet
Tsunamis--Mathematical models
oai:TheScholarship.intra.ecu.edu:10342/40742021-03-03T20:54:28Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Fourier Analysis on SU(2)
Leaser, Tyler
Benson, Chal
The set SU(2) of 2x2 unitary matrices with determinant one forms a compact non-abelian Lie group diffeomorphic to the three dimensional sphere. This thesis surveys general theory concerning analysis on compact Lie groups and applies this in the setting of SU(2). Our principal reference is J. Faraut's book {\em Analysis on Lie Groups}. Fundamental results in representation theory with compact Lie groups include the Peter-Weyl Theorem, Plancherel Theorem and a criterion for uniform convergence of Fourier series.  On SU(2) we give explicit constructions for Haar measure and all irreducible unitary representations. For purposes of motivation and comparison we also consider analysis on U(1), the unit circle in the complex plane. In this context, the general theory specializes to yield classical results on Fourier series with periodic functions and the heat equation in one dimension. We discuss convergence behavior of Fourier series on SU(2) and show that Cauchy problem for the heat equation with continuous boundary data admits a unique solution. Â
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/4074
https://thescholarship.ecu.edu/bitstream/10342/4074/3/Leaser_ecu_0600M_10852.pdf.txt
c378073550842e6efcadaea35f627c90
https://thescholarship.ecu.edu/bitstream/10342/4074/1/Leaser_ecu_0600M_10852.pdf
299205b063b48511d0db63f791abfd7e
https://thescholarship.ecu.edu/bitstream/10342/4074/2/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
Mathematics
Analysis on SU(2)
Heat equation
Lie algebra
Fourier analysis
Lie groups
oai:TheScholarship.intra.ecu.edu:10342/128552023-06-05T13:54:47Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
First Order Definition of Rings Using Group of Units
Kardos, Michael
We discuss the technical background and relevant research regarding the undecidability of $O_{\mathbb{Q}^{\text{ab}}}$. Given an algebraic extension $K/\mathbb{Q}$, we consider the subring defined by
$$R_K=\{x\in O_K\,|\,\forall \varepsilon\in U_K\setminus\{1\}\,\exists\delta\in U_K:\delta-1\equiv x(\varepsilon-1)\bmod(\varepsilon-1)^2\}.$$
We later consider a similar construction over subrings of $\mathbb{Q}$ of characteristic $0$. In doing this, we hope to gain insight into the result of the construction of $R_K$ when $K=\mathbb{Q}^{\text{ab}}$.
East Carolina University
2023-05-03
Master's Thesis
en
http://hdl.handle.net/10342/12855
https://thescholarship.ecu.edu/bitstream/10342/12855/1/KARDOS-MASTERSTHESIS-2023.pdf
a1c92500e49fbe8bf8759cd0bc8dddee
https://thescholarship.ecu.edu/bitstream/10342/12855/2/LICENSE.txt
26585afc479a47ab6219fbecdb05c5cd
https://thescholarship.ecu.edu/bitstream/10342/12855/3/PROQUEST_LICENSE.txt
b17b4b893007d42bdc4ce87285dd94f3
https://thescholarship.ecu.edu/bitstream/10342/12855/4/Blank%20Thesis%20Signature%20Page%20example.docx.pdf
8e6c61bf6c85680dbd36d9418d80debe
https://thescholarship.ecu.edu/bitstream/10342/12855/5/Updated%20NEDL.docx.pdf
d02d64a22e82b6cf4b99536ec379b0f2
number theory
logic
subring
abelian
extension
unit
definability
oai:TheScholarship.intra.ecu.edu:10342/51462021-03-03T21:00:26Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Modeling Tsunami Waves Using Q-Advanced Waves in 2-D
Cook, Cameron
Spurr, Michael J.
A two-dimensional numerical approximation for modeling tsunamis is developed. New q-advanced functions are used to model the forcing due to an earthquake. These results are used to model the Japanese tsunami of 2011, and compared to NOAA data generated by this earthquake and obtained on DART buoys at Wake Island.
