NetNews Usenet Archive 1992 #19/NN_1992_19.iso/spool/sci/math/10641

Internet Message Format | 1992-08-27 | 1.7 KB

Path: sparky!uunet!zaphod.mps.ohio-state.edu!not-for-mail
From: [e-mail] (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Elementary solution needed: harmonic bi-sequences
Date: 27 Aug 1992 14:29:13 -0400
Organization: The Ohio State University, Dept. of Math.
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References: <[e-mail]>
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In article <[e-mail]> [e-mail] (Andrei Prasolov) writes:
>   There is a problem which seems to have a very simple solution.
>I remember that I heard one in my youth. That solution used
>the epsilon-delta language only. However, I cannot recall that.
>
>DEF. Let`s call a bi-sequence (a(i,j), i,j from Z) HARMONIC if for all i,j
>
>(*)       a(i,j) = 1/4 (a(i,j+1) + a(i,j-1) + a(i+1,j) + a(i-1,j)).
>
>PROBLEM. Prove that any bounded (|a(i,j)| < C) harmonic bi-sequence is constant.
>

Here is a proof that I like.  Whether you consider it "elementary" depends
on your background.

The symmetric random walk on the two-dimensional integer lattice Z^2
is recurrent: it visits each node repeatedly (with probability one).
Apply the function a to that random walk.  The result is (by (*)) a
martingale.  But a bounded martingale converges (almost surely),
while _this_ bounded martingale runs repeatedly through all the values of
the function a.  Therefore a is constant.

So maybe it's not elementary, but it is short and cute.

--
  Gerald A. Edgar                Internet:  [e-mail]
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: [phone] (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)