### Comprehensive and Qualifying Exams

# Masters Comprehensive Examinations

## M.A. Comprehensive Examination

The M.A. Comprehensive Examination is a written examination consisting of three parts. Each part is based on topics normally treated in the courses indicated.

- Real and Complex Analysis (5820, 5830, 5880)
- Abstract and Linear Algebra (5330, 5340, 5300, 5310)
- One of:

- Topology (5450, 5460)
- Differential Equations (5800, 5810)
- Probability and Statistics (5680, 5690)
Parts 1 and 2 are each three hours in length and Part 3 is a two hour exam. The parts will normally be written on three consecutive Saturdays in the Fall or in the Spring.

The following rules apply to this examination:

- If a student's score on any part of the written exam is only marginally below the passing level, the examining committee for that part may defer its final decision pending the outcome of a one-hour follow-up oral examination of the student, to be administered within one month of the date of the written exam. This option will not be exercised by the committee if the margin of failure on the entire written exam is deemed to be decisive.
- A student who passes two of the three parts of the exam is credited with those parts and is required to retake only the part failed; a student who fails two or more parts will receive no credit for
anypart.- A student is allowed two attempts on the full three-part examination. A third (and final) attempt on one part of the exam will be permitted if credit has previously been earned for the other two parts.
- A student's first attempt at the examination must occur at one of the two regularly scheduled periods (Fall or Spring). Following an unsuccessful attempt made during one of these periods, a student who is eligible under rule (3) may elect to reattempt the exam (or part thereof as governed by rule (2)) within two months, at a time agreeable to both the student and the exam committee. Otherwise, the retake must take place at the next regularly scheduled examination.
- A student who intends the M.A. degree to be her or his terminal degree in this department may elect to satisfy the degree requirements by submitting and defending a thesis instead of writing the M.S. Comprehensive Examination, but a student may not switch from the exam option to the thesis option after an unsuccessful attempt at the examination.

### Syllabus for M.A. Exam

Real and Complex Analysis

- Real Analysis:
The real number system

Elementary metric space theory

Sequences and series

Differential calculus

Integral calculus (the Riemann integral)

Sequences and sums of functions (Weierstrass Approximation theorem, uniform convergence, Arzela-Ascoli theorem)

The Lebesgue integral (on the real line)

- Complex Analysis:
Complex functions: limits, continuity, properties of elementary functions including branches

Differentiability: derivatives Cauchy-Riemann equations, analyticity, harmonic functions

Integration: Cauchy theorems, Residue theorem, Morera's theorem, Maximum-Modulus theorem

Series: Taylor's theorem, Laurent series expansions

Mappings: elementary functions, properties of conformal maps

Further properties of analytic functions: singular points, zeros, analytic continuation, residues, evaluation of real and complex integrals using residues

Some references:

- Real Analysis:
Rudin,

Principals of Mathematical Analysis, McGraw-HillGoldberg,

Methods of Real Analysis, Blaisdell- Complex Analysis:
Churchill,

Complex Variables and Applications, McGraw-HillSaff & Smider,

Fundamentals of Complex Analysis, Prentice-Hall

Abstract and Linear Algebra

- Abstract Algebra:
Basic concepts of groups and rings including Lagrange's theorem, normal subgroups, factor groups, homomorphisms and isomorphisms, permutations, the field of quotients of an integral domain, polynomial rings, factorization in integral domains.

- Linear Algebra:
Vector spaces, linear transformations and matrices, determinants, canonical forms.

Some references:

- Abstract Algebra:
Herstein,

Topics in Algebra, WileyFraleigh,

A First Course in Abstract Algebra, Addison-Wesley- Linear Algebra:
Hoffman and Kunze,

Linear Algebra, Prentice-HallCurtis,

Linear Algebra, Springer-VerlagFriedberg, Insel, Spence,

Linear Algebra, Prentice-HallOne of:

Topology

- Axiomatization of topological spaces
Different ways of introducing a topological structure.

- Continuous maps
Characterizations of continuity, initial sources and final sinks, discrete and indiscrete spaces.

- Fundamental constructions
Basis for open (or closed) sets, subbase, subspaces, products, quotients, sums. Lattice of topologies on a set.

- Convergence
Sequences; filters and ultrafilters.

- Countability
First and second axiom. Lindelof spaces.

