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1: 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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2: 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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3: 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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4: 14.16 Zeros
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►where , and , .
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►The zeros of in the interval interlace those of .
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has exactly one zero in the interval if either of the following sets of conditions holds:
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►For all other values of and (with ) has no zeros in the interval .
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has no zeros in the interval when , and at most one zero in the interval when .
5: 4.37 Inverse Hyperbolic Functions
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►In (4.37.2) the integration path may not pass through either of the points , and the function assumes its principal value when .
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4.37.16
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4.37.19
,
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►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 .
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4.37.24
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6: 4.3 Graphics
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►In the labeling of corresponding points is a real parameter that can lie anywhere in the interval .
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7: 1.5 Calculus of Two or More Variables
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►A function is continuous at a point
if
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►where and its partial derivatives on the right-hand side are evaluated at , and as .
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has a local minimum (maximum) at if
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►Moreover, if are finite or infinite constants and is piecewise continuous on the set , then
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►where is the image of under a mapping which is one-to-one except perhaps for a set of points of area zero.
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8: 18.40 Methods of Computation
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►Let .
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►Here is an interpolation of the abscissas , that is, , allowing differentiation by .
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18.40.9
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►The PWCF is a minimally oscillatory algebraic interpolation of the abscissas .
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►Further, exponential convergence in , 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 for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
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9: Jim Pitman
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►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.
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►He has published extensively on probability, stochastic processes, combinatorics and is a champion for open access to resources in mathematics.
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10: 1.4 Calculus of One Variable
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►If is continuous at each point , then is continuous on the interval
and we write .
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►If is continuous on and differentiable on , then there exists a point such that
…If () () for all , then is nondecreasing (nonincreasing) (constant) on .
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►Then for continuous on ,
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►A function is convex on if
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