[試題] 103下 王振男 分析導論優二 第一次期中考

[xavier13540 (柊 四千)]

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。

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※ 編輯: xavier13540 (140.112.249.76), 04/01/2015 14:40:15
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