Syllabi for the Qualifier Examinations

• Algebra Qualifier
• Differential Equations Qualifier
• Discrete Qualifier
• Numerical Analysis Qualifier
• Real Analysis Qualifier
• Statistics Qualifier
• Math Biology Qualifier

1. Groups:
Group tables, subgroups, cosets, normal subgroups, quotient groups, Lagrange's Theorem, groups of small order, cyclic groups, permutation, alternating, and dihedral groups, simple groups, homomorphisms, isomorphism theorems, products of groups, finitely generated abelian groups, Sylow theorems.

2. Rings:
Ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, field of quotients, prime and maximal ideals, characteristic, matrix rings, Euclidean rings, polynomial rings, unique factorization theorems, extension fields, degree of an extension, roots of polynomials, finite fields.

3. Linear Algebra:
Linear independence, bases, dual spaces, inner product spaces, linear transformations, matrices, eigenvalues, eigenvectors, Cayley-Hamilton theorem, minimal and characteristic polynomial, Jordan canonical form, orthogonal diagonalization of normal matrices.

Recommended References

Fraleigh, A First Course in Abstract Algebra 5th ed.
Friedberg, Insel, and Spence Linear Algebra 2nd ed.
Exercises in Fraleigh may be too routine to adequately prepare one for the algebra qualifying exam. It is recommended that one work more challenging problems, for example from Herstein's Topics in Algebra.

Ordinary Differential Equations

1. Linear systems (both autonomous and non-autonomous).
2. Existence and uniqueness (via the contraction mapping theorem), and continuation of solutions;
Gronwall's inequality; continuous dependence on initial data.
3. Limit sets for nonlinear systems.
4. Stability of equilibria , Lyapunov stability, Lyapunov-LaSalle theorem.
5. Two-dimensional non-linear autonomous systems, Hamiltonian and gradient systems.
6. Poincaré-Bendixon theorem and DuLac criterion.

Partial Differential Equations

1. Methods of Characteristics: first order quasilinear equations, first order linear systems,
second order linear equations, in particular wave equations.
2. Classification of second order partial differential equations.
3. Fourier Series, including convergence theorems.
4. Fourier transforms.
5. Separation of Variables and applications to Linear Partial Differential Equations.
Introduction to Dirac distributions. Green's identities. Fundamental solutions
and Green's functions.
6. Harmonic Functions, Maximum Principles for the potential and heat equations.

References

ODEs:

F. Brauer & J. Nohel, The Qualitative Theory of Ordinary Differential Equations -
An Introduction, Chapter 1, sections 2.1-2.8, chapter 3, sections 4.1-4.6, sections 5.1-5.5.

K.T. Alligood, T. Sauer, & J.A. Yorke, Chaos, An Introduction to Dynamical Systems, Chapter 8.

M. Hirsch & S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra,
Chapters 3-9, 11, 13.

P. Waltman, A Second Course in Elementary Differential Equations, sections 1.8, 1.9, 1.12, 3.1-3.5,
all of chapter 2.

PDEs:

R. Guenther, J. Lee, Partial Differential Equations of Math. Physics and Integral Equations.
Chapter 1, Sections 2-1, 2-2, 2-6, 3-1 to 3-5, 4-1 to 4-5, 5-1 to 5-5, 8-1 to 8-4, 9-1 to 9-3, 10-5.

Fritz John, Partial Differential Equations. Fourth edition, sections 1.1-1.6, 2.3, 2.4, 4.1-4.3.

This examination is based on the two courses
Mat 415 and Mat 416 and will comprise two parts:

1. Combinatorics:

Pigeonhole principle and Ramsey's theorem;
permutations, combinations, and binomial theorem;
inclusion-exclusion principle, recurrence relations,
and generating functions;
Polya enumeration;
combinatorial design and applications of probabilistic method.

2. Graph theory:

graphs and digraphs;
degrees, paths, cycles, Eulerian circuits,
trees and optimal spanning trees;
matchings, covers, 1-factors, weighted matchings and algorithms;
connectivity, components, blocks, network flows;
planar graphs, Euler's formula, Kuratowski's theorem;
coloring, hamiltonicity, Ramsey theory, and random graphs.

