Several other corrections

src/doc/en/thematic_tutorials/geometry/lectures.rst

@@ -32,46 +32,48 @@ during the SageDays 84 in Olot (Spain).
 Lecture 0: Basic definitions and constructions
 ==============================================
 
-A real :math:`(k\times d)`-matrix :math:`A` and a real vector :math:`b`
-in :math:`\mathbb{R}^d` define a (convex) **polyhedron** :math:`P` as the set of solutions
+A real `(k\times d)`-matrix `A` and a real vector `b`
+in `\mathbb{R}^d` define a (convex) **polyhedron** `P` as the set of solutions
 of the system of linear inequalities:
 
 .. MATH::
 
     A\cdot x + b \geq 0.
 
-Each row of :math:`A` defines a closed half-space of :math:`\mathbb{R}^d`.
+Each row of `A` defines a closed half-space of `\mathbb{R}^d`.
 Hence a polyhedron is the intersection of finitely many closed half-spaces in
-:math:`\mathbb{R}^d`. The matrix :math:`A` may contain equal rows, which may lead to a
+`\mathbb{R}^d`. The matrix `A` may contain equal rows, which may lead to a
 set of *equalities* satisfied by the polyhedron. If there are no redundant rows
 in the above definition, this definition is referred to as the
-:math:`\mathbf{H}` **-representation** of a polyhedron.
+`\mathbf{H}` **-representation** of a polyhedron.
 
-The maximal affine subspace :math:`L` contained in a polyhedron is the
-**lineality** space. Fixing a point :math:`o` of the lineality space to act
-as the *origin*, one can write every point :math:`p` inside a polyhedron as a combination
+A maximal affine subspace `L` contained in a polyhedron is a **lineality** space of
+`P`. Fixing a point `o` of the lineality space `L` to act
+as the *origin*, one can write every point `p` inside a polyhedron as a combination
 
 .. MATH::
 
     p = \ell +\sum_{i=1}^{n}\lambda_iv_i+\sum_{i=1}^{m}\mu_ir_i,
 
-where :math:`\ell\in L` (using :math:`o` as the origin), :math:`\sum_{i=1}^n\lambda_i=1`,
-:math:`\mu_i\geq0`, and :math:`r_i\neq0` for all :math:`0\leq i\leq m` and the
-set of :math:`r_i` 's are positively independent (the origin is not in their positive span).
-There are many equivalent ways to write the above, so one asks :math:`n` and :math:`m`
-to be minimal with that property.
+where `\ell\in L` (using `o` as the origin), `\sum_{i=1}^n\lambda_i=1`,
+`\lambda_i\geq0`, `\mu_i\geq0`, and `r_i\neq0` for all `0\leq i\leq m` and the
+set of `r_i` 's are positively independent (the origin is not in their positive span).
+For a given point `p` there may be many equivalent ways to write the above using 
+different sets `\{v_i\}_{i=1}^{n}` and `\{r_i\}_{i=1}^{m}`. Hence we require the sets 
+to be inclusion minimal sets such that we can get the above property equality
+for any point `p\in P`.
 
-The points :math:`v_i` are called the *vertices* of :math:`P` and the points
-:math:`r_i` are called the *rays* of :math:`P`.
+The vectors `v_i` are called the *vertices* of `P` and the vectors
+`r_i` are called the *rays* of `P`.
 This way to represent a polyhedron is referred to as the
-:math:`\mathbf{V}` **-representation** of a polyhedron. The second sum represents the *convex
-hull* of the vertices :math:`v_i` 's and the second sum represents a *pointed
+`\mathbf{V}` **-representation** of a polyhedron. The first sum represents the *convex
+hull* of the vertices `v_i` 's and the second sum represents a *pointed
 polyhedral cone* generated by finitely many rays.
 
 When the lineality space and the rays are reduced to a point (i.e. no rays and
 no lines) the object is often referred to as a **polytope**.
 
-.. note :: As mentioned in the documentation of the constructor when typing :code:`Polyhedron?`,
+.. note :: As mentioned in the documentation of the constructor when typing :code:`Polyhedron?`:
 
     *You may either define it with vertex/ray/line or
     inequalities/equations data, but not both. Redundant data will
@@ -80,7 +82,7 @@ no lines) the object is often referred to as a **polytope**.
 
     Here is the documentation for the constructor function of :ref:`sage.geometry.polyhedron.constructor`.
 
-:math:`V`-representation
+`V`-representation
 ------------------------
 
 First, let's define a polyhedron object as the convex hull of a set of points
@@ -98,7 +100,7 @@ The string representation already gives a lot of information:
 
  - the dimension of the polyhedron (the smallest affine space containing it)
  - the dimension of the space in which it is defined
- - the base ring (:math:`\mathbb{Z}^2`) over which the polyhedron lives (this specifies the parent class, see :ref:`sage.geometry.polyhedron.parent`)
+ - the base ring (`\mathbb{Z}^2`) over which the polyhedron lives (this specifies the parent class, see :ref:`sage.geometry.polyhedron.parent`)
  - the number of vertices
  - the number of rays
 
@@ -125,7 +127,7 @@ We can also add a lineality space.
 
