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1: 26.13 Permutations: Cycle Notation
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►is in cycle notation.
…In consequence, (26.13.2) can also be written as .
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►For the example (26.13.2), this decomposition is given by
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►If , then is a product of adjacent transpositions:
…Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by : .
2: 7.23 Tables
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Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Finn and Mugglestone (1965) includes the Voigt function , , , 6S.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
3: 14.27 Zeros
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(either side of the cut) has exactly one zero in the interval
if either of the following sets of conditions holds:
…For all other values of the parameters has no zeros in the interval
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4: 26.15 Permutations: Matrix Notation
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►If , then .
The number of derangements of is the number of permutations with forbidden positions .
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►For , denotes after removal of all elements of the form or , .
denotes with the element removed.
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►Let .
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5: 14.16 Zeros
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§14.16(ii) Interval
… ►The zeros of in the interval interlace those of . … ►§14.16(iii) Interval
► has exactly one zero in the interval if either of the following sets of conditions holds: … ► has no zeros in the interval when , and at most one zero in the interval when .6: 3.7 Ordinary Differential Equations
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►where is the matrix
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3.7.6
►and is the vector
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►Let be a finite or infinite interval and be a real-valued continuous (or piecewise continuous) function on the closure of .
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►If is on the closure of , then the discretized form (3.7.13) of the differential equation can be used.
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7: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
… ►Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas. … ►For proofs of these results and further information see Walker (2003).8: 1.4 Calculus of One Variable
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►Suppose is defined on .
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►A function is convex on if
…for any , and .
…A similar definition applies to closed intervals
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9: 32.4 Isomonodromy Problems
10: 26.6 Other Lattice Path Numbers
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is the number of paths from to that are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line and are composed of directed line segments of the form , , or .
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is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form or , and for which there are exactly occurrences at which a segment of the form is followed by a segment of the form .
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is the number of paths from to that stay on or above the diagonal and are composed of directed line segments of the form , , or .
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