# Qualifier Examination Information

All Ph.D. students must pass three examinations in different areas of advanced mathematics or statistics. Following is the relevant part of the Ph.D. program description:

Students must pass three written examinations. Two of these will be chosen from the areas Algebra, Combinatorics and Real Analysis. The third will be chosen from the areas Mathematical Modeling, Applied Statistics and Probability & Mathematical Statistics. Normally, these will be taken within a year of completion of the core coursework. These examinations need not be taken together. Usually, at most two attempts at passing this examination will be permitted. Students who wish to make a third attempt must petition the Graduate Studies Committee of the department for permission to do so.

Every year, the exams will be given in January and August. Each exam will be 3 1/2 hours long. The following examinations will be available in August 2008:

August 04 Mathematical Modeling
August 06 Combinatorics
August 08 Algebras
August 11 Applied Statistics

These examinations will be held in Natural Science Building in NS 333 from 12:00-3:30 PM.

The following exams will be available in May 2008:

May 07 Analysis
May 07 Mathematical Modeling
May 09 Combinatorics
May 12 Algebras
May 14 Applied Statistics

These examinations will be held in Natural Science Building in NS 333 from 12:00-3:30 PM.

Topics students are expected to know for each exam are listed below. Past exams and tips are available.

## Algebra

• Groups: homomorphisms and subgroups, cyclic groups, cosets and counting, normality, quotient groups, symmetric, alternating and dihedral groups, direct product and direct sum, finitely generated abelian groups, group action, the Sylow Theorems, nilpotent and solvable groups, normal and subnormal series.

• Rings: rings and homomorphisms, ideals (prime, maximal), factorization in commutative rings, unique factorization domains, principal ideal domains, euclidean domains, polynomial rings, Eisenstein's criterion.

• Fields and Galois Theory: field extensions, splitting fields, Galois group, separability, cyclic extensions, finite fields, cyclotomic extensions, radical extensions.

#### Suggested Reference:

• Thomas W. Hungerford, Algebra, Springer Verlag, 8th ed. (1997) ISBN: 0387905189, Chapters 1-3 and 5.

## Applied Statistics

• Linear Model: Linear model designs such as (a) crossed, (b) split plot, (c) nested, (d) repeated measures; maximum likelihood estimators and least squares; hypothesis testing; confidence intervals; regression diagnostics, and variable transformations.

• Classification: Nearest neighbor discriminant analysis, logistic regression, neural networks, and C-5 rule induction (decision trees).

• Clustering: K-means, and hierarchical.

#### Suggested References:

• The Analysis of Messy Data, vol. 1 by Miliken and Johnson.
• The Elements of Statistical Learning: Data Mining, Inference, and Prediction by Hastie, Tibshirani, and Friedman.
• Generalized, Linear, and Mixed Models by McCulloch and Searle.

## Combinatorics

• Advanced Counting: sequences, selections, distributions, partitions, lattice paths, Catalan numbers, Stirling numbers, Ferrers diagrams, Ramsey numbers, inclusion-exclusion, derangements, recurrence relations, generating functions (both ordinary and exponential), characteristic polynomial method to solve recurrences, Stirling's approximation, Polya counting

• Elementary Graph Theory: graphs, digraphs, subgraphs, degrees, adjacency matrices, incidence matrices, graph isomorphism, paths, cycles, trees, connectivity, bipartite graphs, edge contraction, subdivisions, linegraphs, independent sets, cliques

• Flows and Related Graph Concepts: network flows, integral flows, Max-Flow/Min-Cut theorem, (edge and vertex) connectivity, vertex cuts, edge cuts, perfect matchings, Menger's theorem, Hall's marriage theorem, Tutte's theorem, factorizations, Petersen graph, Konig-Egervary theorem

• Planar Graphs: drawings in the plane, planar duals, Euler's formula, Kuratowski's theorem, convex embeddings in the plane, coloring planar graphs

• Graph Coloring: vertex and edge coloring, Brooks' theorem, Vizing's theorem, color critical graphs, perfect graphs, Lovasz's Perfect graph theorem, the Strong Perfect Graph Theorem

• Miscellaneous: Ramsey graphs, Hamiltonion cycles, random graphs, Turan's theorem, partially ordered sets, Dilworth's theorem

#### Suggested References

• Kenneth P. Bogart, Introductory Combinatorics, Harcourt-Academic Press, 3rd ed. (2000) ISBN 0121108309
• Douglas B. West, Introduction to Graph Theory, Prentice Hall, 2nd Edition (2000), ISBN: 0130144002
Also useful are the following Schaum's Outlines:
• V. K. Balakrishnan, Schaum's Outline: Graph Theory, McGraw Hill (1997) ISBN: 0070054894
• V. K. Balakrishnan, Schaum's Outline: Combinatorics (including graph theory), McGraw Hill (1995) ISBN: 007003575X

## Mathematical Modeling

• Mechanical vibrations: Newton's laws, spring-mass systems, two-mass oscillators, friction, damping, pendulum, linear stability and equilibria, energy analysis, phase plane analysis, nonlinear oscillations, control oscillations, inverse probllem. [1] pp 1-114 and [2] Chapters 0-2.

• Traffic flow: Velocities and velocity fields, trafffiic flow and density, conservation laws, linear and nonlinear car-following models, steady state, first order partial differenntial equations (the method of characteristics), green light mooddels and rarefaction solution, shock waves (effect of rred light annd slower traffic ahead), highway with entrance (inhomogenneous problem), traffic wave propagation, optimiization problem. [1] pp. 259-394.