Recommended References:

R.A. Brualdi, Introductory Combinatorics, latest ed.
Chapters: 2,3,5,6,7,8,10,14

D. West, Graph Theory, latest ed.
Chapters: 1,2,3,4,5,6,7,8.

Exercises in Brualdi may be too routine to adequately prepare one for
the combinatorics part of the discrete mathematics qualifier. It is
recommended that one work more challenging problems, for example from
R.P. Stanley's book `Enumerative Combinatorics', Volume 1.

1. Number Systems and Errors:
Floating-point arithmetic; error propagation; condition of a problem,stability of an algorithm.

2. Interpolation by Polynomials and Splines:
Existence and uniqueness; Lagrange, Newton, and Hermite interpolation;cubic splines; Bezier curves; errors.

3. The Solution of Nonlinear Equations:
Fixed-point methods and acceleration; Newton-like methods; convergence and order; roots of polynomials; systems of nonlinear equations.

4. Systems of Linear Equations:
Elimination methods; pivoting and stability; error analysis and norms; conditions; residuals, iterative improvement; triangular decomposition; iterative methods.

5. The Matrix Eigenvalue Problem:
Location of eigenvalues; power and inverse power iteration; orthogonal transformations, reduction to Hessenberg form and QR algorithm.

6. Introduction to Numerical Optimization:
Least squares approximations, steepest descent, line search.

7. Differentiation and Integration:
Numerical differentiation; interpolatory quadrature; Gaussian quadrature; adaptive integration; extrapolation.

8. Solution of Initial Value Problems for Differential Equations:
Simple one-step methods; Runge-Kutta methods; errors; step-size control; multistep methods; predictor-corrector methods; stiff systems; stability; explicit methods for the heat equation.

References

K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John Wiley,1989.
J.I. Buchanan and P.R. Turner, Numerical Methods and Analysis, McGraw-Hill,1992.
S. Conte and C. de Boor, Elementary Numerical Analysis, 3rd ed., McGraw-Hill,1980.
B. Welfert, Numerical Analysis Lecture Notes (see author for availability).

PART I: MAT 472

1. The Real Numbers:
R as a complete ordered field; inf and sup of a subset of R; lim inf and lim sup of a sequence; infinite series, tests for convergence, absolute and conditional convergence.

2. Metric Space Topology:

Countable and uncountable sets; open and closed sets, interior, closure; characterization of open subsets of R; normed spaces; continuity of linear maps between Euclidean spaces; Cauchy-Schwartz inequality; Cauchy sequences, completeness; contraction mapping principle; Baire category theorem; compactness, equivalent characterizations: existence of finite subcovers, completeness and total boundedness, Bolzano-Weierstrass property; Heine-Borel theorem in R^n; Cantor sets; connectedness, connectedness of intervals; continuity, uniform continuity, relation with compactness and connectedness; pointwise and uniform convergence, Weierstrass M-test; equicontinuity, Arzela-Ascoli theorem; Weierstrass approximation theorem.

3. Calculus in One Variable:

Derivative, mean value theorem, Taylor's theorem; Riemann integral and integrability, fundamental theorem of calculus; exponential, logarithmic, trigonometric functions; derivative and Riemann integral of uniformly convergent sequences; power series.

PART II: MAT 473

1. Differential Calculus in Rn:

Derivative as a linear map; mean value theorem and mean value inequality; inverse function theorem; implicit function theorem; Taylor's theorem.

2. Lebesgue Integral in Rn:

Sigma-algebras; Lebesgue outer measure; Lebesgue measure, measurable sets, nonmeasurable sets; Borel sets; measurable functions; almost everywhere; simple functions; Lebesgue integral; monotone convergence theorem; Fatou's lemma; dominated convergence theorem; density in L^1 of simple functions, step functions (in one variable), and continuous functions with compact support; characterization of Riemann integrability; Tonelli's theorem; Fubini's theorem; change of variables theorem.