- Separation
Hausdorff, regular, completely regular, normal spaces; $T_i$-spaces, $i=0,1,2,3\frac{1}{2},4$. Urysohn's Lemma.

- Compactness
In Euclidean spaces. Tychonoff Theorem, Stone-Cech compactification. Local compactness in $T$-spaces, Alexandroff compactification.

- Connectedness
Components, local connectedness, path connectedness.

- Metric spaces
Cauchy sequences, completeness. Uniform continuity. Baire's theorem.

- Metrization theorems and paracompactness
The classical theorems of Urysohn and of Nagata-Smirnov-Bing. Stone's theorem.

- Function spaces
Pointwise and compact convergence. The compact-open topology, Ascoli's theorem. Uniform convergence for metric spaces.

- Approximation
Stone-Weierstrass theorem.

Some ReferencesJ. Dugundji,

Topology, Allyn and Bacon, Inc., 1966R. Engelking,

General Topology, PWN-Polish Scientific Publishers, 1977James R. Munkres,

Topology, Prentice-Hall, Inc., 1975Stephen Willard,

General Topology, Addison-Wesley, 1968

Differential Equations

- Ordinary Differential Equations:
Generalities and soluble classes of first order ODE's.

Second order linear ODE's (General theory and explicit solutions in case of constant coefficients).

Systems of linear ODE's with constant coefficients.

Series solutions of second order linear ODE's with analytic coefficients near an ordinary point and near a regular singularity.

Non-linear autonomous ODE's of second order. Phase plane analysis; in particular, equilibrium solutions, their classifications and their stability.

Existence and uniqueness theorems. Nature of dependence on initial conditions.

- Partial Differential Equations:
First order linear (and quasi-linear) equations.

Classification of second order equations and their canonical forms.

Method of separation of variables to solve boundary value problems, in particular the heat equation, the wave equation and the Laplace equation. In this connection: the Convergence theorem for Fourier series. Also Fourier integrals.

Wave equation (or other hyperbolic equations). The Initial (Boundary) value problem. D'Alembert's Principle. (Huygens' principle.)

Heat equation (or other parabolic equations). The Initial (Boundary) value problem. Existence and uniqueness theorem in one-dimensional heat equation.

Laplace equation (or other elliptic equation). Basic properties of harmonic functions, in particular, the Maximum Principle. Boundary value problems. The Dirichlet problem, Green's function and Poisson's formula.

Some References:

- ODE's:
Boyce and DePrima,

Elementary Differential Equations, Wiley.Simmons,

Ordinary Differential Equations, McGraw-Hill.Birkhoff and Rota,

Ordinary Differential Equations.- PDE's:
Zachmano glu and Thoe,

Introduction to Partial Differential Equations, William & Willkins Comp., Baltimore.Colton,

Partial Differential Equations: An Introduction, Random House Birkhaeuser Math Series.Berg and McGregor,

Elementary Partial Differential Equations, Holden-Day, San Francisco.

Probability and StatisticsThis will be based upon a knowledge of the material in Mathematics 5680 and 5690.

## M.S. (Applied Math) Comprehensive Exam

The comprehensive examination for the M.S. in Applied Mathematics will be a written examination consisting of two parts. Each part is based on topics normally treated in the courses indicated.

Real and Complex Analysis (5820, 5830, 5880)

The examination will test the student's knowledge of elementary real and complex analysis.

Differential Equations (6500, 6510)

The examination will test students' knowledge of both ordinary and partial differential equations. The exam will be based upon the more computational aspects of the material in the above courses.

The two parts of the exam will each be three hours in length and will normally be taken on consecutive Saturdays in the Fall or in the Spring.

The following rules apply to this examination:

- The entire (two part) examination can be taken only twice.
- If a student fails only one part of the examination, only that part need be retaken. A third (and final) attempt on one part only will be allowed.
- If a student needs to retake one or both parts, this may be done either within two months or at the next regularly scheduled examination session.
- If a student's score on one or both parts of the examination is only marginally below the passing level, in the examining committee's judgment, the committee may defer its final decision pending the outcome of a one-hour oral examination on that part. Such oral examination(s) are to be given within one month of the date of the written examination and at the mutual convenience of the student and the examining committee.

### Syllabus for M.S. (Applied Math) Exam

Real and Complex Analysis

- Real Analysis:
Completeness of $\mathbb{R}^n$, Sequences and Series.

Compactness of $\mathbb{R}^n$, Connectedness.

Uniform Convergence.

