updates on documentations

docs/make.jl

@@ -19,4 +19,4 @@ makedocs(sitename="ABMEv.jl",
         # "contributing.md",
         ],)
 
-deploydocs(repo= "gitlab.ethz.ch:bvictor/abmev.wiki.git",)
+# deploydocs(repo= "gitlab.ethz.ch:bvictor/abmev.wiki.git",)

docs/src/mathematics/geotrait.md

 # The geotrait
 > This note has been taken from `2Y/articles/geotrait/mathematical_notes.md`
 
+!!! note "'note' admonition"
+    In Landscape Ecology, it is of particular interest to study the geographical position of the lineage through time. This is because richness patterns are thought to arise through allopatric speciation, where populations get separated in space and time. This is the topic of the following section.
+
 ## Julia accessors
 An agents of type `Agent{Ancestors{true}}` stores the values of its ancestors traits. From it, one can thus study the history of the lineage.
 
-In Landscape Ecology, it is of particular interest to study the geographical position of the lineage through time. This is because richness patterns are thought to arise through allopatric speciation, where populations get separated in space and time. This is the topic of the following section.
-
 ## Projection of the geographic history
 
-Consider a set of individuals which lineage geographical history has been stored in a vector $`x^{(i)}(t)`$. If individual $`i`$ was born at time $`t^*`$ then $`x^{(i)}(t>t^*)`$ represents its geographical position, while $`x^{(i)}(t<t^*)`$ represents its ancestors geographical position.
+Consider a set of individuals which lineage geographical history has been stored in a vector ``x^{(i)}(t)``. If individual ``i`` was born at time ``t^*`` then ``x^{(i)}(t>t^*)`` represents its geographical position, while ``x^{(i)}(t<t^*)`` represents its ancestors geographical position.
 
 ### Isolation in time
 We want to account for the isolation in time and possibly in space of the lineage of a given individual. That is, for how long and how distant stay lineages appart?
 
 #### Setting
-Consider a discrete setting, where $`N(t)`$ individuals (or populations) evolve over a set of $`M`$ demes disposed in a linear fashion, such that $`x^i(t) \in \{1,2,...,M\}`$. The population is characterised by the counting measure 
+Consider a discrete setting, where ``N(t)`` individuals (or populations) evolve over a set of ``M`` demes disposed in a linear fashion, such that ``x^i(t) \in \{1,2,...,M\}``. The population is characterised by the counting measure
 ```math
 \nu = \sum_i^{N(t)} \delta_{x^i(t)}
 ```
 
 
-We define the geographical history hamming distance $`\mathfrak{g}`$ as 
+We define the geographical history hamming distance ``\mathfrak{g}`` as
 ```math
 \mathfrak{h}\big(x^{(i)}(t),x^{(j)}(t)\big) = \int_0^t \text{ceil}(\frac{|x^{(i)} - x^{(j)}|(s)}{M-1})ds
 ```
 
-We extend this definition with the measure $`\mathfrak{h}^*`$ which takes into account geographic distance
+We extend this definition with the measure ``\mathfrak{h}^*`` which takes into account geographic distance
 ```math
 \mathfrak{h}^*\big(x^{(i)}(t),x^{(j)}(t)\big) = \int_0^t \Big[x^{(i)} - x^{(j)} \Big]^2(s)\, ds
 ```
 
-Finally, we introduce the geotrait distance $`\mathfrak{g}`$ as
+Finally, we introduce the geotrait distance ``\mathfrak{g}`` as
 ```math
 \mathfrak{g}\big(x^{(i)}(t),x^{(j)}(t)\big) = \Big[\int_0^t (  x^{(i)} - x^{(j)} )(s) \, ds\Big]^2
 ```
@@ -40,15 +41,15 @@ Note that by the triangle inequality we have that
 ```math
 \mathfrak{g}\big(x^{(i)}(t),x^{(j)}(t)\big) \leq \mathfrak{h}^*\big(x^{(i)}(t),x^{(j)}(t)\big)
 ```
-Equality should arises if $`x^{(i)}, x^{(j)}`$ are positively linearly dependent (which should not be the case).
+Equality should arises if ``x^{(i)}, x^{(j)}`` are positively linearly dependent (which should not be the case).
 
-However, what we are eventually interested is a population measure. This measure should be related to the average pairwise distance across the population. Hence we define $`\mathcal{D_d}(\nu,t)`$ such that 
+However, what we are eventually interested is a population measure. This measure should be related to the average pairwise distance across the population. Hence we define ``\mathcal{D_d}(\nu,t)`` such that
 
 ```math
 \mathcal{D_d}(\nu,t) = \frac{1}{2N^2}\sum_i^N  \sum_j^N d(x^{(i)},x^{(j)},t)
 ```
 
-Let $`g^{(i)}(t) = \int_0^t x^{i}(t) dt`$ and $`G(t) = \{g^{(i)}(t), i \in \{1,2,\dots,N(t)\}\}`$. By observing the following
+Let ``g^{(i)}(t) = \int_0^t x^{i}(t) dt`` and ``G(t) = \{g^{(i)}(t), i \in \{1,2,\dots,N(t)\}\}``. By observing the following
 
 ```math
 \frac{1}{2N^2}\sum_{i,j}^N (y_i-y_j)^2 \\=
@@ -56,12 +57,12 @@ Let $`g^{(i)}(t) = \int_0^t x^{i}(t) dt`$ and $`G(t) = \{g^{(i)}(t), i \in \{1,2
  = \frac{1}{2N^2} 2N \sum_i^N(y_i-\bar{y})^2 = \text{Var}(Y)
 ```
 
-Thus we have $`\mathfrak{g}\big(x^{(i)}(t),x^{(j)}(t)\big) = [g^{(i)}(t) - g^{(j)}(t)]^2`$  and hence 
+Thus we have ``\mathfrak{g}\big(x^{(i)}(t),x^{(j)}(t)\big) = [g^{(i)}(t) - g^{(j)}(t)]^2``  and hence
 ```math
     \mathcal{D_\mathfrak{g}}(\nu,t) = \text{Var}(G).
 ```
 
-Also  remark that 
+Also  remark that
 
 ```math
 \mathcal{D_{h^*}}(\nu,t) = \frac{1}{2N^2}\sum_i^N  \sum_j^N h^*(x^{(i)},x^{(j)},t) \\
@@ -71,7 +72,7 @@ Also  remark that
 
 ```
 
-One could also imagine a value $`h^{(i)}(t) = \frac{1}{2N}\sum_j^{N(t)}\mathfrak{h}^*\big(x^{(i)}(t),x^{(j)}(t)\big)`$ and in this case we would have 
+One could also imagine a value ``h^{(i)}(t) = \frac{1}{2N}\sum_j^{N(t)}\mathfrak{h}^*\big(x^{(i)}(t),x^{(j)}(t)\big)`` and in this case we would have
 ```math
     \mathcal{D_\mathfrak{h}}(\nu,t) = \frac{1}{N}\sum_i^{N(t)} h^{(i)}(t)
 ```
@@ -89,4 +90,4 @@ Here is a time average of the speed
 But one could also have a moving average, that is, averaging
 ```math
     \frac{1}{N} \sum_i \sum_j \int_{j\tau}^{j(\tau+1)} \frac{1}{\tau} \partial_s x^{(i)}(s) ds
-```
\ No newline at end of file
+```