Fixes formatting errors

doc/Language/math.pod6

@@ -235,45 +235,45 @@ wherever possible.
 
 =head1 Numerical integration of ordinary differential equations
 
-Perl 6 is an amazing programming language, and of course, you can do a 
-lot of cool maths with it. A great amount of work during an applied 
-mathematician's work is to simulate the models they create. For this 
-reason, in every coding language, a numerical integrator is a must-have. 
-Learning about how to do this in Perl 6 can easily be a good way to have 
-a nice goal while learning a new coding language. 
+Perl 6 is an amazing programming language, and of course, you can do a
+lot of cool maths with it. A great amount of work during an applied
+mathematician's work is to simulate the models they create. For this
+reason, in every coding language, a numerical integrator is a must-have.
+Learning about how to do this in Perl 6 can easily be a good way to have
+a nice goal while learning a new coding language.
 
 =head2 Requirements
 
 In Perl 6 there are some modules in the ecosystem that can do the job we
  want in a very easy way :
 =item L<C<Math::Model>|https://github.com/moritz/Math-Model>
-which lets you write mathematical and physical models in an easy and 
+which lets you write mathematical and physical models in an easy and
 natural way.
 =item L<C<Math::RungeKutta>|https://github.com/moritz/Math-RungeKutta>
-Runge-Kutta integration for systems of ordinary, linear differential 
+Runge-Kutta integration for systems of ordinary, linear differential
 equations.
 
-For this example we are going to use 
+For this example we are going to use
 L<C<Math::Model>|https://github.com/moritz/Math-Model> because its
- useful sintaxis, but remember that this module requires 
+ useful sintaxis, but remember that this module requires
  L<C<Math::RungeKutta>|https://github.com/moritz/Math-RungeKutta> as well.
  Simply install them with I<zef> before using these examples.
 In the future it should be a nice idea to add other modules.
 
 =head2 Malthus model
 
-Let's start with the I<'Hello World'> of mathematical Ecology: 
-L<Malthusian growth model|https://en.wikipedia.org/wiki/Malthusian_growth_model>. 
+Let's start with the I<'Hello World'> of mathematical Ecology:
+L<Malthusian growth model|https://en.wikipedia.org/wiki/Malthusian_growth_model>.
 A Malthusian growth model, sometimes called a simple exponential growth
- model, is essentially exponential growth based on the idea of the 
+ model, is essentially exponential growth based on the idea of the
  function being proportional to the speed to which the function grows.
-The equation, then, looks like this: 
+The equation, then, looks like this:
 
 I<dx/dt = g*x>
 
-I<x(0) = x_0> 
+I<x(0) = x_0>
 
-Where I<g> is the population growth rate, sometimes called Malthusian 
+Where I<g> is the population growth rate, sometimes called Malthusian
 parameter.
 
 How can we translate that into Perl 6? Well Math::Model bring some help
@@ -304,37 +304,37 @@ To fully understand what is going on, let me explain it step by step.
 
 =head3 Step by step explanation.
 
-=begin item 
-First we load the module that make 
+=begin item
+First we load the module that make
 the calculations: L<C<Math::Model>|https://github.com/moritz/Math-Model>.
 
 =begin code :skip-test<Snippet for explanantion purposes>
 use Math::Model;
 =end code
-=end item 
+=end item
 
-=begin item 
+=begin item
 We create the model to add all the information in it.
 =begin code :skip-test<Snippet for explanantion purposes>
 my $m = Math::Model.new(
 =end code
-=end item 
+=end item
 
-=begin item 
+=begin item
 We declare the derivatives that are in our model. In this case, if
-you remember our equation, we have our variable I<x> and its derivative 
+you remember our equation, we have our variable I<x> and its derivative
 I<x'> (usually know as the velocity).
 
 =begin code :skip-test<Snippet for explanantion purposes>
 derivatives => {
     velocity     => 'x',
     },
 =end code
-=end item 
+=end item
 
-=begin item 
-After that, we declare how our models evolve. We just need 
-formulas for the derivatives that are not also integration variables 
+=begin item
+After that, we declare how our models evolve. We just need
+formulas for the derivatives that are not also integration variables
 (in this case, only I<x>), and for other variables we use in the formulas
  (the growth rate).
 
@@ -344,12 +344,12 @@ variables   => {
     growth_constant    => { 1 },   # basal growth rate
 },
 =end code
-=end item 
+=end item
 
 
-=begin item 
-Finally we declare our initial conditions and use I<captures> to 
-tell L<C<Math::Model>|https://github.com/moritz/Math-Model> which 
+=begin item
+Finally we declare our initial conditions and use I<captures> to
+tell L<C<Math::Model>|https://github.com/moritz/Math-Model> which
 variable or variables to record while the simulation is running.
 