 .. end of output
 
-Notice that the base ring changes because of the value :math:`\frac{1}{2}`.
+Notice that the base ring changes because of the value `\frac{1}{2}`.
 Indeed, Sage finds an appropriate ring to define the object.
 
 ::
@@ -281,8 +283,8 @@ without having specified the base ring :code:`RDF` by the user.
 `H`-representation
 ------------------
 
-If a polyhedron object was constructed via a :math:`V`-representation, Sage can provide
-the :math:`H`-representation of the object.
+If a polyhedron object was constructed via a `V`-representation, Sage can provide
+the `H`-representation of the object.
 
 ::
 
@@ -294,9 +296,9 @@ the :math:`H`-representation of the object.
 
 .. end of output
 
-Each line gives a row of the matrix :math:`A` and an entry of the vector :math:`b`.
-The variable :math:`x` is a vector in the ambient space where :code:`P1` is
-defined. The :math:`H`-representation may contain equations:
+Each line gives a row of the matrix `A` and an entry of the vector `b`.
+The variable `x` is a vector in the ambient space where :code:`P1` is
+defined. The `H`-representation may contain equations:
 
 ::
 
@@ -307,8 +309,8 @@ defined. The :math:`H`-representation may contain equations:
 
 .. end of output
 
-The construction of a polyhedron object via its :math:`H`-representation,
-requires a precise format. Each inequality :math:`(a_{i1}, \dots, a_{id})\cdot
+The construction of a polyhedron object via its `H`-representation,
+requires a precise format. Each inequality `(a_{i1}, \dots, a_{id})\cdot
 x + b_i \geq 0` must be written as :code:`[b_i,a_i1, ..., a_id]`.
 
 ::
@@ -360,13 +362,13 @@ the data before defining the polyhedron if possible.
 Lecture 1: Representation objects
 ===================================
 
-Many objects are related to the :math:`H`- and :math:`V`-representations. Sage
+Many objects are related to the `H`- and `V`-representations. Sage
 has classes implemented for them.
 
 `H`-representation
 ------------------
 
-You can store the :math:`H`-representation in a variable and use the
+You can store the `H`-representation in a variable and use the
 inequalities and equalities as objects.
 
 ::
@@ -395,7 +397,7 @@ inequalities and equalities as objects.
 
 .. end of output
 
-It is possible to obtain the different objects of the :math:`H`-representation
+It is possible to obtain the different objects of the `H`-representation
 as follows.
 
 ::
@@ -444,7 +446,7 @@ Similarly, you can access to vertices, rays and lines of the polyhedron.
 
 .. end of output
 
-It is possible to obtain the different objects of the :math:`V`-representation
+It is possible to obtain the different objects of the `V`-representation
 as follows.
 
 ::
@@ -572,7 +574,7 @@ but not algebraic or symbolic values:
 .. end of output
 
 It is possible to get the :code:`cdd` format of any polyhedron object defined
-over :math:`\mathbb{Z}`, :math:`\mathbb{Q}`, or :code:`RDF`:
+over `\mathbb{Z}`, `\mathbb{Q}`, or :code:`RDF`:
 
 ::
 

src/doc/en/thematic_tutorials/geometry/polytutorial.rst

@@ -17,7 +17,7 @@ Basics
 """"""
 
 First, let's define a polytope as the convex hull of a set of points,
-i.e. given  :math:`S` we compute  :math:`P={\rm conv}(S)`:
+i.e. given  `S` we compute  `P={\rm conv}(S)`:
 
 
 ::
@@ -60,11 +60,11 @@ That notation is not immediately parseable, because seriously,
 those do not look like equations of lines (or of halfspaces, which is
 really what they are).
 
-``(-4, 1) x + 12 >= 0`` really means  :math:`(-4, 1)\cdot\vec{x} + 12 \geq 0`.
+``(-4, 1) x + 12 >= 0`` really means  `(-4, 1)\cdot\vec{x} + 12 \geq 0`.
 
 So... if you want to define a polytope via inequalities, you have to
 translate each inequality into a vector.  For example,
-:math:`(-4, 1)\cdot\vec{x} + 12 \geq 0` becomes (12, \-4, 1).
+`(-4, 1)\cdot\vec{x} + 12 \geq 0` becomes (12, \-4, 1).
 
 
 ::

src/doc/en/thematic_tutorials/geometry/tips.rst

@@ -59,7 +59,7 @@ object, you can!
 :code:`repr_pretty_Hrepresentation`
 ==============================================================
 
-If you would like to visualize the :math:`H`-representation nicely and even get
+If you would like to visualize the `H`-representation nicely and even get
 the latex presentation, there is a method for that!
 
 ::