University of Macau / Faculty of Science and Technology. / Department of Mathematics

University of Macau / Faculty of Science and Technology. / Department of Mathematics

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / Among quadratic domains some have unique factorization property and others don't. Gauss' conjecture that there are infinitely many real quadratic fields and only nine imaginary quadratic fields having unique factorization property, has not yet been proved. But there has been success in finding almost all complex quadratic domains having this property. The purpose of this thesis is to discuss unique factorization property of quadratic domains in general, with particular reference to complex quadratic domains. The basic theorem which is proved here is that a quadratic domain is a unique factorization domain if and only if it is a principal ideal domain. Euclidean domain is defined and some Euclidean and non-Euclidean domains are given for illustration. Being a Euclidean domain is only a sufficient but not a necessary condition for unique factorization. By making use of the definition of a Euclidean domain all complex quadratic Euclidean domains are found. Equivalent ideals, ideal classes and class number are also defined. The class number of a field is unity if and only if the domain is a principal ideal domain; i.e., if and only if it is a unique factorization domain. This fact is the key to our discussion here; the fact that the size of the class number gives a measure of how far is our domain from the factorization domain. To determine the class number of a given domain Minkowski's theorem is used. This theorem establishes the existence of an integral ideal in every class such that the norm of the ideal is always less than the absolute value of the square root of the discriminant of the field. Finally a necessary and sufficient condition for a complex qudratic field to have class number unity, is provided by the theorem on Euler's polynomial. In view of the findings of Lehmer, Heilbronn and Linfoot, it is concluded that there are nine complex quadratic fields of class number 1. If at all there is any more, there is only one more d less than zero for which Q (√d)has class number 1. / 2031-01-01

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2031-01-01

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to free-boundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013). / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / October 30, 2017. / backward stochastic differential equations, Hamilton-Jacobi-Bellman equation, Monte Carlo approximation, portfolio optimization, stochastic control, transaction costs / Includes bibliographical references. / Arash Fahim, Professor Directing Dissertation; Jen Atkins, University Representative; Giray Ökten, Committee Member; Lingjiong Zhu, Committee Member.

Biofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like self-produced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque removing process. Differential equations are used for modelling the microfiltration process and the optimal control method is applied to analyze the efficiency of the filtration. The multiphase fluid system is introduced to describe the dental plaque removing process and results are obtained by numerical schemes. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / October 27, 2017. / Includes bibliographical references. / Nick Cogan, Professor Directing Dissertation; Eric Chicken, University Representative; Kyle A. Gallivan, Committee Member; Monica K. Hurdal, Committee Member.

To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$-solver (Section~\ref{3F2 type solution}) and $F_1$-solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,|\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,|\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm~\ref{hom}) of two $D$-modules, where $D$ is a multivariate differential ring, and compute projective homomorphisms (Algorithm~\ref{algo ProjHom}) using the tensor product module and Algorithm~\ref{hom}. As an application, all irreducible order $2$ subsystems from reducible order $3$ systems turn out to come from Gauss hypergeometric function $_2F_1(a,b;c\,|\,x)$ (Chapter~\ref{chapter applications}). / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / November 16, 2017. / Includes bibliographical references. / Mark van Hoeij, Professor Directing Dissertation; Laura Reina, University Representative; Amod Agashe, Committee Member; Ettore Aldrovandi, Committee Member; Paolo Aluffi, Committee Member.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 33-34). / by Hi-joon Chae. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 133-142). / Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work such as multimaterial fiber drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem. The resulting mathematical model, based on linear-stability analysis, can quickly predict the breakup lengthscale and timescale of concentric cylindrical fluids, and provides useful guidance for material selections and design parameters in fiber-drawing experiments. A three-fluid system with competing breakup processes at very different length scales is demonstrated with a full Stokes flow simulation. In the second half of this thesis, we study large-scale PDE-constrained microcavity topology optimization. Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. While most microcavities are designed mostly by hand using some understanding of the physical principles of the confinement, we let the computer discover its own structures. We formulate and solve a full 3d optimization scheme over all possible 2d-lithography patterns in a thin dielectric film. The key to our formulation is a frequency-averaged local density of states (LDOS), where the frequency averaging corresponds to the desired bandwidth, evaluated by a novel technique: solving a single electromagnetic wave scattering problem at a complex frequency. / by Xiangdong Liang. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000. / Includes bibliographical references (p. 100-104). / by Jing-Rebecca Li. / Ph.D.