PhD Qualifying/Comprehensive Examinations
All PhD students must pass qualifying/comprehensive examinations as part of the requirements for a PhD degree. The examinations taken will depend on
Important Note: The Department is in the process of overhauling this portion of the PhD program. In the past, PhD students were required to take three comprehensive examinations over a 13-month period of time. The exams were given over three topics chosen by the students.
To keep everyone up to speed, this section will include information on both systems. The information for the comprehensive examinations will reflect the old system and the information on the qualifying examination portion will reflect where the graduate program is headed.
COMPREHENSIVE EXAMINATION TOPICS
REAL ANALYSIS
Metric Spaces. Open and closed sets. Convergence and limits, dense subsets and separability, completeness, compact metric spaces. Mappings between metric spaces, continuity and uniform continuity.
Topological Spaces. Basic definitions, axioms of countability and separation, convergence, continuous mappings and homeomorphisms, compactness. Real-valued functions on topological spaces, the Stone-Weierstrass and Arzela-Ascoli theorems.
Integration and Measure. Basic properties of measures on abstract spaces. Construction of Lebesgue measure on R, functions of bounded variation and construction of general measures on Rn. The Lebesgue integral and convergence theorems. Product measures and Fubini's theorem.
Differentiation and Integration. Differentiation of monotone functions and functions of bounded variation. Fundamental Theorem of Calculus, absolutely continuous functions, and the Lebesgue decomposition on R. Signed measures, Jordan, Hahn and Lebesgue de-composition theorems. Absolute continuity and singularity of measures, Radon-Nikodym theorem.
Lp Spaces, 1 <= p <= infinity. Hölder and Minkowski inequalities, analysis of LP spaces as normed linear spaces: completeness and representation of linear functionals. Comparison of modes of convergence.
Hilbert Spaces. Classical examples of Hilbert spaces, Schwarz and Bessel inequalities. Orthogonal subspaces and projections, orthonormal sets, representation of linear functionals.
Linear Functional Analysis.
Banach spaces and classical examples, linear transformations and operators. Hahn-Banach and open mapping theorems, uniform boundedness principle. Linear functionals and dual spaces, weak and weak* topologies, representation of linear functionals for classical Banach spaces.References
Real Analysis- Royden
Real Analysis Bruckner, Bruckner, Thomson
Real Analysis Folland
Real and Complex Analysis- Rudin
Real and Abstract Analysis- Hewitt and Stromberg
Related Courses
Math 6210 6220 Real Analysis
ALGEBRA
Linear algebra. Systems of equations, linear transformations, vector spaces, orthogonality.
Matrix Theory. Canonical forms, hermitian and symmetric matrices, vector and matrix norms, no-negative matrices, graphs associated with matrices, field of values, matrix equations, Kronecker and Hadamard products, matrix functions and multilinear functionals.
Groups theory. Subgroups, normal subgroups, cosets, factor groups, homomorphisms and related theorems, finitely generated abelian groups, free groups, group presentations, Sylow theorems.
Rings and Field theory. Ideals, quotient rings, polynomial rings, factorization, field extensions, Galois theory.
Category theory. Functors, natural transformations, equivalences of categories.
Module theory. Jordan-Holder theorem, Krull-Schmidt theorem, projective and injective modules, fi'ee modules, complexes and resolution, derived functors.
Commutative ring theory. Prime and maximal ideal spectrum, localization, Hilbert Basis theorem, Hilbert Nullstellensatz, Noetherian rings.
References
Introduction to Commutative Algebra - Atijah and MacDonald
Contemporary Abstract Algebra - Gallian
A First Course in Abstract Algebra - Fraleigh
Algebra - Hungerford
Topics in Algebra - Herstein
Basic Algebra I and I I- Jacobson
Related Courses
Math 5310-5320 Linear and Modern Algebra
Math 6310-6320 Algebra
TOPOLOGY
Point set topology. Metric spaces, convergence and limits. Topological spaces. Continuous mappings, homeomorphisms, topological properties. Quotient and adjunction spaces. Product topology. Compactness. Connectedness. Local compactness, local connectedness. Separation axioms. Manifolds. Metrization.
Homotopy theory. Path components. Fundamental group. Van Kampen theorem. Homotopy type and homotopy equivalence. Deformation. Topological classification of surfaces. Covering spaces. Covering transformations.