Riemann integral, existence of the integral, uniform convergence and the integral.

Improper integrals.

- Complex Analysis:
Analytic functions and the Cauchy-Riemann Equations.

Elementary conformal mappings.

Cauchy-Goursat theorem, Cauchy integral formula, residue calculus.

Taylor and Laurent series.

Differential Equations

- Ordinary Differential Equations:
Linear systems, calculation of fundamental matrices.

Variation of parameters for systems.

Boundary value problems, eigenvalue problems, Sturm-Liouville theory.

Plane autonomous systems, Liapunov stability.

- Partial Differential Equations:
Method of characteristic for first order equations.

Boundary value problems for the Laplace, Wave and Heat equations, separation of variables.

Greens functions for the Laplace, Wave and Heat equations, Poisson's kernel, Dirichlet problem, method of images.

## M.S. (Statistics) Comprehensive Exam

The purpose of this examination is to ensure that the M.S. graduate has acquired statistical knowledge and skills adequate for a practicing statistician in engineering, science, management, pharmacy, medicine, and other areas.

The examination will be a written examination, including a take-home project. It will consist of two parts:

Probability and Statistical Theory.

This part will normally be based upon a detailed knowledge of the material in Mathematics 5680, 5690 and 6680. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5660, 5700, 6630, 6690, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.

Applied Statistics.

This part will normally be based upon a detailed knowledge of the material in Mathematics 5620, 6630, 6640, and 6690. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5610, 5640, 5660, 5700, 6610, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.

The two parts of the exam will each be three hours in length and will normally be given on consecutive Saturdays in the Fall and/or in the Spring.

The take-home project will be given at least one week before the Applied Statistics portion of the exam, and will be due at the time that the Applied Statistics portion is given. In addition to material from the courses listed above in the description for the Applied Statistics exam, material from Mathematics 5670 may also be covered on the take-home portion of the exam.

The rules applying to this examination are the same as rules (1) through (4) of the M.S. Applied Mathematics exam as delineated in Appendix A2 of this Guide with one additional rule:

- Books and notes (as approved by the examining committee) and calculators may be used during this examination. Your course notes and course textbooks will be approved. Please obtain approval in advance for the use of other written materials during the examination.
Any questions regarding the content of this examination should be directed to Professor Donald White, Head of the Statistics Group.

## M.S. ED. Comprehensive Exam

The M.S. Ed. Comprehensive Examination consists of three parts, each part in an area of mathematics studied by the student. The exact areas are to be arranged with the advisers, but will usually include complex variables and algebra.

# Ph.D. Examinations

The following regulations apply to students entering the Ph.D. program in or after September 1994.

## Ph.D. Qualifying Exam

The Ph.D. qualifying examination is a preliminary examination for the Ph.D. program. It consists of two three-hour parts, each on a separate topic. The two topics on the examination are to be chosen from among the following general areas: algebra, real analysis,topology, differential equations (for students intending to write a dissertation in an applied area) or statistics (for students intending to write a dissertation in the area of statistics). The content of each part will be based on the material presented in the respective required first year-long sequences in algebra (6300, 6310), real analysis (6800, 6810), topology (6400, 6410), differential equations (6500, 6510). See below for more details on the exams syllabus. (Students intending to take the Statistics exam should consult with the statistics program adviser.)

The following rules apply:

- The Ph.D. preliminary exam will generally be offered twice a year, in the Fall and Spring semesters, and will be administered over a period of two weeks.
- Each part of the examination will be graded on a pass/fail basis. Students who fail are required to pass the failed exams by the end of their second year.
- In order to continue in the program both exams must be passed by the end of the student's second year. There will be two opportunities to pass the exam. However, should the student elect to take the exams at the end of their first year this will not count as one of the two opportunities.

