課程名稱︰分析導論優二 課程性質︰數學系大二必修 課程教師︰王振男 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015/03/31 考試時限（分鐘）：180 試題 : n n 1 1. (20%) Let S be an open set in |R . Assume that F: S → |R is a C mapping. Let x ∈ S and the Jacobian matrix DF(x ) is invertible. Show that there 0 0 exists an open set O satisfying x ∈ O ⊂ S such that 0 ∥F(u) - F(v)∥ ≧ c∥u - v∥ ∀u, v ∈ O (1) with some constant c > 0. Hint: A matrix A is invertible iff there exists a n positive constant k > 0 such that ∥Ah∥≧ k∥h∥ for all h ∈ |R . Show that if ∥B - A∥≦ k/2, then ∥Bh∥≧ (k/2)∥h∥. For F, use the mean value theorem. n n 2. (20%) A mapping F satisfies (1) is said to be stable in O. Let F: |R → |R 1 n -1 be a C mapping and be stable in |R . Show that F exists globally, i.e., F n n is a bijective map on |R . Hint: Show that F(|R ) is both open and closed in n n |R . Note that |R is connected. 2 3. (20%) This is a global implicit function theorem. We consider F: |R → |R a 1 C function. Assume that there exists some constant c > 0 such that F (x, y) y 2 1 ≧ c for all x, y ∈ |R . Show that there exists a unique C function g: |R → |R such that F(x, g(x)) = 0 for all x ∈ |R. Hint: Use the injectivity of a strictly monotone function and the usual implicit function theorem. n ×n 1 m 4. (20%) Let A = A(x) ∈ |R , where A is a C function of x ∈ |R . Assume m that for some x ∈ |R , λ(x ) is a simple eigenvalue of A(x ). Show that 0 0 0 1 there exists a C simple eigenvalue λ(x) of A(x) for x near x . Give an 0 ∞ example showing that even for a C matrix-valued function A(x), a non-simple 1 eigenvalue may not be C . 2 2 2 5. (20%) Find the maximum of (x x ...x ) under the restriction x + x + ... + 1 2 n 1 2 2 x = 1. Use the result to derive the following inequality, valid for positive n real numbers a , ..., a : 1 n a + ... + a 1/n 1 n (a ...a ) ≦ ───────. 1 n n -- 2 2 1 ψxavier13540 給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則 ╰x(0) = x° t ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三 次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。 --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.249.76 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1427818183.A.72B.html ※ 編輯: xavier13540 (140.112.249.76), 04/01/2015 14:40:15