Homology theory. Singular homology. Homology of cells and spheres. Applications: Brouwer fixed point theorem, Invariance of domain, Jordan-Brouwer separation. CW complexes. Cellular homology. Homology with coefficients.
Cohomology theory. Singular cohomology. Homological algebra. Universal coefficient theorem for cohomology. Eilenberg-Steenrod axioms for cohomology. Cup product. Cohomology ring. Cohomology of projective spaces. Poineare duality for manifolds.
References
Algebraic topology: a first course -
Greenberg, HarperA Basic Course in Algebraic Topology - Massey
Topology, a First Course - Munkres
Homology theory: an introduction to algebraic topology - Vick
Related Courses
Math 5510 Introduction to Topology
Math 6510-6520 Topology
DIFFERENTIAL EQUATIONS
Initial Value Problems. Existence and uniqueness of solutions, Picard-Lindelöf, contraction mapping principle. Continuation of solutions. Continuous dependence on initial conditions and parameters. Gronwall inequality.
Linear Systems. Fundamental matrices. Nonhomogeneous equations. Constant coefficient systems, canonical forms and exponential matrices. Floquet theory for periodic systems. n-th order linear scalar equations.
Linear Boundary-Value Problems. Green's functions, self adjoint problems, Sturm separation and comparison theorems. Sturm-Liouville theory.
Stability Theory. Phase plane trajectories in two dimensions. Stability types, stable-unstable, uniform, asymptotic, uniform asymptotic and exponential stability, stability in the large. Autonomous and periodic systems, Lyapunov direct method.
Two Dimensional Systems.
Linearization. Asymptotic equivalence. Stability via Lyapunov indirect method. Poincare Bendixson theorem.First Order Partial Differential Equations. Transport equations, first order nonlinear PDEs, characteristics, introduction of conservation laws, shock and entropy conditions. Power series methods, Cauchy-Darboux-Kovalevskaya Theorem.
Second Order Partial Differential Equations. Laplace's equation, fundamental solutions, mean value formula, Green's function, maximum principle, properties of harmonic functions, energy methods. Heat equation, fundamental solution, mean value theorem, properties of solutions for the heat equation, including the maximum principle. Wave equations, solution by spherical means, Duhamel's principle, energy methods.
References
Ordinary Differential Equations- Miller and Michel
Ordinary Differential Equations - Hartman
Theory of Ordinary Differential Equations - Coddington and Levinson
Ordinary Differential Equations- Hale
Ordinary Differential Equations- Reid
Qualitative Theory of Ordinary Differential Equations Brauer and Nohel
Differential Equations, Dynamical Systems and Linear Algebra - Hirsch and Smale
Partial Differential Equations - L.C. Evans
Partial Differential Equations - M. Renardy and R. Rogers
Related Courses
Math 5410 Methods of Applied Mathematics
Math 5420 Partial Differential Equations
Math 6410-6420 Ordinary/Partial Differential Equations I
NUMERICAL ANALYSIS
Linear and Nonlinear Systems. Linear systems: Gaussian elimination, LU and Choleski decompositions. Iterative methods, refinement, Jacobi, Gauss-Seidel, SOR and conjugate gradient. QR and singular value decomposition. Direct sparse methods. Matrix norms and error analysis.
Nonlinear systems: One variable: bisection. Newton, secant, hybrid methods. Rates of convergence. Local convergence analysis. Systems: Newton and quasi-Newton methods and globally convergent modifications. Continuation methods. Fixed point iteration, including the contractive mapping theorem.
Interpolation and Approximation. Interpolation: Polynomial and piecewise polynomial interpolation. The general linear interpolation problem. Multivariate polynomial, spline and trigonometric interpolation, fast Fourier transform. Error analysis in the interpolation process. Approximation: Normed linear spaces and LP approximation (particularly p: 1,2 and infinity). Orthogonal polynomial systems. Hilbert space approximation theory, normal equations, Bessel's inequality. Approximation from convex and strictly convex subspaces. Error analysis. Existence and uniqueness of best approximations. Characterization theorems. Comparison of errors between "best" approximation and interpolatory approximation.