 =begin code :skip-test<Snippet for explanantion purposes>
@@ -358,12 +358,12 @@ initials    => {
 },
 captures    => ('x'),
 =end code
-=end item 
+=end item
 
 
 At this point our model is set. We need to run the simulation and render
  a cool plot about our results:
-=begin code 
+=begin code
 $m.integrate(:from(0), :to(8), :min-resolution(0.5));
 $m.render-svg('population growth malthus.svg', :title('population growth'));
 =end code
@@ -379,19 +379,19 @@ Looks great! But to be honest, it is quit unrepresentative. Let's explore
 
 =head2 Logistic model
 
-Resources aren't infinite and our population is not going to grow 
-forever. P-F Verhulst thought the same thing, so he presented the 
+Resources aren't infinite and our population is not going to grow
+forever. P-F Verhulst thought the same thing, so he presented the
 L<logistic model|https://en.wikipedia.org/wiki/Logistic_function>. This
- model is a common model of population growth, where the rate of 
+ model is a common model of population growth, where the rate of
  reproduction is proportional to both the existing population and the
   amount of available resources, all else being equal.
-It looks like this: 
+It looks like this:
 
 I<dx/dt = g*x*(1-x/k)>
 I<x(0)=x_0>
 
 where the constant g defines the growth rate and k is the carrying
- capacity. 
+ capacity.
 Modifying the above code we can simulate its behaviour in time:
 
 =begin code
@@ -404,7 +404,7 @@ my $m = Math::Model.new(
     variables   => {
         velocity           => { $:growth_constant * $:x - $:growth_constant * $:x * $:x / $:k },
         growth_constant    => { 1 },   # basal growth rate
-	k                  => { 100 }, # carrying capacity
+        k                  => { 100 }, # carrying capacity
     },
     initials    => {
         x       => 3,
@@ -419,27 +419,27 @@ $m.render-svg('population growth logistic.svg', :title('population growth'));
 Let's look at our cool plot:
 L<link to the image|https://rawgit.com/thebooort/perl-6-math-model-tutorial/master/population%20growth%20logistic.svg>
 
-As you can see population growths till a maximum. 
+As you can see population growths till a maximum.
 
 =head2 Alle Effect
 
-Interesting, isn't it? Even if these equations seem basic they are 
-linked to a lot of behaviours in our world, like tumor growth. But, 
+Interesting, isn't it? Even if these equations seem basic they are
+linked to a lot of behaviours in our world, like tumor growth. But,
 before end, let me show you a curious case.
-Logistic model could be accurate but... What happens when, from a 
-certain threshold, the population size is so small that the survival 
+Logistic model could be accurate but... What happens when, from a
+certain threshold, the population size is so small that the survival
 rate and / or the reproductive rate drops due to the individuals' inhability
- to find other ones? 
+ to find other ones?
 
-Well, this is and interesting phenomenon described by W.C.Alle, usually 
+Well, this is and interesting phenomenon described by W.C.Alle, usually
 known as L<Allee effect|https://en.wikipedia.org/wiki/Allee_effect>.
-A simple way to obtain different behaviours associated with this effect 
-is to use the logistic model as a departure, add some terms, and get a 
+A simple way to obtain different behaviours associated with this effect
+is to use the logistic model as a departure, add some terms, and get a
 cubic growth model like this one:
 
 I<dx/dt=r*x*(x/a-1)*(1-x/k)>
 
-where all the constants are the same as before and A is called 
+where all the constants are the same as before and A is called
 I<critical point>.
 
 Our code would be:
@@ -454,8 +454,8 @@ my $m = Math::Model.new(
     variables   => {
         velocity_x           => { $:growth_constant * $:x *($:x/$:a -1)*(1- $:x/$:k) },
         growth_constant    => { 0.7 },   # basal growth rate
-	k                  => { 100 }, # carrying capacity
-	a		   => { 15 },  # critical point
+        k                  => { 100 }, # carrying capacity
+        a		   => { 15 },  # critical point
     },
     initials    => {
         x       => 15,
@@ -467,13 +467,13 @@ $m.integrate(:from(0), :to(100), :min-resolution(0.5));
 $m.render-svg('population growth alle.svg', :title('population growth'));
 =end code
 
-Try to execute this one yourself to see the differents behaviours that 
+Try to execute this one yourself to see the differents behaviours that
 arise when you change the initial condition around the critical point!
 
 =head2 Extra info.
 
-Do you like physics? Check the original post by the creator of the 
-modules that have been used, Moritz Lenz: 
+Do you like physics? Check the original post by the creator of the
+modules that have been used, Moritz Lenz:
 L<https://perlgeek.de/blog-en/perl-6/physical-modelling.html>.