• Dynamical systems: Nonlinear systems in the plane, interacting species, limit cycles, Hamiltonian systems, Liapunov functions and stability, bifurcation theory, three-dimensional autonomous systems and chaos, Poincare maps and nonautonomous systems in the plane, linear discrete dynamical systems. [2] Chapters 3-9, 13.

#### Suggested References

• [1] Richard Haberman, Mathematical Models, Mechanical Vibrations, Population Dynamics and Traffic Flow, SIAM Classics in Applied Mathematics (1987) ISBN: 978-0898714081.
• [2] Stephen Lynch, Dynamical Systems with Applications Using Maple, Springer-Verlag (2000) ISBN: 0-8176-4150-5.

## Probability & Mathematical Statistics

• Basic Probability: Basic axioms, probability spaces, independence of events. Existence of probability measures. Properties of the distribution functions. [2], Ch. 1.1 [3], Ch. 1.1

• Random Variables in R and Rn and their distributions. [2], Ch. 1.2 [3], Ch. 1.3

• Expectation and conditional expectation. Moment inequalities. [3], Ch. 1
• Discrete-time martingales. [2], Ch. 4

• Types of convergence of random variables. Characteristic functions, continuity theorem. [2] Ch. 1 (Sections 5-7) Ch. 2 (Sections 1-3). [3], Ch. 1 (Section 5)

• Laws of large numbers, zero-one law, central limit theorem for independent arrays. [2], Ch. 1 (Sections 1-5, 9), Ch. 2.4 [3], Ch. 1 (Section 5)

• Basic concepts in Mathematical Statistics.Populations, samples, statistics and their sampling distributions. Exact distributions vs approximations via CLT. [1], Ch. 5 [3], Ch. 2 (Section 1 )

• Parametric point estimation and properties of estimators. [1], Ch. 7 [3]. Ch. 3 (Sections 1 and 3)

• Sufficiency, data reduction, Rao-Blackwell & Lehmann-Scheffe theorems. [1], Ch. 6 [3] Ch. 2. (Section 2), Ch. 3. (Sections 1 and 5)

• Interval estimation (confidence, tolerance, prediction). [1], Ch. 9 [3] Ch. 7

• Hypothesis Testing: Neyman-Pearson Lemma, Generalized Likelihood Ratio Test, Uniformly and locally most powerful tests; goodness-of-fit tests. [1], Ch. 10 [3] Ch. 6

• The linear models and regression estimators. [1] Ch. 12 [3] Ch. 3 (Section 3) and Ch. 4 (Section 4)

#### Suggested References

[1] George Casella and Roger L. Berger (1990), Statistical Inference, Brooks/Cole Pub. Co., 2nd ed. (2001) ISBN 0534243126
[2] Richard A. Durret, (1996), Probability: Theory and Examples, Duxbury Press, 2nd ed. (1995) ISBN 0534243185
[3] Jun Shao, Mathematical Statistics, Springer Verlag, 2nd ed. (2003) ISBN 0387953825

## Real Analysis

• Outer measure, measurable sets, sigma-algebras, Borel sets, measurable functions, the Cantor set and function, nonmeasurable sets.

• Lebesgue integration, Fatou's Lemma, the Monotone Convergence Theorem, the Lebesgue Dominated Convergence Theorem, convergence in measure.

• Lp spaces, Hölder and Minkowski inequalities, completeness, dual spaces.

• Abstract measure spaces and integration, signed measures, the Hahn decomposition, the Radon-Nikodym Theorem, the Lebesgue Decomposition Theorem.

• Product measures, the Fubini and Tonelli Theorems, Lebesgue measure on real n-space.

• Equicontinuous families, the Ascoli-Arzela Theorem, Uniform boundedness principle.

• Hilbert spaces, orthogonal complements, representation of linear functionals, orthonormal bases.

#### Suggested References:

• H. L. Royden, Real Analysis, Prentice Hall, 3rd ed. (1988) ISBN: 0024041513 Chapters 1-7, 11, 12.
• Walter Rudin, Real and Complex Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed. (1986) ISBN: 0070542341, Chapters 4-5.

Another book that contains more elementary background information is the following.

• Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed., (1976) ISBN: 007054235X.

## Mathematical Physics

• Particles and fields, recapitulation of quantum mechanics of 1930's, general principles of quantum mechanics including: linear operators, states and observables, fundamental principles of quantum mechanics, compatibility of observations, representations and transformations, time-dependent equation, and the density matrix. Central forces: orbital angular momentum, and motion under central forces. Approximation methods: the method of variation, the method of perturbation, the method of perturbation-variation, and the method of Wentzel-Kramers-Brillouin.

• Rotation and angular momentum, coupling of two angular momenta, scalar, spinor, and vector fields, spin-dependent interactions, polarization of particles with spin, system of particles, LS-coupling and jj-coupling, two-fermion system in LS-coupling, the helium atom, configuration mixing, systems of more than two fermions, calculation of single-particle wave function (Hartree-Fock approximation), time-dependent perturbation theory, scattering, and the method of partial waves.

#### Suggested References

• R. Shankar, Principles of Quantum Mechanics (2nd Ed.), Kluwer Academic/Plenum Publishers, 1994, ISBN: 0306447908.
• N. Zettili, Quantum Mechanics: Concepts and Applications , John Wiley & Sons, 2001, ISBN: 0471489441.