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
hold? Justify your answer.
Question 10 [10 marks]. Let for be real numbers such that for any . Define