# Department of Math and Computer Science

Mathematics Comprehensive Exam Syllabus

## Option I:

## Option III:

### Algebra

- Group theory
- subgroups
- permutation groups
- homomorphisms
- kernels and images
- normal subgroups, quotient groups
- isomorphism theorems

- Ring and field theory
- homomorphisms
- kernels and images
- ideals, quotient rings
- isomorphism theorems
- integral domains
- polynomial rings
- principal ideal domains
- fields
- algebraic field extensions
- Galois theory

- Linear algebra
- vector spaces
- bases and dimension
- matrices and linear transformations
- kernels and images
- eigenvalues
- inner product spaces

#### References:

- Fraleigh: A First Course in Abstract Algebra
- Gallian: Contemporary Abstract Algebra
- Herstein: Topics in Algebra
- Friedberg, Insel, Spence: Linear Algebra

### Complex Analysis

- Holomorphic (or Analytic) Functions of a Complex Variable
- Cauchy-Riemann Conditions and Harmonic Functions
- Elementary Complex Functions (
*e*log^{z}, z^{n}, z^{1/n},*z*) - Complex Integration
- Cauchy - Goursat Theorem
- Cauchy Integral Formula
- Morera's Theorem
- Liouville's Theorem
- Fundamental Theorem of Algebra
- Maximum Principle
- Taylor Series of Holomorphic Functions
- Power Series as Holomorphic Functions
- Meromorphic Functions
- Laurent Series
- Residues and Contour Integration
- Mobius (or Linear Fractional) Transformations
- Conformal Mapping
- Entire Functions and Picard's Little Theorem
- Argument Principle and Rouche's Theorem

#### References:

- Brown and Churchill: Complex Variables and Applications
- Marsden and Hoffman: Basic Complex Analysis
- Ahlfors: Complex Analysis
- Stein and Shakarchi: Complex Analysis
- Hille: Analytic Function Theory
- Spiegel: Schaum's Outline of Complex Variables

### Real Analysis

- Metric spaces
- Convergent sequences
- Cauchy sequences
- Topological ideas
- Open sets
- Closed sets
- Interior, closure, boundary

- Series
- Continuity, uniform continuity
- Compactness
- Connected sets, path-connected sets
- Intermediate Value Theorem
- Extreme Value Theorem
- Differentiation
- Rolle's Theorem
- Mean Value Theorem
- The Riemann integral
- Fundamental theorem of calculus
- Pointwise and uniform convergence
- Weierstrass M Test
- Taylor series
- Differentiation and integration of series
- Sets of measure zero
- Lebesgue's theorem on Riemann integrability

**References:**

- Marsden and Hoffman: Elementary Classical Analysis
- Apostol: Mathematical Analysis

### Topology

- Topological spaces
- Interior, closure, boundary
- Relative topology
- Bases, subbases
- Continuous functions
- Homeomorphisms
- Product spaces
- Quotient spaces
- Connectedness, path-connectedness
- Compactness
- Separation axioms

### Applied Analysis

#### Differential Equations:

- Solving first order and linear nth order equations; Existence, uniqueness, and applications
- Reduction of order
- Power series solutions
- Laplace transforms
- Systems of linear differential equations
- Fourier series

#### References:

- Zill: Differential Equations
- Boyce and DiPrima: Elementary Differential Equations

#### Analysis:

- Metric spaces
- Convergent sequences
- Cauchy sequences
- Topological ideas
- Open sets
- Closed sets
- Interior, closure, boundary

- Series
- Continuity, uniform continuity
- Compactness
- Connected sets, path-connected sets
- Intermediate Value Theorem
- Extreme Value Theorem
- Differentiation
- Rolle's Theorem
- Mean Value Theorem
- The Riemann integral
- Fundamental theorem of calculus
- Pointwise and uniform convergence
- Weierstrass M Test
- Taylor series
- Differentiation and integration of series

#### References:

- Marsden and Hoffman: Elementary Classical Analysis

### Linear Programming

- Formulating linear programming models
- Solving linear programming problems using the simplex method

(and using the two-phase simplex method when appropriate) - The theory of the simplex method; convergence
- The geometry of linear programming; convexity
- Duality theory, including the complementary slackness theorem
- Sensitivity analysis
- The dual simplex method
- The transportation problem
- The assignment problem; the Hungarian method

#### References:

- Thie: An Introduction to Linear Programming and Game Theory
- Winston and Venkataramanan: Introduction to Mathematical Programming

### Numerical Analysis

- Computer arithmetic, error, relative error
- Rootfinding
- Existence and uniqueness of roots
- Bisection
- Newton's method
- Secant method
- Fixed-point iteration
- Determining if an approximation is sufficiently accurate

- Interpolation
- Lagrange form
- Divided differences
- Interpolation Error Theorem

- Numerical Differentiation
- Numerical Integration
- Composite Trapezoidal Rule
- Composite Simpson's Rule

- Solving linear systems by Gaussian Elimination
- Pivoting strategies
- LU decomposition
- Special types of matrices
- Banded matrices
- Diagonal dominance
- Positive definite matrices, Choleski decomposition

- Vector and matrix norms
- Iterative methods for linear systems
- Jacobi's method
- Gauss-Seidel
- General x(k+1) = Tx(k) + c approaches (SOR and others)

- The residual and iterative refinement
- Condition number of a matrix
- Gerschgorin's Theorem
- The Power Method to approximate the dominant eigenvalue
- Least-squares approximation of functions

#### References:

- Burden and Faires: Numerical Analysis
- Timothy Sauer: Numerical Analysis

### Probability

- Calculus of probability
- Sample space
- Addition rule
- Conditional probability
- Independence
- Bayes' Theorem

- Random Variables
- Discrete and continuous univariate and multivariate distributions
- Derived distributions of functions of random variables
- Expectation
- Variance and covariance
- Chebyshev inequality

- Limit theorems
- Convergence in distribution, in probability and almost sure convergence
- Central limit theorem
- Strong and weak laws of large numbers

#### References:

- Wackerly, Mendenhall, Scheaffer: Mathematical Statistics with Applications
- Gut: An Intermediate Course in Probability