East Carolina University
2015-12-15
Master's Thesis
en
http://hdl.handle.net/10342/5146
https://thescholarship.ecu.edu/bitstream/10342/5146/1/COOK-MASTERSTHESIS-2015.pdf
d56b4950298a7a50d76c14999a0877e1
https://thescholarship.ecu.edu/bitstream/10342/5146/2/LICENSE.txt
dc0a4bd85365b2241f10fb1693ed89fd
https://thescholarship.ecu.edu/bitstream/10342/5146/3/PROQUEST_LICENSE.txt
b5f1c0ad6289cc9e27e493ecea8664cc
https://thescholarship.ecu.edu/bitstream/10342/5146/4/signature%20page%20thesis.pdf
3698858abfb0d0c9f839bcd0ab7e42ca
https://thescholarship.ecu.edu/bitstream/10342/5146/5/DOC120215-12022015214721.pdf
4f528905e15f9472863b36fb13981f57
https://thescholarship.ecu.edu/bitstream/10342/5146/6/DOC120215-12022015220014.pdf
29d39edd41ebd926d0d789b91e73b861
https://thescholarship.ecu.edu/bitstream/10342/5146/8/COOK-MASTERSTHESIS-2015.pdf.txt
a69387a810b012bd8cb9610d18ac1c35
KP equation
Crank-Nicholson
Tsunamis--Mathematical models
Tohoku Earthquake and Tsunami, Japan, 2011
oai:TheScholarship.intra.ecu.edu:10342/38472021-03-03T20:54:42Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Diophantine Generation, Galois Theory, and Hilbert's Tenth Problem
Kennedy, Kendra
Shlapentokh, Alexandra
Hilbert's Tenth Problem was a question concerning existence of an algorithm to determine if there were integer solutions to arbitrary polynomial equations over the integers. Building on the work by Martin Davis, Hilary Putnam, and Julia Robinson, in 1970 Yuri Matiyasevich showed that such an algorithm does not exist. One can ask a similar question about polynomial equations with coefficients and solutions in the rings of algebraic integers. In this thesis, we survey some recent developments concerning this extension of Hilbert's Tenth Problem. In particular we discuss how properties of Diophantine generation and Galois Theory combined with recent results of Bjorn Poonen, Barry Mazur, and Karl Rubin show that the Shafarevich-Tate conjecture implies that there is a negative answer to the extension of Hilbert's Tenth Problem to the rings of integers of number fields. Â
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/3847
https://thescholarship.ecu.edu/bitstream/10342/3847/2/Kennedy_ecu_0600M_10632.pdf.txt
57fdc4135a822df5b35c1f0756c31851
https://thescholarship.ecu.edu/bitstream/10342/3847/1/Kennedy_ecu_0600M_10632.pdf
3b5298142000504a18a111b209f8c3a6
https://thescholarship.ecu.edu/bitstream/10342/3847/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
https://thescholarship.ecu.edu/bitstream/10342/3847/4/kennedy.pdf
0a67b26dcdf2220bf671125ba5a3259d
Mathematics
Diophantine undecidability
Diophantine equations
Hilbert's tenth problem
Galois theory
oai:TheScholarship.intra.ecu.edu:10342/40732021-03-03T20:54:21Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Irredundant and Mixed Ramsey Numbers
Clifton, Ann Wells
Hattingh, Johannes H.
The irredundant Ramsey number, s(m,n), is the smallest p such that in every two-coloring of the edges of K[subscript]p using colors red (R) and blue (B), either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element irredundant set. The mixed irredundant Ramsey number, t(m,n), is the smallest number p such that in every two-coloring of the edges of K[subscript]p using colors red (R) and blue (B), either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element independent set. This thesis provides all known results for irredundant and mixed Ramsey numbers. Â
East Carolina University
2013
Master's Thesis
http://hdl.handle.net/10342/4073
https://thescholarship.ecu.edu/bitstream/10342/4073/3/Clifton_ecu_0600M_10863.pdf.txt
5712bf9f3cb73070abe812a4e5df1058
https://thescholarship.ecu.edu/bitstream/10342/4073/1/Clifton_ecu_0600M_10863.pdf
d843f14640ce00fef1543a22f5d41ba7
https://thescholarship.ecu.edu/bitstream/10342/4073/2/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
Mathematics
Irredundant
Mixed
Ramsey numbers
oai:TheScholarship.intra.ecu.edu:10342/27982022-12-09T15:26:44Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
ESTIMATION OF THE PROBABILITY A BROWNIAN BRIDGE CROSSES A CONCAVE BOUNDARY
Yang, Fan
Carolan, Christopher (Christopher A.)
