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fix i.e. and a.k.a.

This commit is contained in:
Andreas Fabri 2012-08-29 11:06:54 +00:00
parent 6f4d993dd7
commit 9f54f4360f
6 changed files with 8 additions and 8 deletions
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6
Algebraic_foundations/doc/Algebraic_foundations/CGAL/number_utils.h
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@ -22,7 +22,7 @@ namespace CGAL {
\ingroup PkgAlgebraicFoundations \ingroup PkgAlgebraicFoundations
The template function `compare` compares the first argument with respect to The template function `compare` compares the first argument with respect to
the second, i.e. it returns `CGAL::LARGER` if \f$ x\f$ is larger then \f$ y\f$. the second, i.e.\ it returns `CGAL::LARGER` if \f$ x\f$ is larger then \f$ y\f$.
In case the argument types `NT1` and `NT2` differ, In case the argument types `NT1` and `NT2` differ,
`compare` is performed with the semantic of the type determined via `compare` is performed with the semantic of the type determined via
@ -138,9 +138,9 @@ namespace CGAL {
/*! /*!
\ingroup PkgAlgebraicFoundations \ingroup PkgAlgebraicFoundations
The function `integral_division` (a.k.a. exact division or division without remainder) The function `integral_division` (a.k.a.\ exact division or division without remainder)
maps ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$ maps ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$
exists (i.e. if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking exists (i.e.\ if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking
this operation is undefined. Since the ring represented is an integral domain, this operation is undefined. Since the ring represented is an integral domain,
\f$ z\f$ is uniquely defined if it exists. \f$ z\f$ is uniquely defined if it exists.

2
Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--Gcd.h
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@ -9,7 +9,7 @@ The greatest common divisor (\f$ gcd\f$) of ring elements \f$ x\f$ and \f$ y\f$
ring element \f$ d\f$ (up to a unit) with the property that any common divisor of ring element \f$ d\f$ (up to a unit) with the property that any common divisor of
\f$ x\f$ and \f$ y\f$ also divides \f$ d\f$. (In other words: \f$ d\f$ is the greatest lower bound \f$ x\f$ and \f$ y\f$ also divides \f$ d\f$. (In other words: \f$ d\f$ is the greatest lower bound
of \f$ x\f$ and \f$ y\f$ in the partial order of divisibility.) We demand the \f$ gcd\f$ to be of \f$ x\f$ and \f$ y\f$ in the partial order of divisibility.) We demand the \f$ gcd\f$ to be
unit-normal (i.e. have unit part 1). unit-normal (i.e.\ have unit part 1).
\f$ gcd(0,0)\f$ is defined as \f$ 0\f$, since \f$ 0\f$ is the greatest element with respect \f$ gcd(0,0)\f$ is defined as \f$ 0\f$, since \f$ 0\f$ is the greatest element with respect
to the partial order of divisibility. This is because an element \f$ a \in R\f$ is said to divide \f$ b \in R\f$, iff \f$ \exists r \in R\f$ such that \f$ a \cdot r = b\f$. to the partial order of divisibility. This is because an element \f$ a \in R\f$ is said to divide \f$ b \in R\f$, iff \f$ \exists r \in R\f$ such that \f$ a \cdot r = b\f$.

2
Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IntegralDivision.h
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@ -7,7 +7,7 @@
Integral division (a.k.a. exact division or division without remainder) maps Integral division (a.k.a. exact division or division without remainder) maps
ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$ ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$
exists (i.e. if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking exists (i.e.\ if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking
this operation is undefined. Since the ring represented is an integral domain, this operation is undefined. Since the ring represented is an integral domain,
\f$ z\f$ is uniquely defined if it exists. \f$ z\f$ is uniquely defined if it exists.

2
Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IsSquare.h
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@ -44,7 +44,7 @@ typedef Hidden_type second_argument;
/// @{ /// @{
/*! /*!
returns <TT>true</TT> in case \f$ x\f$ is a square, i.e. \f$ x = y*y\f$. returns <TT>true</TT> in case \f$ x\f$ is a square, i.e.\ \f$ x = y*y\f$.
\post \f$ unit\_part(y) == 1\f$. \post \f$ unit\_part(y) == 1\f$.
*/ */

2
Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--UnitPart.h
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@ -8,7 +8,7 @@ element.
The mathematical definition of unit part is as follows: Two ring elements \f$ a\f$ The mathematical definition of unit part is as follows: Two ring elements \f$ a\f$
and \f$ b\f$ are said to be associate if there exists an invertible ring element and \f$ b\f$ are said to be associate if there exists an invertible ring element
(i.e. a unit) \f$ u\f$ such that \f$ a = ub\f$. This defines an equivalence relation. (i.e.\ a unit) \f$ u\f$ such that \f$ a = ub\f$. This defines an equivalence relation.
We can distinguish exactly one element of every equivalence class as being We can distinguish exactly one element of every equivalence class as being
unit normal. Then each element of a ring possesses a factorization into a unit unit normal. Then each element of a ring possesses a factorization into a unit
(called its unit part) and a unit-normal ring element (called its unit part) and a unit-normal ring element

2
Algebraic_foundations/doc/Algebraic_foundations/Concepts/IntegralDomainWithoutDivision.h
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@ -6,7 +6,7 @@
This is the most basic concept for algebraic structures considered within CGAL. This is the most basic concept for algebraic structures considered within CGAL.
A model `IntegralDomainWithoutDivision` represents an integral domain, A model `IntegralDomainWithoutDivision` represents an integral domain,
i.e. commutative ring with 0, 1, +, * and unity free of zero divisors. i.e.\ commutative ring with 0, 1, +, * and unity free of zero divisors.
<B>Note:</B> A model is not required to offer the always well defined integral division. <B>Note:</B> A model is not required to offer the always well defined integral division.