推 t0444564 : 已收錄 05/12 23:09
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）︰2015/05/12
考試時限（分鐘）：180
試題 :
1. Evaluate the following integrals.
(a) (10%) ∬ [x+y] dA, where Q = [0, 2] ×[0, 2], and [t] is the greatest
Q
integer ≦ t.
(b) (10%) ∬ f(x, y) dA, where Q = [0, 1] ×[0, 1] and
Q
╭ 0, if at least one of x, y is irrational,
f(x, y) = ╯
╰ 1/n, if y is rational and x = m/n,
where m and n are relatively prime integers with n > 0.
2. (15%) Give an example to show that φ(f(x)) may not be measurable if φ and f
are measurable.
n
3. (20%) Let f be an extended real-valued measurable function defined on |R .
Show that there exists a Borel function g such that f = g a.e. Hint: First
consider f ≧ 0 and use the approximation of simple functions. For the
+ -
general case, we write f = f - f .
4. (25%) Show that there exists a measurable set which is not Borel. You could
use the fact that a properly-defined inverse of Cantor function, P: [0, 1] →
[0, 1], is measurable, 1-1, and the values of P lie entirely in the Cantor
set. Also, you may need the fact that if f is a real measurable function on
-1
|R and B is a Borel set, then f (B) is a measurable set.
1 1
5. (20%) This is a simple form of Sard's theorem. Let f: I → |R be a C
1
function, where I is an open set of |R . Denote C = {x∈I: f'(x) = 0}. Show
that f(C) is of measure zero. Hint: Let K be a closed interval contained in
I. consider the function g: K ×K → |R
╭ f(y) - f(x) - f'(x)(y-x)
│ ────────────, for x ≠ y,
g(x, y) = ╯ y - x
│
╰ 0, for x ＝ y.
Then show that g is uniformly continuous. Thus for any ε > 0 there exists
δ > 0 such that |g(x, y)| < ε whenever |x-y| < δ. partition K into
subintervals with norm less then δ. Then consider those subintervals that
have nonempty intersection with C.
註: 第四題原題目敘述並不清晰，這裡的題目敘述是修改後的。
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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), 05/12/2015 13:21:59