This thesis studies a new method to estimate the probability that a Brownian bridge crosses a concave boundary. We show that a Brownian bridge crosses a concave boundary if and only if its least concave majorant crosses said concave boundary. As such, we can equivalently simulate the least concave majorant of a Brownian bridge in order to estimate the probability that a Brownian bridge crosses a concave boundary.  We apply these theoretical results to the problem of estimating joint confidence intervals for a true CDF at every point. We compare this method to a traditional method for estimating joint confidence intervals for the true CDF at every point which is based upon the limiting distribution of what is often called the Kolmogorov-Smirnov distance, the sup-norm distance between the empirical and true CDFs. We indicate the disadvantages of the traditional approach and demonstrate how our approach addresses these weaknesses. Â
East Carolina University
2010
Master's Thesis
en_US
http://hdl.handle.net/10342/2798
https://thescholarship.ecu.edu/bitstream/10342/2798/3/YANG_ecu_0600M_10127.pdf.txt
7ce44415c5b8238a6bdedb210a75acd2
https://thescholarship.ecu.edu/bitstream/10342/2798/2/embargo20120415.txt
bc949ea893a9384070c31f083ccefd26
https://thescholarship.ecu.edu/bitstream/10342/2798/1/YANG_ecu_0600M_10127.pdf
2e6d47b3adac575af9f10c39d2c630c4
Statistics
Gaussian processes
Brownian bridges (Mathematics)
oai:TheScholarship.intra.ecu.edu:10342/38452021-03-03T20:54:18Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Idempotents in Cyclic Codes
Brame, Benjamin
Robinson, Zachary
Cyclic codes give us the most probable method by which we may detect and correct data transmission errors. These codes depend on the development of advanced mathematical concepts. It is shown that cyclic codes, when viewed as vector subspaces of a vector space of some dimension n over some finite field F, can be approached as polynomials in a ring. This approach is made possible by the assumption that the set of codewords is invariant under cyclic shifts, which are linear transformations. Developing these codes seems to be equivalent to factoring the polynomial x[superscript]n-x over F. Each factor then gives us a cyclic code of some dimension k over F. Constructing factorizations of x[superscript]n-x is accomplished by using cyclotomic polynomials and idempotents of the code algebra. The use of these two concepts together allows us to find cyclic codes in F[superscript]n. Hence, the development of cyclic codes is a journey from codewords and codes to fields and rings and back to codes and codewords.
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/3845
https://thescholarship.ecu.edu/bitstream/10342/3845/2/Brame_ecu_0600M_10674.pdf.txt
4b4b55cb4d89e338cacd47a7c3341537
https://thescholarship.ecu.edu/bitstream/10342/3845/1/Brame_ecu_0600M_10674.pdf
26b48976b7addddb1907efb22f6c4f59
https://thescholarship.ecu.edu/bitstream/10342/3845/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
https://thescholarship.ecu.edu/bitstream/10342/3845/4/Brame.pdf
690c85c4f6c7071fa66aea7a0d4afa34
Mathematics
Codes
Coding
Cyclic
Cyclotomic
Idempotents
Coding theory
Error-correcting codes (Information theory)
oai:TheScholarship.intra.ecu.edu:10342/27972021-03-03T20:54:19Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
A PRIMER FOR THE FOUNDATIONS OF ALGEBRAIC GEOMETRY
Hampton, Earl F.
Ravi, M. S.
