博士班資格考試科目

• 科目︰ 數值分析

參考書︰

1. J. M. Ortega : Numerical Analysis, a Second Course, SIAM, Philadephia, 1990.
2. H. R. Schwarz : Numerical Analysis, a Comprehensive Introduction, John Wiley & Sons, 1989.
範圍 :
1. Interpolation and approximation of functions.
2. Numerical integrations.
3. Solutions of linear equations.
4. Solutions of nonlinear systems.
5. Matrix eigenvalue problems.
6. Solution of least square problems.
7. Numerical ordinary differential equations.
8. Numerical partial differential equations.
9. Numerical optimation.
10. Error and stability analysis.

• 科目︰ 微分方程

Part I. Partial Differential Equations.
參考書︰

1. F. John, Partial Differential Equations, Springer-Verlag. (Chapter 1-4, 7)
2. E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications, Dover. (Chapter 1-9)
範圍 :
1. Initial value problems for first order partial differential equations.
2. Classification for second order equations.
3. The three basic partial differential equations (Laplace, Wave and Heat)-methods of solving, existence and uniqueness of solutions.
4. Maximum principles and Fourier expansion.
5. Separation of variables and Fourier expansion.
Part II. Ordinary Differential Equations.
參考書︰
1. G. Birkhoff and G. C. Rota : Ordinary Differential Equations, Wiley. (Chapter 1-6, 10, 11)
2. J. Hale : Ordinary Differential Equations, Interscience. (Chapter 1, 3, 6, 10)

• 科目︰ 矩陣理論

參考書︰

1. R. A. Horn and C. A. Johnson : Matrix Analysis.
2. G. Strang : Linear Algebra and Its Applications.
範圍 :
1. Relations : similarity, diagonalizable, unitary equivalence, unitary diagonalizable, congruence, Sylvester's law of inertia.
2. Canonical forms : Jordan and rational canonical forms, Shur normal form, ploar and singular value decompositions, LU, QR, and Cholesky factorizations.
3. Matrix norms : Frobenius norm, maximum column (row) sum norm, spectral norm, spectral radius, convergence of matrices.
4. Location of eigenvalues : Gersohgorin disk theorem, diagonally dominant matrices, perturbation of eigenvalues.
5. Hermitian Matrices : spectral theorem, Rayleigh quotient principle, Courant-Fischer minmax theorem, Weyl's theorem, Cauchy interlace theorem, majorization of eigenvalues.
6. Positive definite matrices : quadratic forms, principal monors, eigenvalues, pivots, positive semidefinite ordering.
7. Nonnegative matrices : induced partial ordering, Perron's theorem, Frobenius theorem.

• 科目︰ 離散數學

參考書︰

1. R. J. McEliece, R. B. Ash and C. Ash : Introduction to Discrete Mathematics.
2. R. L. Graham, D. E. Knuth and O. Patashnik : Concrete Mathematics. (Chapter 1-3, 7-9)

               1.  The theory of counting.

1. The principle of inclusion and exclusion.
2. Permutations involving indistinguishable objects.

1. Spanning trees.
2. Matching and flow.
3. vertex coloring.
4. Planar graphs.
5. Eulerian graphs and Hamiltonian graphs.

• 科目︰ 泛函分析

參考書︰

1. J. Conway : A Course in Functional Analysis, 2nd ed., Springer-Verlag. (Chapter 1-9)
範圍 :
1. Hahn-Banach extension theorem.
2. Open mapping theorem.
3. Closed graph theorem.
4. Uniformly boundedness principle.
5. Separation theorem.
6. Dualities.
7. Bannah-Alaoglu theorem.
8. Riesz representation theorem for bounded linear functions on Hilbert spaces, L^p space (1≦p＜∞) and C₀(X).
9. Bipolar theorem.
10. Basic concept of Banah spaces, Hilbert spaces, Banah algrbras, C^*-algrbras, von Neumann algrbras, and their classical machinaries including
1. bases, weak and weak^* topologies.
2. Gelfand representation.
3. Functional calculus (with analytic, continuous or integrable functions).
4. Gelfand-Naimark-Segal construction.
5. Compact operators and the spectral theory.

• 科目︰ 計算機原理

參考書︰

1. T. H. Cormen, C. E. Leiserson and R. L. Rivest : Introduction to Algorithms.
2. R. C. T. Lee, R. C. Chang and S. S. Tseng : Introduction to Algorithms : Fundamental and Advanced.
3. John E. Hopcroft and Jeffrey D. Ulllman : Introduction to Automata Theory, Languages and Computation.