Numerical Calculus and Differential Equations. Integration and Differentiation. Newton-Cotes and Gaussian quadrature, extrapolation methods and adaptive quadrature. Error analysis. Ordinary differential equations: Initial value problems: implicit and explicit inethods, Runge-Kutta and multistep methods, step control, order and convergence. Stability theory (Zero, absolute, relative, A-stable). Boundary value problems: finite differences, collocation, shooting. Error analysis. Partial differential equations: Finite difference and finite element methods, order and convergence. Special techniques for each type of PDE, i.e., elliptic, parabolic, hyperbolic. Use of inaximum principles. Variational methods, including time-dependent problems. The Bramble-Hilbert lemma. Error analysis.
References
Elementary Numerical Analysis: An Algorithmic Approach- Conte and de Boor
Introduction to Numerical Analysis- Stoer and Bulksch
Matrix Computations - Golub and Van Loan
A First Course in Numerical Analysis - Ralston and Rabinowitz
Numerical Methods - Dahlquist and Bjork
Numerical Methods for Constrained Optimization and Nonlinear Equations - Dennis and Schnabel
Approximation Theory and Numerical Analysis - Watson
Computational Methods in Ordinary Differential Equations - Lambert
The Finite Difference Methods in Partial Differential Equations - Mitchell and Griffiths
The Finite Element Method in Partial Differential Equations Mitchell and Wait
Related Courses
Math 5610 5620 Numerical Analysis
Math 6610-6620 Numerical Analysis
PROBABILITY
Probability. Probability measures and probability spaces: Discrete and absolutely continuous measures, decompositions. Cumulative distributions and densities, Radon-Nikodym theorem. Characterizations, moment generating, probability generating and characteristic functions. Conditional probability and independence. Classical discrete and continuous univariate and multivariate distributions. Functions of random variables. Expectation. Types of convergence of random variables and their relationships. Law of large numbers, central limit theorem.
Stochastic Processes. Basic definitions, Kolomogorov existence theorem. Simple random walk, Markov chains, Poisson and stationary processes and Brownian motion.
References
A First Course in Probability - Ross
An Introduction to Probability Theory arid Its Applications, Vol. 1 & 2 - Feller
Probability and Measure- Billingsley
Stochastic Processes - Ross
Probability Theory: Independence, Interchangeability, Martingales - Chow and Teicher
Related Courses
Math 5710 Introduction to Probability
Math 6750-6760 Probability
MATHEMATICAL STATISTICS
Probability. Markov's, Chebychev's inequalities; convergence in probability and the Weak Law of Large Numbers: convergence with probability 1 and the Strong Law of Large Numbers: moment generating fimctions, the Central Limit Theorem, the Random Scatter Theorem; conditional probability, order statistics.
Estimation. Loss functions and risk; sufficiency, unbiasedness and UMVU estimation: maximum likelihood estimation in univariate and multivariate models, optimality theory for maximum likelihood estimators, asymptotic relative efficiency; Bayesian inference, admissibility and minimaxity.
Confidence Intervals and Tests. MP and UMP tests; the Neyman-Pearson Lemma; the likelihood ratio principle: robustness; parametric and non-parametric confidence procedures.
References
Theory of Point Estimation - E.L. Lehmann.
Testing Statistical Hypotheses - E.L. Lehmann
Mathematical Statistics: Basic Ideas and Selected Topics - Bickel and Doksum
Statistical lnference - Casella and Berger
Related Courses
Statistics 6710-6720 Mathematical Statistics
NON-PARAMETRIC STATISTICS
NonParametric Statistics. The nature, advantages and disadvantages of NonParametric analysis; rank transformation methods for estimation and testing in one and two sample location and dispersion models; non parametric analysis of variance, multiple comparisons, and simple linear regression; the jackknife and bootstrap.
Reliability. Survey of common survival distributions including their hazard functions and the properties of systems associated with each distribution; hypothesis testing, point and interval estimation of parameters of the distributions under complete and censored sampling; analysis of series and parallel systems; reliability growth concepts in engineering systems and software.
Robust Methods. The robust philosophy; influence curves; breakdown point; L-, R- and M-estimators of location; M-estimation of location and scale; M-estimation in regression models; rubust testing; optimal B-robustness; regression diagnostics; equileverage designs; asymptotic theory; minimum distance methods; bootstrapping to get standard errors.
References
Nonparametric Statistical Methods- Hollander and Wolfe. Wiley, 1973
Reliability: Management Methods and Mathematics- Lloyd and Lipow. American Society for Quality Control, 1984.
Robust Estimation and Testing - Staudte and Sheather. Wiley, 1990.