The purpose of this thesis is to define the basic objects of study in algebraic geometry, namely, schemes and quasicoherent sheaves over schemes. We start by discussing algebraic sets as common zeros of polynomials and prove Hilbert's Nullstellensatz to establish a correspondence between algebraic sets and ideals in a polynomial ring. We then discuss just enough category theory to define a sheaf as a contravariant functor and then introduce ringed spaces, the spectrum of a ring, and the definition of affine schemes. We then discuss sheaves of modules over schemes. We then define projective varieties as ringed spaces. We end by proving Hilbert's syzygy theorem that can be used to study the equations defining projective varieties. Â
East Carolina University
2010
Master's Thesis
en_US
http://hdl.handle.net/10342/2797
https://thescholarship.ecu.edu/bitstream/10342/2797/2/HamptonIII_ecu_0600M_10175.pdf.txt
a24e9de0b6cb156b20d968af0e064ba7
https://thescholarship.ecu.edu/bitstream/10342/2797/1/HamptonIII_ecu_0600M_10175.pdf
9ad7abd0a9dbc11c5af493b6ab7b74e3
https://thescholarship.ecu.edu/bitstream/10342/2797/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
Mathematics
Geometry, Algebraic
Schemes (Algebraic geometry)
Categories (Mathematics)
Sheaf theory
oai:TheScholarship.intra.ecu.edu:10342/128532023-06-05T13:54:30Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives
Crumpler, Charles Wingate
In this paper, we will be concerned with developing an inferential structure over the field with four elements in characteristic $2$. We begin by discussing the historical context in which this research occurs. In subsequent sections, we will construct the field, called $\F_4$ and describing the algebraic structure over $\F_4$. We then define the connectives $\wedge$, $\vee$, and $\neg$ over $\F_4$ by extending their standard definition over $\F_2$. We define the basic syntax and semantics of $\F_4$. We show that $\wedge$ and $\vee$ are dual over $\F_4$ with respect to $\neg$ and that $\F_4$ is functionally complete over $\{ \ \wedge, \vee, \neg \ \} \cup \F_4$. We develop a notion of inferences over $\F_4$ by imbuing it with a partial order, defining validity and the material implication, and defining a proof. Upon completing this, we prove the Deduction, Soundness, and Completeness Theorems, thereby showing that inferences over $\F_4$ behaves in ways comparable to, but not equivalent to, those over a field of two values in characteristic $2$.
East Carolina University
2023-05-03
Master's Thesis
en
http://hdl.handle.net/10342/12853
https://thescholarship.ecu.edu/bitstream/10342/12853/1/CRUMPLER-MASTERSTHESIS-2023.pdf
c20a575dd4824957b8daabd5d187c17e
https://thescholarship.ecu.edu/bitstream/10342/12853/2/LICENSE.txt
0fbb1b5beb0e083c065dd371e0acf410
https://thescholarship.ecu.edu/bitstream/10342/12853/3/PROQUEST_LICENSE.txt
8f6cb37cb03496fd8e121bc7afd05bdc
https://thescholarship.ecu.edu/bitstream/10342/12853/4/Blank%20Thesis%20Signature%20Page%20example.docx.pdf
ab2395a9cbbc4d9012048342366e594f
https://thescholarship.ecu.edu/bitstream/10342/12853/5/Updated%20NEDL.docx.pdf
c84abe04f22e8c6a76152cfac9ebb9e8
logic, many-valued logic, soundness, completeness
oai:TheScholarship.intra.ecu.edu:10342/17622021-03-03T20:55:43Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Financial Market Analysis Using a Kinetics Model
Brown, Frank R., Jr.
Pravica, David W.
Over the past several decades physicists have used   models and techniques that were developed in the  sciences in order to analyze the price and volume behavior of financial markets.   These models and techniques include  the application of nonextensive thermodynamic statistics, information entropy, and  detrended fluctuation analysis.   This thesis extends the aforementioned approaches to   include the use of chemical kinetics concepts.  We create two-state models of stock market trading records -   models where the stock is treated as being in either an increasing (I) state  or a decreasing (D) state.  We then treat the transition from one state to the other   using standard reaction kinetic methodologies.  We supplement the kinetic analysis with analysis of the autocorrelation function.  We apply this approach to both closing prices and to trading volumes.  In both the closing price and the volume models,   we find that that the processes are not strictly Markovian   but instead exhibit some perturbation due to memory effects.  The closing price model shows evidence of momentum effects in stock pricing  while the volume model captures autocorrelations centered around  the quarterly earnings report cycle. Â
East Carolina University
2013
Master's Thesis
http://hdl.handle.net/10342/1762
https://thescholarship.ecu.edu/bitstream/10342/1762/1/BrownJr_ecu_0600M_10895.pdf
3973fb41f0b4434e88eba1fa34475580
https://thescholarship.ecu.edu/bitstream/10342/1762/2/BrownJr_ecu_0600M_10895.pdf.txt
ce2adb900f338c39ca4b881f85b1a0c8
Applied mathematics
Finance
Econophysics
Financial markets
Kinetics
Stochastic processes
Weibull
Mathematical analysis
Capital market
Chemical kinetics
Stock exchanges
oai:TheScholarship.intra.ecu.edu:10342/38442021-03-03T20:54:35Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
The Effect of the Mathematics of Finance on the Dynamics of a Credit Economy
Bennett, Jessica J.
