Mathematics Commons^™

The Polynomial Chaos Method With Applications To Random Differential Equations, Juan Antonio Licea Salazar

Mathematics Dissertations - Archive

The role of randomness in mathematical models is of paramount importance, with emphasis placed upon the accuracy and reliability of predictions a rational approach is the use of differential equations with random parameters to describe natural phenomena. Well known methods such as Monte Carlo methods and the method of moments have been implemented to approximate the solutions to random differential equations in the last few decades. In this work, analytic solutions to a particular Riccati type dierential equation and discrete delay dierential equation with random coefficients are derived, also, due to its spectral rate of convergence and simplicity, the polynomial …

Numerical Studies For M-Matrix Algebraic Riccati Equations, Weichao Wang

Mathematics Dissertations - Archive

A new doubling algorithm - Alternating-Directional Doubling Algorithm (ADDA) - is developed for computing the unique minimal nonnegative solution of an M-Matrix Algebraic Riccati Equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms - SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393-412) and SDA-ss of Bini, Meini, and Poloni (Numer. Math., 116 (2010), pp. 553-578) for the same purpose. A deflation technique is then presented for an irreducible singular M-matrix Algebraic Riccati Equation (MARE). The technique improves the rateof convergence of a doubling …

Dns Study For The Origin Of The Chaos In Late Boundary Layer Transition Over A Flat Plate, Manoj Kumar Thapa

Mathematics Dissertations - Archive

This dissertation is devoted to the investigation of the origin and mechanism of chaos ( asymmetric or disorganized flow) in late boundary layer transition over a flat plate without pressure gradient. This kind of flow not only exists at the very late stage of transition but also is a crucial component for turbulence formation- still not well understood. According to existing theories, the formation of the chaos in late boundary layer transition was considered due to big background disturbances (noises) and use of non-periodic spanwise boundary condition during numerical simulations. It was further assumed that universal coherent structures (ring-like vortices) …

Toric Varieties And Cobordism, Andrew Wilfong

Theses and Dissertations--Mathematics

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950's, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension …

Regularity And Uniqueness Of Some Geometric Heat Flows And It's Applications, Tao Huang

Theses and Dissertations--Mathematics

This manuscript demonstrates the regularity and uniqueness of some geometric heat flows with critical nonlinearity.

First, under the assumption of smallness of renormalized energy, several issues of the regularity and uniqueness of heat flow of harmonic maps into a unit sphere or a compact Riemannian homogeneous manifold without boundary are established.

For a class of heat flow of harmonic maps to any compact Riemannian manifold without boundary, satisfying the Serrin's condition,

the regularity and uniqueness is also established.

As an application, the hydrodynamic flow of nematic liquid crystals in Serrin's class is proved to be regular and unique.

The natural …

Relative Perturbation Theory For Diagonally Dominant Matrices, Megan Dailey

Theses and Dissertations--Mathematics

Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well conditioned L factor. We then establish relative perturbation bounds for the inverse that are entrywise and independent of the condition number. This allows us to also present relative perturbation bounds for the linear system Ax=b that are independent of the condition number. Lastly, we continue the work of Ye to provide relative perturbation bounds …

When Cp(X) Is Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

Let M be a metrizable group. Let G be a dense subgroup of M^X . If G is domain representable, then G = MX . The following corollaries answer open questions. If X is completely regular and C_p(X) is domain representable, then X is discrete. If X is zero-dimensional, T₂ , and C_p(X;D) is subcompact, then X is discrete.

An Implicit Interface Boundary Integral Method For Poisson’S Equation On Arbitrary Domains, Catherine Kublik, Nicolay M. Tanushev, Richard Tsai

Mathematics Faculty Publications

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions.

An Example Employing Convexity In Functional Fixed Point Arguments, Richard I. Avery, Paul W. Eloe, Johnny Henderson

Mathematics Faculty Publications

This paper illustrates a convexity technique to be employed with functional fixed point theorems in proving the existence of solutions to boundary value problems.

Hjb Equation And Statistical Arbitrage Applied To High Frequency Trading, Yonggi Park

Electronic Theses and Dissertations

In this thesis we investigate some properties of market making and statistical arbitrage applied to High Frequency Trading (HFT). Using the Hamilton-Jacobi-Bellman(HJB) model developed by Guilbaud, Fabien and Pham, Huyen in 2012, we studied how market making works to obtain optimal strategy during limit order and market order. Also we develop the best investment strategy through Moving Average, Exponential Moving Average, Relative Strength Index, Sharpe Ratio.