Robust Statistics: The Approach Based on Influence Functions Hampel, Ronchetti, Rousseeuw, and Stahel. Wiley, 1986.
Robust Statistics - by P.J. Huber. Wiley, 1981.
Related Courses
Stat 6520 Nonparametric Density Estimation and Smoothing
Stat 6510 Resampling Methods
LINEAR MODELS
Experimental Design Types of experiments and experimental units; completely randomized design; simultaneous comparison methods; model comparisons; assessing and accounting for inhomogeneity of variance: assumptions and residuals; transformations; regression using dummy variables and orthogonal polynomials in ANOVA; randomized block de-sign; Tukey's test for non-additivity; latin squares; sets of orthogonal latin squares; crossover designs: balanced fractional designs: balanced and unbalanced two-way designs and higher order designs; variance components; split plot, repeated measures and nested designs.
Regression Theory Simple linear regression; least squares; residual analysis and checking regression assumptions; transformations; inference for regression coefficients, the regression line and predictions: dummy variables in regression, ANOVA and ANCOVA; matrix theory; the multiple linear regression model; expectation and covariance of random vectors; means and variances of quadratic form; quadratic forms in normal variables; least squares estimation - geometric, analytic and algebraic approaches; properties of least square estimators (including the Gauss-Markov theorem); estimation of Ct2; distribution theory; orthogonal structure in the design matrix; optimal design; weighted and general least squares; introducing additional regressors; design matrices of less than full rank: estimable functions; estimation with linear restrictions; non-negative estimation; biased estimation and ridge regression; robust estimation; hypothesis testing and confidence procedures for regression coefficients and regression surfaces; multiple correlation; simultaneous inference; bias due to under and over fitting; random regressors; changepoint regression; polynomial regression and response surfaces; piecewise polynomial regression and splines; variable subset selection in regression; computational techniques.
Applied Regression Analysis Detecting and correcting for multi-collinearity, heteroscedasticity, non-normality of errors and influential observations in multiple regression; partial correlation and regresion, and partial leverage plots; non-parametric regression: robust regression; generalized linear models (GLMs) including binary and polytomous response models and log-linear models; GLMs for survival data; regression with autocorrelated residuals.
References
Fundamental Concepts in the Design of Experiments - Hicks
Analysis of Messy Data: Designed Experiments - Milliken and Johnson
Linear Regression Analysis - G. A. F. Seber
Ceneralized Linear Models - McCullagh and Nelder
Regression with Graphics - L.C. Hamilton
Related Courses
Stat 5110 Theory of Linear Models
Stat 5100 Linear Regression and Time Series
Stat 5200 Design of Experiments
Stat 6200 Analysis of Unbalanced Data and Complex Experimental Design
Stat 6120 Generalized Linear Models
MULTIVARIATE ANALYSIS & TIME SERIES
Study Guide
The following matherial is testable. Conceptual, mathematical and applied understanding is important: detailed formulae are not. Computations will be minimal, although you should have a calculator.
Multivariate Analysis
Text: Multivariate Observations, G.A.F. Seber, Wiley, 1984.
Chapter 1 All sections.
Chapter 2 Sections 2.1, 2.2, 2.3.1, 2.3.2, 2.4.1, 2.5.4, 2.5.5, 2.6.
Chapter 3 Sections 3.1.3.2, 3.3.3.4.1, 3.4.4, 3.6.1, 3.6.2, 3.6.3.
Chapter 4 Sections 4.1, 4.2, 4.3.
Chapter 5 Sections 5.1, 5.2.1.5.2.2, 5.2.3.5.2.4, 5.4.1, 5.4.2, 5.5, 5.7.1, 5.7.2, 5.7.3.
Chapter 6 Sections 6.1, 6.2, 6.3.1, 6.3.2a-c, 6.4.
Chapter 7 Sections 7.1, 7.2, 7.3, 7.5.7.7, 7.8.
Chapter 8 Sections 8.1.8.2, 8.3.8.4, 8.5, 8.6.1, 8.6.2, 8.6.3.
Chapter 9 Sections 9.1.9.2.1, 9.2.2, 9.2.3, 9.2.6, 9.3, 9.4, 9.5.
Time Series
Text: Applied Statistical Time Series Analysis, R.H. Shumway, Prentice-Hall, 1988.
Chapter 1 Omit Section 8.
Chapter 2 Omit Section 8 and Section ll.
Chapter 3 Omit Section 5.