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The Ok Corral and the Power of the Law (A Curious Poisson‐Kernel Formula for a Parabolic Equation)
Author(s) -
Williams David,
Mcilroy Paul
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397004062
Subject(s) - mathematics , combinatorics , kernel (algebra) , boundary (topology) , mathematical analysis , mathematics subject classification
Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ( m , n ) be the expected number of survivors. Clearly, we have boundary conditions: μ ( m , 0 ) = m , μ ( 0 , n ) = n .We also have the equation μ ( m , n ) = m m + n μ ( m , n − 1 ) + n m + n μ ( m − 1 , n ) ( m , n ⩾ 1 ) .This is because the probability that the first successful shot is made by the side with m gunmen is m /( m + n ). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which m = 8192 + k , n = 8192 − k , d ( m , n ) = ( m 2 − n 2 ) = 128 ( 2 k ) .1991 Mathematics Subject Classification 60F05.