diff --git a/source/numerics.tex b/source/numerics.tex index a49328e4e0..28e606e6b6 100644 --- a/source/numerics.tex +++ b/source/numerics.tex @@ -2980,7 +2980,7 @@ of \state{x}{i} consists of the values of - $X_{i-n}, \ldots, X_{i-1}$, + $X_{i-n}, \dotsc, X_{i-1}$, in that order. \indexlibrary{\idxcode{mersenne_twister_engine}!constructor}% @@ -2991,7 +2991,7 @@ \begin{itemdescr} \pnum\effects Constructs a \tcode{mersenne_twister_engine} object. Sets $X_{-n}$ to $\tcode{value} \bmod 2^w$. - Then, iteratively for $i = 1\!-\!n,\ldots,-1$, + Then, iteratively for $i = 1\!-\!n,\dotsc,-1$, sets $X_i$ to \[% @@ -3018,7 +3018,7 @@ and $a$ an array (or equivalent) of length $ n \cdot k $, invokes \tcode{q.generate($a+0$, $a+n \cdot k$)} - and then, iteratively for $i = -n,\ldots,-1$, + and then, iteratively for $i = -n,\dotsc,-1$, sets $X_i$ to $ \left(\sum_{j=0}^{k-1}a_{k(i+n)+j} \cdot 2^{32j} \right) \bmod 2^w $. Finally, @@ -3135,7 +3135,7 @@ \indextext{\idxcode{subtract_with_carry_engine}!textual representation}% \indextext{textual representation!\idxcode{subtract_with_carry_engine}} consists of the values of - $X_{i-r}, \ldots, X_{i-1}$, + $X_{i-r}, \dotsc, X_{i-1}$, in that order, followed by $c$. @@ -3147,7 +3147,7 @@ \begin{itemdescr} \pnum\effects Constructs a \tcode{subtract_with_carry_engine} object. Sets the values of - $ X_{-r}, \ldots, X_{-1} $, + $ X_{-r}, \dotsc, X_{-1} $, in that order, as specified below. If $X_{-1}$ is then $0$, sets $c$ to $1$; @@ -3161,7 +3161,7 @@ 40014u,0u,2147483563u> e(value == 0u ? default_seed : value); \end{codeblock} Then, to set each $X_k$, - obtain new values $ z_0, \ldots, z_{n-1} $ + obtain new values $ z_0, \dotsc, z_{n-1} $ from $n = \lceil w/32 \rceil $ successive invocations of \tcode{e} taken modulo $2^{32}$. Set $X_k$ to $ \left( \sum_{j=0}^{n-1} z_j \cdot 2^{32j}\right) \bmod m$. @@ -3184,7 +3184,7 @@ and $a$ an array (or equivalent) of length $ r \cdot k $, invokes \tcode{q.generate($a+0$, $a+r \cdot k$)} - and then, iteratively for $i = -r, \ldots, -1$, + and then, iteratively for $i = -r, \dotsc, -1$, sets $X_i$ to $ \left(\sum_{j=0}^{k-1}a_{k(i+r)+j} \cdot 2^{32j} \right) \bmod m $. If $X_{-1}$ is then $0$, @@ -3604,7 +3604,7 @@ each constructor% \indexlibrary{\idxcode{shuffle_order_engine}!constructor} that is not a copy constructor -initializes $\tcode{V[0]}, \ldots, \tcode{V[k-1]}$ and $Y$, +initializes $\tcode{V[0]}, \dotsc, \tcode{V[k-1]}$ and $Y$, in that order, with values returned by successive invocations of \tcode{e()}.% \indextext{random number generation!engines|)} @@ -3879,7 +3879,7 @@ returns $0.0$. Otherwise, returns an entropy estimate\footnote{If a device has $n$ states whose respective probabilities are - $ P_0, \ldots, P_{n-1} $, + $ P_0, \dotsc, P_{n-1} $, the device entropy $S$ is defined as\\ $ S = - \sum_{i=0}^{n-1} P_i \cdot \log P_i $.} for the random numbers returned by \tcode{operator()}, @@ -4050,7 +4050,7 @@ \item With $m$ as the larger of $s + 1$ and $n$, transform the elements of the range: - iteratively for $ k = 0, \ldots, m-1 $, + iteratively for $ k = 0, \dotsc, m-1 $, calculate values \begin{eqnarray*} r_1 & = & @@ -4077,7 +4077,7 @@ \item Transform the elements of the range again, beginning where the previous step ended: - iteratively for $ k = m, \ldots, m\!