Tuesday, April 01, 2008

april fools

perhaps this is just illustrates the inner geek in me more fully, but the following things have greatly amused me today:

       
  • NASA’s photo of the day has an awesome caption.
       
  • the Google empire has gone crazy... gmail offers a "send emails from the past" option, google calendar is offering a "wake up kit" and an "i’m feeling lucky" option to set up appointments with celebrities, youtube is rickrolling all the featured videos... and apparently they’re starting a colony on mars. hmmm... and google australia is letting you search the future with gDay.  the testimonials are funny.
       
  • my advisor, already famous for his online opinions, continues his almost-annual april fool’s opinion tradition.
       
  • and finally..., speaking of my advisor, yesterday he sent out the following email:


    Dear Math 640 students,

    I just finished posting two homework sets. The first one is due tomorrow (please slide them under my door), and the second one is due Thurs. I am rather disappointed at some of you who have been late handing-in the homework (and I commend those who always hand them in on time).

    Anyway, the homework problem set that is due tomorrow should be completed by tomorrow, 11:59pm, or else you would get zero, and be in danger of failing this class (i.e. getting a B).


    this is already suspicious, since we know he loves april fools pranks, but it was worth figuring out anyhow... the problem set he posted for today is:
    ~~~~~~~~~~~~

    First Homework Set for March 31, 2008 class, Due April 1, 2008
    [No extensions!]



    • Recall that for any set of non-negative integers A, mex(A) is
      the smallest  non-negative integer not in A. For example,
      mex({2,4,5})=0, mex({0,1,2,5,8})=3, etc.

      Define a sequence ai recursively by a1=2, and for i >= 1  by:
      ai=mex({0,1} U { j ar, j >= 1 , 1 <= r < i}),

      Prove the following properties of ai

      1. There are infinitely many i such that ai+1-ai=2
      2. Every even intger n >= 6 can be written as
        ai+aj, for some pos. integers i and j.
      3. Define a sequence F(n) by,
        F(ai1 ai1 ai2  ...air)=(-1)r if n can be expressed as a product of distinct ai’s , and 0 otherwise.
        Let G(n)=add(F(i), i=0..n)
        Prove that |G(n)| <= Cn.999, for some fixed constant C.

    • Remember that Euler’s pentagonal product
      eta(q)=(1-q)(1-q2)(1-q3) ... ,
      when expanded, has lots of 0-coefficients and the rest are 1 or -1.
      Condider the 24-th power of that
      eta(q)24=[(1-q)(1-q2)(1-q3) ...]24,
      and let’s call the coeff. of qn,  tau(n). Prove that tau(n) is never zero.

    ~~~~~~~~~~~~~~~
    thus, almost instantly, emails started flying between his ph.d. students as follows:
    ~~~~~~~~~~~

    is it an april fools joke? the homework that he posted looks hard. i don’t want to do it.
    -em

    ~~~~~~~~~~~

    i assume it is.  i was just wondering if it was equivalent to one of the clay problems or something equally ridiculous ;) why else would it be due on april 1?
    lara

    ~~~~~~~~~~~
    and finally...

    Don’t read below if you haven’t looked at the homework.  To make a buffer, here’s the Millenium Falcon:

                     c==o

                   _/____.._

            _.,--’" ||^ || "..z._

           /_/^ ___..||  || _/o.. "..-._

         _/  ]. L_| || .||  .._/_  . _..--._

        /_~7  _ . " ||. || /] .. ]. (_)  . "..--.

       |__7~.(_)_ []|+--+|/____T_____________L|

       |__|  _^(_) /^   __..____ _   _|

       |__| (_){_) J ]K{__ L___ _   _]

       |__| . _(_) ..v     /__________|________

       l__l_ (_). []|+-+-<..^   L  . _   - ---L|

        ..__..    __. ||^l  ..Y] /_]  (_) .  _,--’

          ..~_]  L_| || ... ...../~.    _,--’"

           .._.. . __/||  |..  ....-+-<’"

             "..---._|J__L|X o~~|[....     

                ..____/ ..___|[//     

                     ..--’   ..--+-’
    "Millenium Falcon" Modified Corellian YT-1300 Transport

    (No, I did not draw that - I don’t have the patience for ASCII art).

    It’s certainly April Fool’s Joke.  I didn’t recognize that a(n) is the nth prime at first, but now I see that his problems are:
          1. The Twin Prime Conjecture
          2. Goldbach’s conjecture
          3. His G is the Merten’s function (I had to look this up).  This result
          would be equivalent to the Riemann Hypothesis, apparently.

    The partition-related function is Ramanujan’s tau function.  That tau(n) is never zero is another unsolved conjecture.  It’s a shame - I was kind of hoping this one would be a ridiculously easy problem, but that everyone would be scared off by the first three.

    -Baxter



hooray for clever fun. ;)

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