CAM Ph.D. Qualifying Exam in Numerical Analysis CONTENTS

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1 CAM Ph.D. Qualifying Exam in Numerical Analysis CONTENTS Preliminaries Round-off errors and computer arithmetic, algorithms and convergence Solutions of Equations in One Variable Bisection method, fixed-point iteration, the Newton-Raphson method, error analysis for iteration methods, accelerating convergence, zeros of polynomials and Muller's method Interpolation and Polynomial Approximation Interpolation and the Lagrange polynomial, divided differences, Hermite interpolation, cubic spline interpolation Numerical Differentiation and Integration Numerical differentiation, Richardson's extrapolation, element of numerical intergration, composite numerical integration, Gaussian quadrature Initial-Value Problems for ODEs Elementary theory of initial value problems, Euler's method, higher-order Taylor methods, Runge-Kutta method, multistep methods, higher-order equations and systems of differential equations Methods for Solving Linear Systems Gaussian elimination (including pivoting strategies), matrix factorization, iterative techniques (including Jacobi's, GS, SOR) and error estimate Approximation Theory Least square approximation, Orthogonal polynomials, Chebyshev polynomials, rational function approximation, trigonometric polynomial approximation Approximating Eigenvalues Eigenvalue and eigenvectors, power methods Numerical Solutions of Nonlinear Systems of Equations Fixed points for functions of several variables, Newton's method

2 Boundary Value Problems for ODEs The linear shooting method, finite difference methods for linear problems REFERENCES Burden, Richard L. and Faires, Douglas J., Numerical Analysis, 8th ed., Thomson Brooks/Cole, Sample Exam Problem from Math 414 Sample Exam Problem from Math 415

3 CAM Ph.D. Qualifying Exam in Partial Differential Equations First Order Equations CONTENTS Linear first-order equation; Quasi-linear equation; Method of characteristics Second Order Equations Classification of equations; Canonical forms of the hyperbolic, parabolic and elliptic equations; Equations of mathematical physics: wave equation, heat equation and Laplace's equations Elements of Fourier Analysis Fourier series of a function; Convergence of Fourier series; Sine and Cosine expansions; Fourier Sine and Cosine transforms; Fourier Integral; Fourier Transform Wave Equation Cauchy problem and d'alembert's solution; d'alembert's solution as a sum of forward and backward waves; Domain of dependence and the characteristic triangle; Wave equation on a half-line; Wave equation on a half-line with moving end; Nonhomogeneous wave equation on the real line; Fourier series solutions on a closed interval; Nonhomogeneous problem on a closed interval; Cauchy problem by Fourier integral; Wave equation in two space dimensions Heat Equation Weak maximum principle; Heat equation on bounded intervals; Heat equation on the real line; Heat equation on the half-line; Nonhomogeneous heat equation; Heat equation in two space variables Dirichlet and Neumann Problems Harmonic functions; Dirichlet problem for a rectangle; Dirichlet problem for a disk; Possion s integral representation for a disk; Dirichlet problem for the upper half-plane; Dirichlet problem for a rectangular box; Neumann problem for a rectangle; Neumann problem for a disk REFERENCES P.V. O'Neil, Beginning Partial Differential Equations, 2 nd Edition, Wiley, New York, Sample Exam Problems 1. Use the method of characteristics to find a solution of sec that passes through the given curve defined by, 0.

4 2. Solve the Dirichlet problem, 0 for 9, and, for 9.

5 Axioms of Probability CAM Ph.D. Qualifying Exam in Statistics CONTENTS Random experiments, sample space and events, probability function, rules of probability Combinatorial Methods Permutations and combinations, ordered and unordered samples Random Variables Discrete random variables, continuous random variables, distribution functions, moments, probability generating function, special distributions: binomial, Poisson, hypergeometric geometric, negative Binomial, normal, Lognormal, negative exponential, uniform, Gamma and Chi-square, Beta Random Vectors Bivariate and multivariate distributions, multinomial distribution, marginal distributions and independence Distributions of Functions of Random Variables Sums of random variables, Jacobians, the t and F distributions, distributions of order statistics, expectations of functions of random variables Limit Theorems Chebyshev s inequality and Weak Law of Large Numbers, Strong Law of Large Numbers, Central Limit Theorem, convergence in distribution and convergence in probability Conditional Distributions and Expectations Conditional densities and probability functions; conditional probability and independence; Conditional expectations. Estimation Point estimation; Bayesian estimates, confidence intervals for means and variances, sufficient statistics, maximum likelihood estimates, properties of maximum likelihood estimates, Rao- Cramer lower bound Statistical Hypotheses

6 Certain best tests, uniformly most powerful tests, likelihood ratio tests, sequential probability ratio test, Chi-square tests, T and F tests, noncentral F distributions, power of a test statistics, least squares, simple and multiple regression, analysis of variance Normal Distribution Theory The multivariate normal distribution, the distribution of contain quadratic forms, the independence of certain quadratic forms REFERENCES 1. First Course in Probability, A, 7/Edition, Sheldon Ross, ISBN-10: , ISBN-13: , Publisher: Prentice Hall. 2. Introduction to Mathematical Statistics, Robert V. Hogg, Allen Craig, Joseph W. McKean, Prentice Hall; 6th edition, ISBN-10: , ISBN-13: Sample Exam Problems 1. A bin of 50 manufactured parts contains three defective parts and 47 nondefective parts. A sample of six parts is selected from the 50 parts. Selected parts are not replaced. That is, each part can only be selected once and the sample is a subset of the 50 parts. How many different samples are the of size six that contain exactly 2 defective parts? 2. In a certain assembly plant, three machines A, B and C make 30%, 45% and 25% respectively, of the products. It is known from past experience that 2%, 3% and 2% of the products made by each machine respectively are defective. Now suppose that a finished product is randomly selected. What is the probability that the product is defective?

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