Pravica, David W.
The general equilibrium theory of J.M. Keynes was developed in the 1930's to help explain the great depression and prevent future economic downturns.   Out of this came the IS-LM (investment saving/liquid money) model, introduced by J.R. Hicks in 1936.   There is controversy about the success of Hicks's approach,   not the least of which is the lack of dynamical aspects in the theory.   The thesis considers three interest groups identified as bankers, capitalists and workers.   A coupled system of differential equations describes the flow of money, capital and credit over time.   A mathematical analysis of the high dimensional system reveals the existence of equilibrium points.   Their stability properties are determined.  By modifying the equations,   cycles appear corresponding to periods of boom and bust.   Thus, shocks to the system, which are theorized by neoclassical economists to be due to external events, are shown to be possible using dynamical endogenous equations. Â
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/3844
https://thescholarship.ecu.edu/bitstream/10342/3844/2/Bennett_ecu_0600M_10646.pdf.txt
197ac5f3df20e2e4875f3589e4a4f163
https://thescholarship.ecu.edu/bitstream/10342/3844/1/Bennett_ecu_0600M_10646.pdf
c831cc70b9ba03c9962476e31bce0462
https://thescholarship.ecu.edu/bitstream/10342/3844/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
https://thescholarship.ecu.edu/bitstream/10342/3844/4/bennett.pdf
acb7d8ed24434672f12545db4db9f416
Mathematics
Economics
Finance
Credit economy
Delay differential equations
Dynamics
Eigenvalues
Economics--Mathematical models
Keynesian economics
IS-LM model (Macroeconomics)
oai:TheScholarship.intra.ecu.edu:10342/36032021-03-03T20:54:34Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Random Walks on Finite Fields and Heisenberg Groups
Zhu, Yi
Benson, Chal
Let H be a finite group and [mu] a probability measure on H. This data determines an invariant random walk on H beginning from the identity element. The probability distribution for the state of the random walk after n steps is given by the n'th convolution power of the probability measure [mu]. The random walk and measure [mu] are said to be ergodic if the support of this distribution is the entire group for n sufficiently large. In this case a specialization of the Markov Ergodic Theorem ensures that the distribution after n steps converges point-wise to the uniform distribution. One employs the total variation distance on probability measures to analyze the rate of convergence to equilibrium. Suppose now that a finite group K acts on H by automorphisms. We say that the action pair K : H is ergodic when the K-invariant probability measure [mu] supported on some K-orbit is ergodic. We call, moreover, K : H a Gelfand action pair when the convolution algebra of K-invariant functions on H is commutative. Specializing the theory of spherical functions to the context of Gelfand action pairs we obtain a version of the Diaconis-Shahshahani Upper Bound Lemma, controlling the total variation distance to equilibrium for the random walk determined by [mu].  The main results in this thesis concern invariant random walks on finite fields and three dimensional Heisenberg groups over finite fields. Let F be a finite field of odd characteristic and K a subgroup of the multiplicative group for F with even order. We obtain a necessary and sufficient condition for ergodicity of the action pair K : F and an explicit summation formula for the upper bound on total variation distance to equilibrium guaranteed by the Upper Bound Lemma. Let F[~] be a quadratic extension field for F and U denote the kernel of the norm mapping from F[~] to F. An application of our field theoretic criterion for ergodicity shows that U : F[~] is an ergodic action pair and we specialize our upper bound result to this context. Forming the three dimensional Heisenberg group H = F[~] x F over F the action of U on F[~] induces an action of U on H by automorphisms. Benson and Ratcliff have shown that U : H is a Gelfand action pair and determined the associated spherical functions. We prove that the pair U : H is ergodic and make explicit the bound given by the Upper Bound Lemma. Â
East Carolina University
2011
Master's Thesis
http://hdl.handle.net/10342/3603
https://thescholarship.ecu.edu/bitstream/10342/3603/2/Zhu_ecu_0600M_10410.pdf.txt
4be939c662e3e015163758ea58276338
https://thescholarship.ecu.edu/bitstream/10342/3603/1/Zhu_ecu_0600M_10410.pdf
c07a0d367b77c7dcce19936f5e785a7f
https://thescholarship.ecu.edu/bitstream/10342/3603/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
Mathematics
Gelfand pairs
Heisenberg group
Random walks (Mathematics)
Spherical functions
Finite fields (Algebra)
Invariants
oai:TheScholarship.intra.ecu.edu:10342/49542021-03-03T20:56:38Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Eigenvalues for Sums of Hermitian Matrices
Taylor, James M.