Invisibility: A Mathematical Perspective, Austin G. Gomez

CMC Senior Theses

The concept of rendering an object invisible, once considered unfathomable, can now be deemed achievable using artificial metamaterials. The ability for these advanced structures to refract waves in the negative direction has sparked creativity for future applications. Manipulating electromagnetic waves of all frequencies around an object requires precise and unique parameters, which are calculated from various mathemat- ical laws and equations. We explore the possible interpretations of these parameters and how they are implemented towards the construction of a suitable metamaterial. If carried out correctly, the wave will exit the metamaterial exhibiting the same behavior as when it had entered. …

Nonnegativity Of Exact And Numerical Solutions Of Some Chemotactic Models, Patrick De Leenheer, Jay Gopalakrishnan, Erica Zuhr

Mathematics and Statistics Faculty Publications and Presentations

We investigate nonnegativity of exact and numerical solutions to a generalized Keller–Segel model. This model includes the so-called “minimal” Keller–Segel model, but can cover more general chemistry. We use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative. We then derive numerical methods, using finite element techniques, for the generalized Keller–Segel model. Adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting, we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions. One of the numerical methods is …

On The Persistence Properties Of The Cross-Coupled Camassa-Holm System, David Henry, Darryl Holm, Rossen Ivanov

Articles

In this paper we examine the evolution of solutions, that initially have compact support, of a recently-derived system of cross-coupled Camassa-Holm equations. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.

Particle Trajectories In Extreme Stokes Waves Over Inifinte Depth, Tony Lyons

Articles

We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of the particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid being zero there.

Mathematical Modelling At Secondary School: The Macsi-Clongowes Wood College Experience, J.P.F. Charpin, S. O'Hara, Dana Mackey

Articles

In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students. Participants developed simple mathematical models for everyday life problems with an open-ended answer. The format and content of the workshop are described and feedback from both students and participating teachers is provided. For nearly all participants, this workshop was an enjoyable experience which showed mathematics and other STEM components in a very positive way.

Weyl-Titchmarsh Type Formula For Periodic Schrodinger Operator With Wigner-Von Neumann Potential, Sergey Simonov, Pavel Kurasov

Articles

Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for this operator is proven.

Mass Transfer From A Vertical Flat Plate Due To Natural Convection With A Constant Counterflow, David Mcdonnell, Brendan Redmond, Lawrence J. Crane

Articles

This paper first examines the mass transfer from a vertical flat surface of a soluble material due to natural convection. A perturbation term is then introduced into the stream function to model the introduction of a constant counterflow. The effect this counterflow has on both the overall mass transfer and the overall velocity profile is studied in detail.

Bayesian Model Selection For Exponential Random Graph Models, Alberto Caimo, Nial Friel

Articles

Exponential random graph models are a class of widely used exponential family models for social networks. The topological structure of an observed network is modelled by the relative prevalence of a set of local sub-graph configurations termed network statistics. One of the key tasks in the application of these models is which network statistics to include in the model. This can be thought of as statistical model selection problem. This is a very challenging problem---the posterior distribution for each model is often termed ``doubly intractable'' since computation of the likelihood is rarely available, but also, the evidence of the posterior …

Endomorphisms Of Abelian Groups With Small Algebraic Entropy, D. Dikranjan, Ketao Gong, P. Zanardo

Articles

We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means ). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group , where an automorphism ϕ with must have all eigenvalues in the open circle of radius 2, centered at 0 and ϕ must leave invariant a lattice in , i.e., be essentially an automorphism of . In particular, all eigenvalues of an automorphism ϕ with must be roots of unity. This is a particular case of a more general fact known …

On Characterizations Of Exponential Distribution Through Order Statistics And Record Values With Random Shifts, M. Ahsanullah, Imtiyaz A. Shah, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

Distributional relations of the form Y d= X +T where X, Y, and T are record values or order statistics and the random translator T is independent from X are considered. Characterizations of the exponential distribution when the ordered random variables are non-neighboring are proved. Corollaries for Pareto and power function distributions are also derived.