+\!n\!-\!1 $, + iteratively for $ k = m, \dotsc, m\!+\!n\!-\!1 $, calculate values \begin{eqnarray*} r_3 & = & @@ -4195,7 +4195,7 @@ \pnum\effects Invokes \tcode{g()} $k$ times - to obtain values $ g_0, \ldots, g_{k-1} $, respectively. + to obtain values $ g_0, \dotsc, g_{k-1} $, respectively. Calculates a quantity \[ S = \sum_{i=0}^{k-1} (g_i - \tcode{g.min()}) @@ -5782,7 +5782,7 @@ \indextext{discrete probability function!\idxcode{discrete_distribution}}% \indextext{\idxcode{discrete_distribution}!discrete probability function}% \[% - P(i\,|\,p_0,\ldots,p_{n-1}) + P(i\,|\,p_0,\dotsc,p_{n-1}) = p_i \; \mbox{.} \] @@ -5790,7 +5790,7 @@ \pnum Unless specified otherwise, the distribution parameters are calculated as: - $p_k = {w_k / S} \; \mbox{ for } k = 0, \ldots, n\!-\!1$ , + $p_k = {w_k / S} \; \mbox{ for } k = 0, \dotsc, n\!-\!1$ , in which the values $w_k$, commonly known as the \techterm{weights}% @@ -5910,7 +5910,7 @@ let $ w_0 = 1 $. Otherwise, let $ w_k = \tcode{fw}(\tcode{xmin} + k \cdot \delta + \delta / 2) $ - for $ k = 0, \ldots, n\!-\!1 $. + for $ k = 0, \dotsc, n\!-\!1 $. \pnum\complexity The number of invocations of \tcode{fw} shall not exceed $n$. @@ -5925,7 +5925,7 @@ \pnum\returns A \tcode{vector} whose \tcode{size} member returns $n$ and whose $ \tcode{operator[]} $ member returns $p_k$ - when invoked with argument $k$ for $k = 0, \ldots, n\!-\!1 $. + when invoked with argument $k$ for $k = 0, \dotsc, n\!-\!1 $. \end{itemdescr} @@ -5947,7 +5947,7 @@ \indextext{probability density function!\idxcode{piecewise_constant_distribution}}% \indextext{\idxcode{piecewise_constant_distribution}!probability density function}% \[% - p(x\,|\,b_0,\ldots,b_n,\;\rho_0,\ldots,\rho_{n-1}) + p(x\,|\,b_0,\dotsc,b_n,\;\rho_0,\dotsc,\rho_{n-1}) = \rho_i \; \mbox{,} \mbox{ for } b_i \le x < b_{i+1} @@ -5961,13 +5961,13 @@ \indextext{interval boundaries!\idxcode{piecewise_constant_distribution}}% , shall satisfy the relation $ b_i < b_{i+1} $ -for $i = 0, \ldots, n\!-\!1 $. +for $i = 0, \dotsc, n\!-\!1 $. Unless specified otherwise, the remaining $n$ distribution parameters are calculated as: \[% \rho_k = \; \frac{w_k}{S \cdot (b_{k+1}-b_k)} - \; \mbox{ for } k = 0, \ldots, n\!-\!1, + \; \mbox{ for } k = 0, \dotsc, n\!-\!1, \] in which the values $w_k$, commonly known as the \techterm{weights}% @@ -6089,10 +6089,10 @@ and $ b_1 = 1 $. Otherwise, let $\bigl[\tcode{bl.begin()}, \tcode{bl.end()}\bigr)$ - form a sequence $ b_0, \ldots, b_n $, + form a sequence $ b_0, \dotsc, b_n $, and let $ w_k = \tcode{fw}\bigl(\bigl(b_{k+1} + b_k\bigr) / 2\bigr) $ - for $ k = 0, \ldots, n\!-\!1 $. + for $ k = 0, \dotsc, n\!-\!1 $. \pnum\complexity The number of invocations of \tcode{fw} shall not exceed $n$. @@ -6120,8 +6120,8 @@ \pnum\effects Constructs a \tcode{piecewise_constant_distribution} object with parameters taken or calculated from the following values: - Let $ b_k = \tcode{xmin} + k \cdot \delta $ for $ k = 0, \ldots, n $, - and $ w_k = \tcode{fw}(b_k + \delta / 2) $ for $ k = 0, \ldots, n\!-\!1 $. + Let $ b_k = \tcode{xmin} + k \cdot \delta $ for $ k = 0, \dotsc, n $, + and $ w_k = \tcode{fw}(b_k + \delta / 2) $ for $ k = 0, \dotsc, n\!