中央研究院數學研究所 招考八十四年度研習員筆試試題 及錄取名單

中央研究院數學研究所

招考八十四年度研習員筆試試題 及錄取名單

試題:

1. (a) Let pi >0, qi >0, i = 1, · · · , n and ^Pn_i=1p_i =^Pn_i=1q_i = 1. Then

−

Xn i=1

p_ilog pi ≤ −

Xn i=1

p_ilog qi

with equality iff pi = qi for all i.

(b) Use (a) to prove the inequality between the arithmetic and geometric means: Let x1, · · · , x_n be arbitrary positive numbers, let a1, · · · , a_n > 0 and ^Pn_i=1a_i = 1. Then

x^a₁¹x^a2₂ · · · x^a_nⁿ ≤

Xn i=1

a_ix_i

with equality iff all xi are equal.

(c) Let f : [0, 1] −→ R be a continuous function. Prove the following continuous version of the inequality between the arithmetic and geometric means

exp

Z ₁

0 f(x)dx ≤

Z ₁

0 exp f (x)dx.

2. Find the conditions for α, β, such that

Z ∞ 0

e^−xdx

x^α+ x^β, converge.

1

2 數學傳播 十八卷二期 民83年6月

3. Prove or disprove (give a counterexample) the following statements:

(a) Σ^∞_n=1a_n<+∞, an≥0 implies limn→∞a_n = 0.

(b) ^R₀^∞f(x)dx < +∞ where f : [0, ∞) −→ R be a nonnegative continuous function implies limx→∞f(x) = 0.

4. Let fn be a bounded sequence of holomorphic functions on the unit disk △ in C such that

n→∞lim f_n^(k)(0) = 0 for all k.

Show that fn →0 uniformly on any compact subset of △.

5. Let M be an n × n matrix over C

(a) State a necessary and sufficient condition for M to be diagonalizable over C.

(b) Prove your statement.

錄取名單:

林昭廷: 國立台灣大學數學研究所 林秀穎: 國立清華大學數學研究所