I have just passed QE held in August 2009 for my first attendance, I hereby show you the analysis paper

**Question 1** [10 marks]. Suppose and are both measurable functions on the interval such that for all

Assume that and both are monotone decreasing and continuous from left. Can you conclude that for all ? Give the reason to support your answer.

**Question 2** [10 marks]. Compute the volume of the region bounded by

where the determinant of the matrix is NOT equal to zero.

**Question 3** [10 marks]. Let be a measureable set in with finite measure. Suppose is a real valued continuous function on such that for almost every , is a continuous function of and for every real number , is measurable function of . If is a sequence of measurable functions on that converges to in measure, show that converges to in measure.

**Question 4** [10 marks]. Find the function

if is a constant. Justify your answer.

**Question 5** [10 marks]. Compute the intergal

where is a constant.

**Question 6** [10 marks]. Supposet is a holomorphic function on the complex plane . If locally keeps the area invariant, what will the function be?

**Question 7** [10 marks]. Is there an analytic function on (unit disk in the complex plane with center ) such that for with and ? If so, find such an . Is it unique?

**Question 8** [10 marks]. Let be two positive integers and and be open subsets in and respectively. Does there exist a map such that and the inverse of are both ? Justify your answer.

**Question 9** [10 marks]. Is there a square integrable function on such that both inequalities

and

hold? Justify your answer.

**Question 10** [10 marks]. Let for be real numbers such that for any . Define

Show that

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