cycle notation
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1: 26.13 Permutations: Cycle Notation
§26.13 Permutations: Cycle Notation
… ►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that immediately follows or, if is the last listed element of the cycle, then is the first element of the cycle. … ►is in cycle notation. …They are often dropped from the cycle notation. … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by : .2: 21.1 Special Notation
§21.1 Special Notation
►(For other notation see Notation for the Special Functions.) ►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 . |
3: 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 . …4: 4.13 Lambert -Function
§4.13 Lambert -Function
… ►Alternative notations are for , for , both previously used in this section, the Wright -function , which is single-valued, satisfies … ►For the definition of Stirling cycle numbers of the first kind see (26.13.3). … ►
4.13.10
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4.13.11
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5: 32.1 Special Notation
6: 6.1 Special Notation
§6.1 Special Notation
►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the argument. …7: 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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