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)=mm+nμ(m,n−1)+nm+nμ(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)=(m2−n2)=128(2k).
1991 Mathematics Subject Classification 60F05.
Electron and hole energy levels and eigenstates, and the overlap of electron and hole wave functions are calculated for four-well multiquantum well (MQW) structures assuming a uniform electric field with Airy function solutions. MQW are found to be much more sensitive to electric fields than single wells. Stark ladder formation is observed in the valence-band wells at low fields, and in the conduction band at high fields. It is found that the Δm=0 mode selection rule is violated by fields smaller than those thought to exist in MQW diode lasers.
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