### Syllabus for Ph.D. Qualifying Exams

Differential Equations

- Ordinary Differential Equations

- General theory for first order equations
Existence and uniqueness of solutions

Continuous dependence on parameter and initial conditions

Infinite series solutions and method of majorants

- Linear Systems
General theory of linear systems

Linear periodic systems

- Second Order linear equations
Boundary value problems, Green's functions, Sturm-Liouville theory

Comparison theorems

- Qualitative theory of ordinary differential equations in the plane
Limit cycles, Poincaré-Bendixson Theorem

Stability, Liapunov's method

- Partial Differential Equations
Cauchy-Kowalevski Theorem

- Hyperbolic systems

- Existence and uniqueness of solutions
Method of characteristics

Energy estimates

Fourier transforms, Green's functions

- Second order elliptic equations
Application of the Maximum Principle

- Elementary Sobolev space theory:
$H^1(\mathbb{R}^n)$ and $H^2(\mathbb{R}^n)$

- Existence and uniqueness of solutions
Dirichelet Principle

Perron's Method

Eigenvalues of the Laplace operator

- Heat equation

- Existence and uniqueness of solutions
Fundamental solutions

Energy estimates

Maximum principle

Algebra

- Background material

- Linear Algebra

- Vector spaces and linear transformations
Determinants

Canonical forms

Diagonalization of quadratic forms

- General
Homomorphism theorems

Jordan-Hölder Theorem

Fittings's Lemma

Krull-Schmidt Theorem

- Groups

- Group actions
Fundamental counting theorem

- Permutation groups
Transitivity and primitivity

Simplicity of $A_n$ for $n \ge 5$

Class equation

Frattini Argument

Sylow's theorems and $p$-groups

- Group constructions
Direct products

Semidirect products

Structure of finitely generated Abelian groups

- Derived series and central series
Solvable groups and nilpotent groups

- Fields
Simple extensions (algebraic and transcendental)

Galois Group of an extension

Algebraic closure

Separable and inseparable extensions

Normal extensions

Fundamental theorem of Galois Theory

Finite fields

- Rings
Projective and Injective modules

- Commutative Rings
Factorization

Localization with respect to multiplicatively closed sets

- Simple modules and primitive rings
Jacobson radical

Jacobson Density Theorem

- Artinian Rings
Wedderburn-Artin Theorems

Real Analysis

- Background material
Infinite series and products

Power series

Elementary theory of the derivative

Monotone function and functions of bounded variation

- Metric spaces

- Topology
Completeness

Connectedness

Compactness and totally boundedness

Uniform convergence

Baire Category Theorem

Ascoli-Arzela Theorem

Stone-Weierstrass Theorem

- Integration

- Measure Theory on the real line
Outer measure

Relation between measure and content

Measurable function, Egoroff and Lusin Theorems

Lebesgue integral on the real line

Monotone and bounded convergence theorems, Fatou's Lemma

Riemann integral and relation to Lebesgue integral

Improper Riemann and Lebesgue integrals

Some References:T. Apostol,

Mathematical AnalysisR. Goldberg,

Methods of Real AnalysisH.L. Royden,

Real AnalysisW. Rudin,

Principles of Mathematical Analysis

Topology

- Background material
Cardinality and countability

Axiom of Choice, Well Ordering, Maximum Principle

Basic facts about the ordinals

- General topology
Open set, closed set, closure, interior, and neighborhood systems

Convergence of filters and nets

Separation and countability axioms

Continuous functions

Connectedness and path connectedness

Compactness

Product spaces and Tychonov Theorem

Urysohn Lemma and Tietze Extension Theorem

Local compactness and paracompactness

Quotient spaces

- Algebraic topology
Homotopy of maps and homotopy equivalence

Fundamental group

Covering spaces and classification

Seifert-van Kampen Theorem

Some references:W.S. Massey,

A Basic Course in Algebraic TopologyJ.R. Munkres,

Topology, A First CourseI.M. Singer and J.A. Thorpe,

Lecture Notes on Elementary Topology and GeometryS. Willard,

General Topology

## Ph.D. Oral Examination

A student shall take the Ph.D. oral examination upon successful completion of the Ph.D. qualifying examination, or at the request of a faculty member whom the student has asked to be his/her thesis supervisor and who has not yet accepted the student as an advisee. The oral examination will be administered by a committee of three faculty members. The exam will be concerned with the general area of specialization of the student. It is the student's responsibility to ask a faculty member to agree to serve as the chair of the examination committee. The examination will consist of two parts. Part I shall be a talk by the student on a topic at a level sufficient to demonstrate the student's ability to engage in mathematical research. The topic will be chosen in consultation with and approval of the committee chair. Part II shall be an examination of the student by the committee to ascertain the level of the student's understanding of the topic and related background material. Parts I and II will be administered at the same session.

The following rules apply:

- The oral exam will be graded on a pass/fail basis.
- The student must pass the oral examination within one year of passing the Ph.D. qualifying exam or by the end of the second year, which ever is later.