91短视频

1. Group theory
   1. subgroups
   2. permutation groups
   3. homomorphisms
   4. kernels and images
   5. normal subgroups, quotient groups
   6. isomorphism theorems
2. Ring and field theory
   1. homomorphisms
   2. kernels and images
   3. ideals, quotient rings
   4. isomorphism theorems
   5. integral domains
   6. polynomial rings
   7. principal ideal domains
   8. fields

References:

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

• Metric spaces, sequence
• Open and Closed sets, Limits and Continuity in metric spaces
• Connectedness, Completeness and Compactness and relation to Continuity. Uniform Continuity
• Riemann Integration - definition, properties, sets of measure zero, Riemann-Lebesgues Theorem
• Derivatives, Rolle's Theorem and Mean Value Theorem
• Sequences of Functions, Pointwise versus Uniform Convergence and relation to continuity and derivatives
• Series of Functions, Weierstrass M test, relation to continuity, integration and derivatives. 

References:

• Richard Goldberg, Methods of Real Analysis, 2nd edition
• Marsden and Hoffman: Elementary Classical Analysis
• Apostol: Mathematical Analysis

1. Rootfinding
   • Existence and uniqueness of roots
   • Bisection
   • Newton's method
   • Fixed-point iteration
   • Determining if an approximation is sufficiently accurate
2. Finite difference approximations and partial differential equations
   • Derivative approximation formulas
   • Explicit and implicit methods for the heat equation and related PDEs
3. Linear systems - Direct methods
   • Gaussian elimination
   • LU Decomposition and back substitution
   • Positive definite matrices and Choleski
   • Banded/sparse systems
4. Vector and matrix norms
5. Linear systems - Iterative methods
   • Jacobi's method
   • Gauss-Seidel
   • General matrix splitting

References:

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

1. Holomorphic (or Analytic) Functions of a Complex Variable
2. Cauchy-Riemann Conditions and Harmonic Functions
3. Elementary Complex Functions ( e^z, zⁿ, z^1/n, logz)
4. Complex Integration
5. Cauchy - Goursat Theorem
6. Cauchy Integral Formula
7. Morera's Theorem
8. Liouville's Theorem
9. Fundamental Theorem of Algebra
10. Maximum Principle
11. Taylor Series of Holomorphic Functions
12. Power Series as Holomorphic Functions
13. Meromorphic Functions
14. Laurent Series
15. Residues and Contour Integration
16. Mobius (or Linear Fractional) Transformations
17. Conformal Mapping
18. Entire Functions and Picard's Little Theorem
19. 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

1. Topological spaces
2. Interior, closure, boundary
3. Relative topology
4. Bases, subbases
5. Continuous functions
6. Homeomorphisms
7. Product spaces
8. Quotient spaces
9. Connectedness, path-connectedness
10. Compactness
11. Separation axioms

Differential Equations:

1. Power series solutions
2. Laplace transforms
3. Homogeneous and non-homogenous systems of linear differential equations
4. Fourier series
5. Matrix exponential

References:

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

• 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

References:

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