-\!1 $. \pnum\complexity The number of invocations of \tcode{fw} shall not exceed $n$. @@ -6136,7 +6136,7 @@ \pnum\returns A \tcode{vector} whose \tcode{size} member returns $n + 1$ and whose $ \tcode{operator[]} $ member returns $b_k$ - when invoked with argument $k$ for $k = 0, \ldots, n $. + when invoked with argument $k$ for $k = 0, \dotsc, n $. \end{itemdescr} \indexlibrarymember{densities}{piecewise_constant_distribution}% @@ -6148,7 +6148,7 @@ \pnum\returns A \tcode{vector} whose \tcode{size} member returns $n$ and whose $ \tcode{operator[]} $ member returns $\rho_k$ - when invoked with argument $k$ for $k = 0, \ldots, n\!-\!1 $. + when invoked with argument $k$ for $k = 0, \dotsc, n\!-\!1 $. \end{itemdescr} @@ -6170,7 +6170,7 @@ \indextext{probability density function!\idxcode{piecewise_linear_distribution}}% \indextext{\idxcode{piecewise_linear_distribution}!probability density function}% \[% - p(x\,|\,b_0,\ldots,b_n,\;\rho_0,\ldots,\rho_n) + p(x\,|\,b_0,\dotsc,b_n,\;\rho_0,\dotsc,\rho_n) = \rho_i \cdot {\frac{b_{i+1} - x}{b_{i+1} - b_i}} + \rho_{i+1} \cdot {\frac{x - b_i}{b_{i+1} - b_i}} \; \mbox{,} @@ -6185,10 +6185,10 @@ \indextext{interval boundaries!\idxcode{piecewise_linear_distribution}}% , shall satisfy the relation $ b_i < b_{i+1} $ -for $i = 0, \ldots, n\!-\!1 $. +for $i = 0, \dotsc, n\!-\!1 $. Unless specified otherwise, the remaining $n+1$ distribution parameters are calculated as -$ \rho_k = {w_k / S} \; \mbox{ for } k = 0, \ldots, n $, +$ \rho_k = {w_k / S} \; \mbox{ for } k = 0, \dotsc, n $, in which the values $w_k$, commonly known as the \techterm{weights at boundaries}% \indextext{\idxcode{piecewise_linear_distribution}!weights at boundaries}% @@ -6310,10 +6310,10 @@ and $ b_1 = 1 $. Otherwise, let $\bigl[\tcode{bl.begin(),} \tcode{bl.end()}\bigr)$ - form a sequence $ b_0, \ldots, b_n $, + form a sequence $ b_0, \dotsc, b_n $, and let $ w_k = \tcode{fw}(b_k) $ - for $ k = 0, \ldots, n $. + for $ k = 0, \dotsc, n $. \pnum\complexity The number of invocations of \tcode{fw} shall not exceed $n+1$. @@ -6341,8 +6341,8 @@ \pnum\effects Constructs a \tcode{piecewise_linear_distribution} object with parameters taken or calculated from the following values: - Let $ b_k = \tcode{xmin} + k \cdot \delta $ for $ k = 0, \ldots, n $, - and $ w_k = \tcode{fw}(b_k) $ for $ k = 0, \ldots, n $. + Let $ b_k = \tcode{xmin} + k \cdot \delta $ for $ k = 0, \dotsc, n $, + and $ w_k = \tcode{fw}(b_k) $ for $ k = 0, \dotsc, n $. \pnum\complexity The number of invocations of \tcode{fw} shall not exceed $n+1$. @@ -6357,7 +6357,7 @@ \pnum\returns A \tcode{vector} whose \tcode{size} member returns $n + 1$ and whose $ \tcode{operator[]} $ member returns $b_k$ - when invoked with argument $k$ for $k = 0, \ldots, n $. + when invoked with argument $k$ for $k = 0, \dotsc, n $. \end{itemdescr} \indexlibrarymember{densities}{piecewise_linear_distribution}% @@ -6369,7 +6369,7 @@ \pnum\returns A \tcode{vector} whose \tcode{size} member returns $n$ and whose $ \tcode{operator[]} $ member returns $\rho_k$ - when invoked with argument $k$ for $ k = 0, \ldots, n $. + when invoked with argument $k$ for $ k = 0, \dotsc, n $. \end{itemdescr}% % \indextext{random number distributions!sampling|)}%