Benson, Chal
In this thesis we explore how the eigenvalues of nxn Hermitian matrices A,B relate to the eigenvalues of their sum C=A+B. We mainly focus on inequalities bounding sums of r eigenvalues for C by sums of r eigenvalues for A with r eigenvalues for B, for some r less than n.     We begin by using linear algebra to establish some classical results, including a result by R.C. Thompson that allows us to reformulate the eigenvalue problem in terms of nonempty intersections in the Grassmannian manifold of r-planes in complex n-dimensional space. In particular, every nonempty triple intersection of Schubert varieties in a Grassmannian yields an eigenvalue inequality. Such nonempty intersections correspond to nonzero cup products in the cohomology ring of the Grassmannian, and consequently, to nonzero Littlewood-Richardson coefficients. The Littlewood-Richardson rules provide us with an efficient method of detecting when these coefficients are nonzero, and hence of finding eigenvalue inequalities which necessarily hold for all nxn Hermitian matrices A,B,C=A+B.    For the remainder of this thesis, we turn our attention to particular inequalities of the above form that Alfred Horn conjectured would completely determine the possible eigenvalues of A,B,C=A+B. Horn's conjecture, formulated in 1962, was resolved in the affirmative during the late 1990's in celebrated work of A. Knutson and T. Tao, building on results of A. Klyachko and others. We will develop the connection between Horn's inequalities and the earlier parts of this thesis. In particular, we will see that each Horn inequality corresponds to a nonzero cup product that lies in the top degree cohomology group of the Grassmannian.     An alternate formulation of Horn's Theorem shows that indices yield a Horn inequality if and only if certain associated partitions occur as the eigenvalues for some rxr Hermitian matrices A, B, C=A+B. We will prove that when r=n-2 there are necessarily diagonal rxr matrices satisfying this condition. Â
East Carolina University
2015
Master's Thesis
http://hdl.handle.net/10342/4954
https://thescholarship.ecu.edu/bitstream/10342/4954/1/Taylor_ecu_0600O_11437.pdf
f0c87f1c4554907e64eab61c151a9002
https://thescholarship.ecu.edu/bitstream/10342/4954/2/TaylorNEDL.pdf
13da88a169c20ee898392a4ef983ee9f
https://thescholarship.ecu.edu/bitstream/10342/4954/3/Taylor_ecu_0600O_11437.pdf.txt
4b1ea20b629c472f3b1d4848d4afbebe
Mathematics
Hermitian matrices
Horn inequalities
Matrices
Symmetric matrices
Eigenvalues
oai:TheScholarship.intra.ecu.edu:10342/38462021-03-03T20:54:20Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Mathematical Analysis of Tsunami and Rogue Waves
Eidschun, Bradley
Pravica, David W.
In this thesis both forced and non-linear wave equations will be studied.   Actual data from tsunami and rogue waves will be used and a signal analysis will be performed using wavelets. Main results show that a different choice of wavelet leads to different efficiencies occurring in the signal recovery process. Â
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/3846
https://thescholarship.ecu.edu/bitstream/10342/3846/2/Eidschun_ecu_0600M_10675.pdf.txt
8b15e2a29506d177e94e1733fc00e0f8
https://thescholarship.ecu.edu/bitstream/10342/3846/1/Eidschun_ecu_0600M_10675.pdf
da5f6d38288841b5fbb9bbeaa76c4e27
https://thescholarship.ecu.edu/bitstream/10342/3846/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
https://thescholarship.ecu.edu/bitstream/10342/3846/4/Eidschun.pdf
bd7ff4c7b46cd9302be5a7d0965572e6
Mathematics
Wave equation
Nonlinear wave equations
Mathematical analysis
Tsunamis
Rogue waves
oai:TheScholarship.intra.ecu.edu:10342/53442021-03-03T21:02:49Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
The Spectral Theorem for Self-Adjoint Operators
Chilcoat, Kenneth
Ratcliff, Gail Dawn Loraine
The Spectral Theorem for Self-Adjoint Operators allows one to define what it means to evaluate a function on the operator for a large class of functions defined on the spectrum of the operator. This is done by developing a functional calculus that extends the intuitive notion of evaluating a polynomial on an operator. The Spectral Theorem is fundamentally important to operator theory and has applications in many fields, especially harmonic analysis on locally compact abelian groups. This thesis represents a merging of two traditional treatments of the Spectral Theorem and includes an extended example highlighting an important application in harmonic analysis.
