typos

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<AS>::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.

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.