| 科目 | 參考書籍及範圍 |
| 分析通論 | 1. Measure, Intergration, and Functional Analysis (by Ash)
2. Real Analysis (by Royden)
3. Real and Complex Analysis (by Rudin)
Topics:
(i) measure space and measurable functions
(ii) Lebesgue integration (including Riemann-Stieltjes and Lebesgue-Stieltjes integrals)
(iii) differentiation and Radon-Nikodym
(iv) convegence concepts and convergence theorems
(v) L^p spces and basic inequalities such as Minkowski inequality, Hölder inequality, Jensen's inequality etc.
(vi) fundamentals of functional analysis on normed linear space and applications to real analysis |
| 代數通論 | Algebra (by Hungerford) |
| 機率論 | 1. A Course in Probability Theory (by Kai Lai Chung)
2. Probability Theory (by Yuan Shih Chow and Henry Teicher)
(i) distribution funcation
(ii) classes of sets, measure and probability spaces
(iii) random variable, expectation, independence
(iv) convergence concepts
(v) law of large numbers, random series
(vi) characteristic function
(vii) conditional expectation, conditional independence, introduction to martingales |
| 數理統計 | 1. Theory of Point Estimation (by Lehmenn)
2. Testing Statistical Hupotheses (by Lehmann)
Topics:
(i) group families, exponential families, sufficient statistics, completeness
(ii) UMVU estimators, performance of the estimators, the information inequality
(iii) location-scale families, the principle of equivariance
(iv) Bayes estimation, minimax estimation, minimaxity and admissibility
(v) convergence in probability and in law, large-sample comparisons of estimators, the median as an estimator of location, trimmed mean
(vi) asymptotic efficiency, efficient likelihood estimations
(vii) the Neyman-Pearson fundamental lemmas, distributions with monotones likelihood ratio
(viii) unbiasedness for hypothesis testing, UMP unbiased test
(ix) confidence sets, unbiased confidence sets, Bayes confidence sets
(x) symmetry and invariance, maximal invariants, most powerful invariant test |
| 泛函分析 | 1. Real Analysis (by Rudin)
2. Functional Analysis (by Rudin)
3. Introduction to Functional Analysis (by Taylor and Lay)
4. Functional Analysis (by Yosida)
Topics:
(i) fundamentals of topological linear spaces (including Hahn-Banach theorem, open mapping theorem, closed graph theorem, uniform bounded principle and fixed point theorem etc.)
(ii) Fourier transform and its applications
(iii) basic theory of bounded linear operators
(iv) compact operators and spectral theory (including trace class operator and Hilbert-Schmidt operator on Hilbert spaces)
(v) Spectral mapping theorems
(vi) fundamentals of unbounded operators |
| 代數拓樸 | Elements of Algebraic Topology (by Munkres) |
| 微分方程 | Partial Differential Equations (by Fritz John) |
| 數值分析 | Numerical Analysis (by Burden and Faires) ch.1-ch.9 |
| 數學規劃 | 1. Linear Optimization and Extensions Theory and Algorithms (by Fang and Puthenpura)
2. Linear and Nonlinear Programming (by Luenberger)
3. Nonilinear Programming Theory and Algorithms (by Bazaraa)
Topics:
(i) simplex method and interior point methods
(ii) complexity analysis and the ellipoid methods
(iii) convex analysis
(iv) duality and KKT conditions
(v) contraint qualifications and the saddle point theory
(vi) unconstrained and contrained minimizations |