East Carolina University
2016-04-25
Master's Thesis
en
http://hdl.handle.net/10342/5344
https://thescholarship.ecu.edu/bitstream/10342/5344/1/CHILCOAT-MASTERSTHESIS-2016.pdf
3b05630dd43312bb60d8860e0d8a1326
https://thescholarship.ecu.edu/bitstream/10342/5344/2/LICENSE.txt
1a1f7adc94d9e3bd77650fd2614a01b0
https://thescholarship.ecu.edu/bitstream/10342/5344/3/PROQUEST_LICENSE.txt
76840d449ea00bf44afb880f271c0021
https://thescholarship.ecu.edu/bitstream/10342/5344/4/Signature%20Page.pdf
4397398f54633856c3ccfeb15ae77c97
https://thescholarship.ecu.edu/bitstream/10342/5344/5/Distribution%20License%20and%20Options.pdf
7bd978169285a1d52a50f32311b221f5
https://thescholarship.ecu.edu/bitstream/10342/5344/7/CHILCOAT-MASTERSTHESIS-2016.pdf.txt
a45e60d5d0dda4d7c5c3c71fa66da8d5
Hilbert Space
Spectral theory (Mathematics)
Operator theory
Harmonic analysis
oai:TheScholarship.intra.ecu.edu:10342/44062021-03-03T20:55:47Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Mathematical Aspects of Image Processing
Kirk, Samantha
Ratcliff, Gail Dawn Loraine
In this thesis, image processing is explored from a mathematical point of view. After defining a digitized image, techniques for adjusting resolution are discussed. Image transformations defined on a neighborhood centered about a pixel and their relationships to convolution are considered. The Fourier transform and the discrete Fourier transform are introduced in both one and two dimensions. Properties of the Fourier transform are demonstrated with analysis of the power spectrum of an image. A degradation model is used to study image restoration, in the cases where distortion is due to noise and motion blur. Other approaches to image restoration employ the processes of inverse and Wiener filtering. Â
East Carolina University
2014
Master's Thesis
http://hdl.handle.net/10342/4406
https://thescholarship.ecu.edu/bitstream/10342/4406/3/Kirk.pdf
544859fdea02c5d2c0d04184d6335a1c
https://thescholarship.ecu.edu/bitstream/10342/4406/1/Kirk_ecu_0600O_11133.pdf
f361f90cfdd8909cffa09c192288c629
https://thescholarship.ecu.edu/bitstream/10342/4406/2/Kirk_ecu_0600O_11133.pdf.txt
eb5c6d5b07e79fa88ac8d354941271d8
Mathematics
Transform
Image processing--Mathematical models
Fourier transformations
oai:TheScholarship.intra.ecu.edu:10342/49532021-03-03T20:56:38Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Markov Chains, Random Walks, and Card Shuffling
Outlaw, Nolan
Ratcliff, Gail Dawn Loraine
A common question in the study of random processes pertains to card shuffling. Whether or not a deck of cards is random can have huge implications on any game being played with those particular cards. This thesis explores the question of randomness by using techniques established through analysis of Markov chains, random walks, computer simulations, and some basic shuffling models. Ultimately, the aim is to explore the cutoff phenomenon, which asserts that at some point during the shuffling process there is a sharp decline in the shuffled deck's distance from random. Â
East Carolina University
2016
Master's Thesis
http://hdl.handle.net/10342/4953
https://thescholarship.ecu.edu/bitstream/10342/4953/1/Outlaw_ecu_0600O_11494.pdf
434bc13ebaefe9500f199c859d83d887
https://thescholarship.ecu.edu/bitstream/10342/4953/2/OutlawNEDL.pdf
c8a7dfab070b9cc78a7b39965e6e1c05
https://thescholarship.ecu.edu/bitstream/10342/4953/3/Outlaw_ecu_0600O_11494.pdf.txt
ae7efe05fefaa81d79fac0d5b924a7fe
Mathematics
Markov chains
Markov processes
Random walks (Mathematics)
Card shuffling
oai:TheScholarship.intra.ecu.edu:10342/17642021-03-03T20:52:59Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Applications of Stochastic Processes to Cancer Research
Steely, Kristin Michelle
Pravica, David W.
