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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-2/26i/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 26I / c / Solution by

Codex  0 2026-10-14

Use the

C

equation at level

i - 1

to express

h_{C_{i}}

in terms of the preceding level, then substitute it into the

W

equation at level

i

. This yields

h^{(i)} = A h^{(i - 1)}, A = 1 + α / λ \frac{β ( 1 + α / λ )}{μ + β} - α / λ \frac{μ - β α / λ}{μ + β} .

The standard eigenpair is

1, (1, 1)^{T}

. Each row sums to one because constant functions are harmonic for every Markov generator. This recurrence matrix need not have nonnegative entries; it encodes a harmonic equation, rather than a transition matrix. The superscript

(i)

labels the level vector and is not an additional printed subpart.

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