From 9fcb63096705a4daaece0c5a61fb5388f14ef4c2 Mon Sep 17 00:00:00 2001 From: Andreas Fabri Date: Mon, 5 Feb 2007 11:14:50 +0000 Subject: [PATCH] typos --- .../algebraic_structures.tex | 18 +++++++++--------- .../doc_tex/Algebraic_foundations/intro.tex | 2 +- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/Algebraic_foundations/doc_tex/Algebraic_foundations/algebraic_structures.tex b/Algebraic_foundations/doc_tex/Algebraic_foundations/algebraic_structures.tex index bbddc8b709b..37228a44155 100644 --- a/Algebraic_foundations/doc_tex/Algebraic_foundations/algebraic_structures.tex +++ b/Algebraic_foundations/doc_tex/Algebraic_foundations/algebraic_structures.tex @@ -8,7 +8,7 @@ motivated by their well known counter parts in traditional algebra, but we also had to pay tribute to existing types an their restrictions. To keep the interface minimal, it was not desirable to cover all known algebraic structures, -e.g. we did not introduce concepts for such basic structures as {\em groups} or +e.g., we did not introduce concepts for such basic structures as {\em groups} or exceptional structures as {\em skew fields}. \begin{figure}[htbp] @@ -58,13 +58,13 @@ fulfills is encoded in the tag An algebraic structure is at least \ccc{Assignable}, \ccc{CopyConstructible}, \ccc{DefaultConstructible} and \ccc{EqualityComparable}. Moreover, we require that it is -constructible from \ccc{int}, for any int in the range from -128 to 127. +constructible from \ccc{int}, for any int in the range from \ccc{-128} to \ccc{127}. For ease of use and since their semantic is sufficiently standard to presume their existence, the usual arithmetic and comparison operators are required to be realized via \CC\ operator overloading. The division operator is reserved for division in fields. -All other unary (e.g. sqrt) and binary functions -(e.g. gcd, div) must be models of the well known \stl-concepts +All other unary (e.g., sqrt) and binary functions +(e.g., gcd, div) must be models of the well known \stl-concepts \ccc{AdaptableUnaryFunction} or \ccc{AdaptableBinaryFunction} concept and local to the traits class (e.g., \ccc{Algebraic_structure_traits::Sqrt()(x)}). @@ -74,9 +74,9 @@ two-pass template compilation problems experienced with the old design using overloaded functions. However, for ease of use and backward compatibility all functionality is also accessible through global functions defined within namespace \ccc{CGAL}, -e.g. \ccc{CGAL::sqrt}. This is realized via function templates using -the according functor of the traits class. For an overview see section -\ref{caf_ref::classified_refernce_pages} in the reference manual. +e.g., \ccc{CGAL::sqrt}. This is realized via function templates using +the according functor of the traits class. For an overview see +Section~\ref{caf_ref::classified_refernce_pages} in the reference manual. %Dispatching For dispatching \ccc{Algebraic_structure_traits} provides the tags: @@ -88,7 +88,7 @@ algebraic concept a type fulfills and is one of \ccc{Unique_factorization_domain_tag}, \ccc{Euclidean_ring_tag} or even \ccc{Null_tag} in case the type is not a model of an algebraic structure concept. The tags are derived from each other such that they reflect the hierarchy of the algebraic -structure concept, e.g. \ccc{Field_with_sqrt_tag} is derived from \ccc{Field_tag}. \\ +structure concept, e.g., \ccc{Field_with_sqrt_tag} is derived from \ccc{Field_tag}. \\ \ccc{Is_exact} and \ccc{Is_numerical_sensitive} are both either \ccc{Tag_true} or \ccc{Tag_false}. An algebraic structure is considered exact, @@ -97,7 +97,7 @@ of two algebraic expressions is always correct. An algebraic structure is considered as numerically sensitive, if the performance of the type is sensitive to the condition number of an algorithm. %performance includes both rounding errors or runtime. -Note that there is really a difference among this two notions, e.g. the fundamental type \ccc{int} +Note that there is really a difference among this two notions, e.g., the fundamental type \ccc{int} is not numerical sensitive but considered inexact due to overflow. Conversely, types as \ccc{leda_real} or \ccc{CORE::Expr} are exact but sensitive to numerical issues due to the internal use of multi precision floating point arithmetic. diff --git a/Algebraic_foundations/doc_tex/Algebraic_foundations/intro.tex b/Algebraic_foundations/doc_tex/Algebraic_foundations/intro.tex index 5f08c546614..8052dccf12f 100644 --- a/Algebraic_foundations/doc_tex/Algebraic_foundations/intro.tex +++ b/Algebraic_foundations/doc_tex/Algebraic_foundations/intro.tex @@ -6,7 +6,7 @@ in particular objects defined on algebraic curves and surfaces. As a consequence types representing polynomials, algebraic extensions and finite fields play a more important role in related implementations. This package has been introduced to stay abreast of these changes. -Since in particular polynomials must be supported by the introduces framework +Since in particular polynomials must be supported by the introduced framework the package avoids the term {\em number type}. Instead the package distinguishes between the {\em algebraic structure} of a type and whether a type is embeddable on the real axis, or {\em real embeddable} for short.