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1: 26.13 Permutations: Cycle Notation
is ( 1 , 3 , 2 , 5 , 7 ) ( 4 ) ( 6 , 8 ) in cycle notation. …In consequence, (26.13.2) can also be written as ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) . … For the example (26.13.2), this decomposition is given by ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) = ( 1 , 3 ) ( 2 , 3 ) ( 2 , 5 ) ( 5 , 7 ) ( 6 , 8 ) . If j < k , then ( j , k ) is a product of 2 k - 2 j - 1 adjacent transpositions: …Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) = ( 2 , 3 ) ( 1 , 2 ) ( 4 , 5 ) ( 3 , 4 ) ( 2 , 3 ) ( 3 , 4 ) ( 4 , 5 ) ( 6 , 7 ) ( 5 , 6 ) ( 7 , 8 ) ( 6 , 7 ) : inv ( ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) ) = 11 .
2: 3.7 Ordinary Differential Equations
where A ( τ , z ) is the matrix
3.7.6 A ( τ , z ) = [ A 11 ( τ , z ) A 12 ( τ , z ) A 21 ( τ , z ) A 22 ( τ , z ) ] ,
and b ( τ , z ) is the vector … Let ( a , b ) be a finite or infinite interval and q ( x ) be a real-valued continuous (or piecewise continuous) function on the closure of ( a , b ) . … If q ( x ) is C on the closure of ( a , b ) , then the discretized form (3.7.13) of the differential equation can be used. …
3: 26.15 Permutations: Matrix Notation
If ( j , k ) B , then σ ( j ) k . The number of derangements of n is the number of permutations with forbidden positions B = { ( 1 , 1 ) , ( 2 , 2 ) , , ( n , n ) } . … For ( j , k ) B , B [ j , k ] denotes B after removal of all elements of the form ( j , t ) or ( t , k ) , t = 1 , 2 , , n . B ( j , k ) denotes B with the element ( j , k ) removed. … Let B = { ( j , j ) , ( j , j + 1 ) |  1 j < n } { ( n , n ) , ( n , 1 ) } . …
4: 14.27 Zeros
P ν μ ( x ± i 0 ) (either side of the cut) has exactly one zero in the interval ( - , - 1 ) if either of the following sets of conditions holds: …For all other values of the parameters P ν μ ( x ± i 0 ) has no zeros in the interval ( - , - 1 ) . …
5: 32.4 Isomonodromy Problems
32.4.4 A ( z , λ ) = ( 4 λ 4 + 2 w 2 + z ) [ 1 0 0 - 1 ] - i ( 4 λ 2 w + 2 w 2 + z ) [ 0 - i i 0 ] - ( 2 λ w + 1 2 λ ) [ 0 1 1 0 ] ,
32.4.5 B ( z , λ ) = ( λ + w λ ) [ 1 0 0 - 1 ] - i w λ [ 0 - i i 0 ] .
32.4.6 A ( z , λ ) = - i ( 4 λ 2 + 2 w 2 + z ) [ 1 0 0 - 1 ] - 2 w [ 0 - i i 0 ] + ( 4 λ w - α λ ) [ 0 1 1 0 ] ,
32.4.7 B ( z , λ ) = [ - i λ w w i λ ] .
32.4.8 A ( z , λ ) = [ 1 4 z 0 0 - 1 4 z ] + [ - 1 2 θ u 0 u 1 1 2 θ ] 1 λ + [ v 0 - 1 4 z - v 1 v 0 ( v 0 - 1 2 z ) / v 1 1 4 z - v 0 ] 1 λ 2 ,
6: 26.6 Other Lattice Path Numbers
D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
7: 14.16 Zeros
where m , n and δ μ , δ ν ( 0 , 1 ) . … The zeros of Q ν μ ( x ) in the interval ( - 1 , 1 ) interlace those of P ν μ ( x ) . … P ν μ ( x ) has exactly one zero in the interval ( 1 , ) if either of the following sets of conditions holds: … For all other values of μ and ν (with ν - 1 2 ) P ν μ ( x ) has no zeros in the interval ( 1 , ) . Q ν μ ( x ) has no zeros in the interval ( 1 , ) when ν > - 1 , and at most one zero in the interval ( 1 , ) when ν < - 1 .
8: 4.37 Inverse Hyperbolic Functions
In (4.37.2) the integration path may not pass through either of the points ± 1 , and the function ( t 2 - 1 ) 1 / 2 assumes its principal value when t ( 1 , ) . … It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on ( - , 1 ] . …
9: 24.1 Special Notation
j , k , , m , n integers, nonnegative unless stated otherwise.
( k , m ) greatest common divisor of m , n .
( k , m ) = 1 k and m relatively prime.
10: Jim Pitman
Pitman has devoted much effort to promote the development of open access resources in the fields of probability and statistics. As a member of the Executive Committee of the Institute of Mathematical Statistics (IMS) from 2005 to 2008, he guided the IMS through implementation of a policy to promote open access to all of its professional journals, through systematic deposit of peer-reviewed final versions of all articles on arXiv. … He has published extensively on probability, stochastic processes, combinatorics and is a champion for open access to resources in mathematics. …