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►Cycles of length one are fixed points.
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►An element of with fixed points, cycles of length
cycles of length
, where , is said to have cycle type
.
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►A transposition is a permutation that consists of a single cycle of length two.
…A permutation that consists of a single cycle of length
can be written as the composition of two-cycles (read from right to left):
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►Figure 19.3.6:
as a function of and for , .
…Its value tends to as by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as .
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►To code a message by this method, we replace each letter by two digits, say , , , , and divide the message into pieces of convenient length smaller than the public value .
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►Computer algebra systems use exact rational arithmetic with rational numbers , where and are multi-length integers.
During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length.
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►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)):
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