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##### 1: 26.15 Permutations: Matrix Notation
If $(j,k)\in B$, then $\sigma(j)\neq k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\ldots,(n,n)\}$. … For $(j,k)\in B$, $B\setminus[j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\ldots,n$. $B\setminus(j,k)$ denotes $B$ with the element $(j,k)$ removed. … Let $B=\{(j,j),(j,j+1)\>|\>1\leq j. …
##### 2: 14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. …
##### 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 $m$, $n\in\mathbb{Z}$ and $\delta_{\mu}$, $\delta_{\nu}\in(0,1)$. … The zeros of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ interlace those of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$. … $P^{\mu}_{\nu}\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds: … For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$) $P^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$. $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$, and at most one zero in the interval $(1,\infty)$ when $\nu<-1$.
##### 5: 4.37 Inverse Hyperbolic Functions
In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function $(t^{2}-1)^{1/2}$ assumes its principal value when $t\in(1,\infty)$. …
4.37.16 $\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.37.19 $\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$,
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 $(-\infty,1]$. …
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 6: 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 $R_{n}/(\lambda^{2}+\mu^{2})^{n/2}\to 0$ as $(\lambda,\mu)\to(0,0)$. $f(x,y)$ 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 $(a,b)\times(c,d)$, then … where $D$ is the image of $D^{*}$ under a mapping $(u,v)\to(x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …
##### 7: 18.40 Methods of Computation
Let $x^{\prime}\in(a,b)$. … Here $x(t,N)$ is an interpolation of the abscissas $x_{i,N},i=1,2,\dots,N$, that is, $x(i,N)=x_{i,N}$, allowing differentiation by $i$. …
18.40.9 $x(t,N)=\cfrac{x_{1,N}}{1+\cfrac{a_{1}(t-1)}{1+\cfrac{a_{2}(t-2)}{1+\cdots}}}% \frac{a_{N-1}(t-(N-1))}{1},$ $t\in(0,\infty)$,
The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\dots,N$. … 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 $x\in[-1,1]$ and $(-\infty,\infty)$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
##### 8: 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. …
##### 9: 1.4 Calculus of One Variable
If $f(x)$ is continuous at each point $c\in(a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. … If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c\in(a,b)$ such that …If $f^{\prime}(x)\geq 0$ ($\leq 0$) ($=0$) for all $x\in(a,b)$, 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 …
##### 10: 28.17 Stability as $x\to\pm\infty$
If all solutions of (28.2.1) are bounded when $x\to\pm\infty$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … However, if $\Im\nu\neq 0$, then $(a,q)$ always comprises an unstable pair. … For real $a$ and $q$ $(\neq 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. …