[分析] 幾題高微

[paperbattle (跳針猴王)]

1.請問要如何造一個bijective map f:(0,1)->[1,0]

2.Let a_0 = 0 , a_n+1 = 1/(2+a_n) for all n>0

已經知道這數列bounded 但不是monotone

如何證明a_n收斂?

3.Let T_A (X) = AX - XA  , 其中A是可對角化矩陣 A和X都是n階方陣

證明T_A可對角化

4.Let u:R^2→R be analytic and nonconstant.
Suppose u_y=u_xx for x,y in R.
Can the function u have a local maximum point at (x_0,y_0) in R^2?
Prove or disprove your answer.

請高手指點 謝謝

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