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1: 26.13 Permutations: Cycle Notation
§26.13 Permutations: Cycle Notation
… ►is in cycle notation. Cycles of length one are fixed points. … ►For the example (26.13.2), this decomposition is given by … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by : .2: 26.2 Basic Definitions
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Cycle
►Given a finite set with permutation , a cycle is an ordered equivalence class of elements of where is equivalent to if there exists an such that , where and is the composition of with . …If, for example, a permutation of the integers 1 through 6 is denoted by , then the cycles are , , and . …3: 21.7 Riemann Surfaces
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►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann
surface. All compact Riemann surfaces can be obtained this
way.
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►On this surface, we choose
cycles (that is, closed oriented curves, each with at most a finite number of singular points) , , , such that their intersection indices satisfy
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►Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles
, , .
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4: 21.1 Special Notation
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positive integers. | |
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intersection index of and , two cycles lying on a closed surface. if and do not intersect. Otherwise gets an additive contribution from every intersection point. This contribution is if the basis of the tangent vectors of the and cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is . | |
line integral of the differential over the cycle . |
5: 4.13 Lambert -Function
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►For the definition of Stirling cycle numbers of the first kind see (26.13.3).
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4.13.10
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4.13.11
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6: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the number of permutations of with
cycles of length 1,
cycles of length 2, , and
cycles of length :
…(The empty set is considered to have one permutation consisting of no cycles.)
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7: Bibliography G
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General approach to few-cycle intense laser interactions with complex atoms.
Phys. Rev. A 76, pp. 053411.
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8: 3.6 Linear Difference Equations
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►If agreement is not within a prescribed tolerance the cycle is continued.
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9: 26.8 Set Partitions: Stirling Numbers
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denotes the Stirling number of the first kind: times the number of permutations of with exactly
cycles.
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