Anant's Official Faculty Picture

For Potential Graduate Students

I get far too many emails from potential graduate students who want to apply to Berkeley. I do not have the time to respond to everyone individually and so will put the responses to the most common queries here:

Challenge Problems

After reading the above, some of you still might want to contact me regarding joining the graduate program here. That is fine, but do not feel bad if I do not have a chance to respond to you. If you want to catch my eye, you have two choices. First, look through my online papers or preprints and see if you can find a bug. If you do, mention "paper bug" in the subject line and tell me what the error is. The other choice is to solve one or more of the following challenge problems and mention "challenge problem solution" in your subject line:
  1. Abbey and Beth are friends. So are Carl and Dan. They are having a little dispute about fairness and they ask you to resolve it.

    Whenever Carl and Dan go out to eat each Saturday, they take strict turns paying the tab. If Carl payed last time, Dan will pay this time and vice versa. But they don't bother about consciously keeping track of money and will order a big dish randomly from the menu and share it equally. Big dish prices are uniformly distributed on [5,10]

    Abbey and Beth take a different approach to fairness. They too go out to eat and take strict turns paying the bill. But they order small dishes separately. Whenever one of them calculates that she is currently more than $10 behind in her fair share of payments, she will order the salad for $1. Otherwise she randomly picks a dish from the small dish menu where the prices are uniformly distributed on [2.5,5].

    Carl and Dan believe that Abbey and Beth are being overly cautious and advise them to trust in the power of averaging to make things work out in the end. Abbey and Beth are skeptical and do not want to risk their friendship.

    Assume that if at any time the balance of payments becomes skewed by more than M times the maximum price of a dish in one direction, then unconscious feelings of being cheated will manifest and the friendship will grow distant.

  2. You are looking to detect a weak x(t) signal in a particular 2 MHz frequency band: [f - 1MHz, f + 1MHz]. You have access to a Nyquist-sampled version of y(t) = a x(t) + n(t) where n(t) is white gaussian noise of unit intensity and a is much smaller than 1. Your challenge is to give an algorithm to decide whether x(t) is present with a false alarm probability of at most 1/1000 and a probability of missed detection of at most 1/10. While working out the above problems, you are allowed to make whatever assumptions you feel are necessary that preserve the essential character of the problem.