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[rand] Use \dotsc, not \ldots for comma-separated lists #2033

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Apr 15, 2018
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[rand] Use \dotsc, not \ldots for comma-separated lists #2033

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68 changes: 34 additions & 34 deletions source/numerics.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -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}%
Expand All @@ -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
\[%
Expand All @@ -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,
Expand Down Expand Up @@ -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 $c$.


Expand All @@ -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 $1$;
Expand All @@ -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$.
Expand All @@ -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 $0$,
Expand Down Expand Up @@ -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|)}
Expand Down Expand Up @@ -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()},
Expand Down Expand Up @@ -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 & = &
Expand All @@ -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 & = &
Expand Down Expand Up @@ -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()})
Expand Down Expand Up @@ -5782,15 +5782,15 @@
\indextext{discrete probability function!\idxcode{discrete_distribution}}%
\indextext{\idxcode{discrete_distribution}!discrete probability function}%
\[%
P(i\,|\,p_0,\ldots,p_{n-1})
P(i\,|\,p_0,\dotsc,p_{n-1})
= p_i
\; \mbox{.}
\]

\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}%
Expand Down Expand Up @@ -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 $n$.
Expand All @@ -5925,7 +5925,7 @@
\pnum\returns A \tcode{vector<double>}
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}


Expand All @@ -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}
Expand All @@ -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}%
Expand Down Expand Up @@ -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 $n$.
Expand Down Expand Up @@ -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 $n$.
Expand All @@ -6136,7 +6136,7 @@
\pnum\returns A \tcode{vector<result_type>}
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}%
Expand All @@ -6148,7 +6148,7 @@
\pnum\returns A \tcode{vector<result_type>}
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}


Expand All @@ -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{,}
Expand All @@ -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}%
Expand Down Expand Up @@ -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 $n+1$.
Expand Down Expand Up @@ -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 $n+1$.
Expand All @@ -6357,7 +6357,7 @@
\pnum\returns A \tcode{vector<result_type>}
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}%
Expand All @@ -6369,7 +6369,7 @@
\pnum\returns A \tcode{vector<result_type>}
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|)}%
Expand Down