The purpose of this thesis is to implement stochastic models that are currently used to analyze the impact of different drug treatments on cancer and to model drug resistance by cancer cells. Mathematical models are used to compare single-cancer treatment results with those that involve multiple drugs at the same time. Using various parameters for the model, the probability of treatment success was calculated. A comparison was made with a probability theory approach and good agreement was obtained. Â
East Carolina University
2013
Master's Thesis
http://hdl.handle.net/10342/1764
https://thescholarship.ecu.edu/bitstream/10342/1764/1/Steely_ecu_0600M_10928.pdf
d390dcd573fcaaaa67dfeea1289b438e
https://thescholarship.ecu.edu/bitstream/10342/1764/2/Steely_ecu_0600M_10928.pdf.txt
05e063fa81e61d94359aa39487d9a223
Mathematics
Applied mathematics
Medicine
Kolmogorov forward equation
Markov processes
Ordinary differential equations
Partial differential equations
Probability generating function
Stochastic processes
Stochastic analysis
Drug resistance in cancer cells
oai:TheScholarship.intra.ecu.edu:10342/38482021-03-03T20:54:21Zcom_10342_122com_10342_55com_10342_1col_10342_124col_10342_65
Newton Polygons on p-adic Number Fields
Teller, Jacek
Robinson, Zachary
This thesis offers a clear introduction to p-adic number fields, and the method of Newton polygons to approximate the size of roots of polynomials in the completion of the algebraic closure of p-adic number fields. Ostrowski's theorem is also proved herein. The thesis is intended to serve as an aperitif to further study in the area. Â
East Carolina University
2012
Master's Thesis
http://hdl.handle.net/10342/3848
https://thescholarship.ecu.edu/bitstream/10342/3848/2/Teller_ecu_0600M_10676.pdf.txt
688434942b7919dfbdd0c2ec05db9240
https://thescholarship.ecu.edu/bitstream/10342/3848/1/Teller_ecu_0600M_10676.pdf
b563ba24a1bbbdd66ed9994fc1356f23
https://thescholarship.ecu.edu/bitstream/10342/3848/3/license.txt
48d772d9ef478e9dd063ea202fa5e0d9
https://thescholarship.ecu.edu/bitstream/10342/3848/4/teller.pdf
fa5307be28943c592207014be18a1fa5
Mathematics
Computer science
Physics
Fields
Hensel's lemma
Newton polygons
Ostrowski's theorem
Roots
P-adic numbers
Newton diagrams
oai:TheScholarship.intra.ecu.edu:10342/68322022-12-09T15:26:30Zcom_10342_1com_10342_55col_10342_72col_10342_65
IDENTIFYING SUBJECTIVE VALUE IN WOMEN’S COLLEGE GOLF RECRUITING REGARDLESS OF SOCIO-ECONOMIC CLASS
Allred, Victoria
Carolan, Christopher (Christopher A.)
College athletics have grown into a major industry and athletic departments are pushing coaches to recruit the top talent. To recruit golfers, college coaches depend on multiple ranking systems. These systems are bias towards more expensive, national tournaments over inexpensive, state tournaments. In response, players who come from a lower socio-economic class will have fewer financial aid opportunities than someone from a higher economic class. Analyzing junior girls’ golf statistics, has led to the creation of an objective methodology to compare golfers without regard to socio-economic imbalances.
East Carolina University
2018-05-03
Honors Thesis
http://hdl.handle.net/10342/6832
https://thescholarship.ecu.edu/bitstream/10342/6832/1/ALLRED-HONORSTHESIS-2018.pdf
2267ea13afe4eac0456f935824e72181
https://thescholarship.ecu.edu/bitstream/10342/6832/2/LICENSE.txt
64b00975446fcc69c3daeaf501c5245b
https://thescholarship.ecu.edu/bitstream/10342/6832/4/ALLRED-HONORSTHESIS-2018.pdf.txt
b888b81b846a86198c9fecdabd50c5d9
sports economics
golf
golf statistics