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1: 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 ) } . …
2: 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 ) . …
3: 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 ) . …
4: 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 .
5: 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 , ) . …
4.37.16 arcsinh z = ln ( ( z 2 + 1 ) 1 / 2 + z ) , z / i ( , 1 ) ( 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 ] . …
6: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and e x . Parallel tangent lines at ( 1 , 0 ) and ( 0 , 1 ) make evident the mirror symmetry across the line y = x , demonstrating the inverse relationship between the two functions. Magnify
In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …
7: 1.5 Calculus of Two or More Variables
A function f ( x , y ) is continuous at a point ( a , b ) if … where f and its partial derivatives on the right-hand side are evaluated at ( a , b ) , and R n / ( λ 2 + μ 2 ) n / 2 0 as ( λ , μ ) ( 0 , 0 ) . f ( x , y ) has a local minimum (maximum) at ( a , b ) if … Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then … where D is the image of D under a mapping ( u , v ) ( x ( u , v ) , y ( u , v ) ) which is one-to-one except perhaps for a set of points of area zero. …
8: 18.40 Methods of Computation
Let x ( a , b ) . … Here x ( t , N ) is an interpolation of the abscissas x i , N , i = 1 , 2 , , N , that is, x ( i , N ) = x i , N , allowing differentiation by i . …
18.40.9 x ( t , N ) = x 1 , N 1 + a 1 ( t 1 ) 1 + a 2 ( t 2 ) 1 + a N 1 ( t ( N 1 ) ) 1 , t ( 0 , ) ,
The PWCF x ( t , N ) is a minimally oscillatory algebraic interpolation of the abscissas x i , N , i = 1 , 2 , , N . … Further, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
9: 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. …
10: 1.4 Calculus of One Variable
If f ( x ) is continuous at each point c ( a , b ) , then f ( x ) is continuous on the interval ( a , b ) and we write f C ( a , b ) . … If f ( x ) is continuous on [ a , b ] and differentiable on ( a , b ) , then there exists a point c ( a , b ) such that …If f ( x ) 0 ( 0 ) ( = 0 ) for all x ( a , b ) , then f is nondecreasing (nonincreasing) (constant) on ( a , b ) . … Then for f ( x ) continuous on ( a , b ) , … A function f ( x ) is convex on ( a , b ) if …