# a posteriori

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##### 1: 3.2 Linear Algebra
The $p$-norm of a matrix $\mathbf{A}=[a_{jk}]$ is … Then we have the a posteriori error bound … If $\mathbf{A}$ is an $n\times n$ matrix, then a real or complex number $\lambda$ is called an eigenvalue of $\mathbf{A}$, and a nonzero vector $\mathbf{x}$ a corresponding (right) eigenvector, if …
###### §3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix
has the same eigenvalues as $\mathbf{A}$. …
##### 2: 26.16 Multiset Permutations
Let $S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}$ be the multiset that has $a_{j}$ copies of $j$, $1\leq j\leq n$. $\mathfrak{S}_{S}$ denotes the set of permutations of $S$ for all distinct orderings of the $a_{1}+a_{2}+\cdots+a_{n}$ integers. The number of elements in $\mathfrak{S}_{S}$ is the multinomial coefficient (§26.4) $\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}$. … The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with $S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}$ we have …
##### 4: 1.1 Special Notation
 $x,y$ real variables. … a testing function. … or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$. … determinant of the square matrix $\mathbf{A}$ trace of the square matrix $\mathbf{A}$ …
In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$, $a$ being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$.
##### 5: Alexander A. Its
###### Profile Alexander A. Its
Alexander A. Its (b. …  Belokolos, A. … Fokas, A.  A. …
##### 6: Sidebar 9.SB2: Interference Patterns in Caustics
A thin beam of light refracted by an irregularity in bathroom-window glass produces this image on a distant screen. The bright sharp-edged triangle is a caustic, that is a line of focused light. …
##### 7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
###### Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …
##### 8: 8.13 Zeros
• (a)

one negative zero $x_{-}(a)$ and no positive zeros when $1-2n;

• (b)

one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when $-2n.

• For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to-\infty$ see Tricomi (1950b), with corrections by Kölbig (1972b). …
• (a)

two zeros in each of the intervals $-2n when $x<0$;

• When $x>x_{n}^{*}$ a pair of conjugate trajectories emanate from the point $a=a_{n}^{*}$ in the complex $a$-plane. …
##### 9: Annie A. M. Cuyt
###### Profile Annie A. M. Cuyt
Annie A. M. Cuyt (b. …As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications. …In 2013 she was elected a life-time member of the Flemish Royal Society of the Sciences and Arts. In November 2015, Cuyt was named a Senior Associate Editor of the DLMF.
##### 10: 12.4 Power-Series Expansions
where the initial values are given by (12.2.6)–(12.2.9), and $u_{1}(a,z)$ and $u_{2}(a,z)$ are the even and odd solutions of (12.2.2) given by
12.4.3 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$
12.4.4 $u_{2}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(z+(a+\tfrac{3}{2})\frac{z^{3}}{3!}+(a+% \tfrac{3}{2})(a+\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$