Math Comps Syllabus

Option I:

Option III:

1. Metric spaces
2. Convergent sequences
3. Cauchy sequences
4. Topological ideas
- a. Open sets
- b. Closed sets
- c. Interior, closure, boundary
5. Series
6. Continuity, uniform continuity
7. Compactness
8. Connected sets, path-connected sets
9. Intermediate Value Theorem
10. Extreme Value Theorem
11. Differentiation
12. Rolle's Theorem
13. Mean Value Theorem
14. The Riemann integral
15. Fundamental theorem of calculus
16. Pointwise and uniform convergence
17. Weierstrass M Test
18. Taylor series
19. Differentiation and integration of series
20. Sets of measure zero
21. Lebesgue's theorem on Riemann integrability

1. Solving first order and linear nth order equations; Existence, uniqueness, and applications
2. Reduction of order
3. Power series solutions
4. Laplace transforms
5. Systems of linear differential equations
6. Fourier series

1. Metric spaces
2. Convergent sequences
3. Cauchy sequences
4. Topological ideas
- a. Open sets
- b. Closed sets
- c. Interior, closure, boundary
5. Series
6. Continuity, uniform continuity
7. Compactness
8. Connected sets, path-connected sets
9. Intermediate Value Theorem
10. Extreme Value Theorem
11. Differentiation
12. Rolle's Theorem
13. Mean Value Theorem
14. The Riemann integral
15. Fundamental theorem of calculus
16. Pointwise and uniform convergence
17. Weierstrass M Test
18. Taylor series
19. Differentiation and integration of series

1. Formulating linear programming models
2. Solving linear programming problems using the simplex method
(and using the two-phase simplex method when appropriate)
3. The theory of the simplex method; convergence
4. The geometry of linear programming; convexity
5. Duality theory, including the complementary slackness theorem
6. Sensitivity analysis
7. The dual simplex method
8. The transportation problem
9. The assignment problem; the Hungarian method

1. Computer arithmetic, error, relative error
2. Rootfinding
- a. Existence and uniqueness of roots
- b. Bisection
- c. Newton's method
- d. Secant method
- e. Fixed-point iteration
- f. Determining if an approximation is sufficiently accurate
3. Interpolation
- a. Lagrange form
- b. Divided differences
- c. Interpolation Error Theorem
4. Numerical Differentiation
5. Numerical Integration
- a. Composite Trapezoidal Rule
- b. Composite Simpson's Rule
6. Solving linear systems by Gaussian Elimination
7. Pivoting strategies
8. LU decomposition
9. Special types of matrices
- a. Banded matrices
- b. Diagonal dominance
- c. Positive definite matrices, Choleski decomposition
10. Vector and matrix norms
11. Iterative methods for linear systems
- a. Jacobi's method
- b. Gauss-Seidel
- c. General x(k+1) = Tx(k) + c approaches (SOR and others)
12. The residual and iterative refinement
13. Condition number of a matrix
14. Gerschgorin's Theorem
15. The Power Method to approximate the dominant eigenvalue
16. Least-squares approximation of functions

1. Calculus of probability
- a. Sample space
- b. Addition rule
- c. Conditional probability
- d. Independence
- e. Bayes' Theorem
2. Random Variables
- a. Discrete and continuous univariate and multivariate distributions
- b. Derived distributions of functions of random variables
- c. Expectation
- d. Variance and covariance
- e. Chebyshev inequality
3. Limit theorems
- a. Convergence in distribution, in probability and almost sure convergence
- b. Central limit theorem
- c. Strong and weak laws of large numbers