References:

W. Fleming, "Functions of Several Variables", Ch. 1-5.
G. Folland, "Real Analysis" (for the Change of Variables Theorem).
F. Jones, "Lebesgue Integration on Euclidean Space".
M. Rosenlicht, "Introduction to Analysis", Ch. II-VII and IX.
W. Rudin, "Principles of Mathematical Analysis", Ch. 1-5, parts of 10, 11.

1. Foundations:
Counting principles, sample spaces, probability set functions.

2. Random Variables:
Discrete and continuous random variables; marginal and joint distributions, distribution and density functions; conditional distributions and independence; sampling distribution.

3. Expectation:
Properties of mathematical expectation; moments and moment-generating functions; conditional expectation.

4. Transformations:
Distributions and expectations of functions of random variables; probability integral transforms, change of variable methods; order statistics.

5. Limit Theorems:
Convergence in probability and in distribution; laws of large numbers, central limit theorem.

6. Point Estimation:
Unbiasedness, consistency, efficiency; method of moments; maximum likelihood estimators; Bayesian estimators.

7. Sufficiency and Completeness:
Sufficient statistics, complete families, minimal sufficiency; Rao-Blackwell and Lehmann-Scheffe Theorems; distributions of exponential class.

8. Interval Estimation:
Confidence intervals; pivotal quantities.

9. Tests of Hypotheses:
Significance and power, Neyman-Pearson lemma and most powerful tests; (generalized) likelihood-ratio method.

10. Selected Topics:
Chi-square tests for contingency and goodness-of-fit; the nonparametric sign and signed-rank tests.

Selected References

Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical Statistics (2nd ed.), PWS-Kent, Boston, 1992.
Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics (5th ed.), Prentice Hall, New York, 1995.
Bernard W. Lindgren, Statistical Theory (4th ed.), Chapman & Hall, New York, 1993.
Sheldon Ross, A First Course in Probability (4th ed.) Macmillan, New York, 1994.

Part 1. Introduction to Mathematical Population Biology

1. Deterministic single species population dynamics. Methods for the study of linear and nonlinear discrete and continuous dynamical systems. Cobwebbing with 1 dimensional difference equations; periodic points and their linearized stability. Global stability for some scalar difference models.


2. Population dynamics in structured populations. Deterministic matrix and partial differential equations formulation of birth and death processes. Perron-Frobenius Theorem and discussion of dynamics of $x_{n+1}=L x_n$ for low dimensional nonnegative Leslie matrix L.


3. Population dynamics of interacting species. Host parasitoid interactions, predator prey systems, competition. Nullclines/Direction Fields for planar systems of ODE.


4. Bendixson-Dulac criterion and the fact that a periodic orbit surrounds an equilibrium point. Liapunov-LaSalle theorem and its applications.


5. Standard models in epidemiology. Basic reproduction number. Discussion of questions associated with co-existence, co-evolution, control and eradication.

Prerequisites
MAT 272, 274/275, 342/343, or the approval of the instructor.

Texts
Leah Edelstein, Mathematical Models in Biology, SIAM Classics in Applied Mathematics 46, 2004.
Brauer Fred and Carlos Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40, Springer Verlag 2001

Part 2. Introduction to Mathematical Physiology

1. Biochemical reactions and enzyme kinetics. Law of Mass Action and use for deriving systems of differential equations. The quasi-steady-state assumption and relation to singular perturbation problems. Modeling the regulation of cell function including gene networks.


2. Ficks Law and the derivation of the diffusion equation. The linear (passive) cable equation and cable theory. Analytic solution to steady-state and transient conduction problems for branched and unbranched neurons. Finite difference methods and compartmental modeling.


3. Electrodiffusion models: multi-ion flux and the Poisson-Nernst-Planck Equations. Constant field assumption and the Goldman-Hodkin-Katz equations. The Hodgkin-Huxleymodel.


4. Elementary bifurcation analysis techniques with application to excitable cell dynamics and chemical oscillators. Multiple-scale perturbation method for analyzing nonlinear oscillators. Quasi-static reduction of relaxation oscillators coupled to passive cables.

Prerequisites
MAT 272, 274/275, 342/343, or the approval of the instructor.

Text
James Keener and James Sneyd, Mathematical Physiology, Springer, 1998.