Minor fixes for 15-2

docs/en/week15/15-2.md

@@ -37,7 +37,7 @@ $$F_\beta(y)\dot{=}-\frac{1}{\beta} \log \frac{1}{|\mathcal{Z}|}{\int}_\mathcal{
 where $\beta=(k_B T)^{-1}$ is the inverse temperature, consisting of the Boltzmann constant multiplied by the temperature. If the temperature is very high, $\beta$ is going to be extremely small and if the temperature is cold, then $\beta\rightarrow \infty$.
 **Simple discrete approximation:**
 $$\tilde{F}_\beta(y)=-\frac{1}{\beta} \log \frac{1}{|\mathcal{Z}|}\underset{z\in\mathcal{Z}}{\sum} \exp[{-\beta}E(y,z)]\Delta z$$
-Here, we define $-\frac{1}{\beta} \log \frac{1}{|\mathcal{Z}|}\underset{z\in\mathcal{Z}}{\sum} \exp[{-\beta}E(y,z)]$ to be the $\underset{z}{\text{softmin}}_\beta[E(y,z)]$, such that the relaxation of the zero temperature limit free energy becomes the *actual*-softmin.
+Here, we define $-\frac{1}{\beta} \log \frac{1}{|\mathcal{Z}|}\underset{z\in\mathcal{Z}}{\sum} \exp[{-\beta}E(y,z)]$ to be the $\smash{\underset{z}{\text{softmin}}}_\beta[E(y,z)]$, such that the relaxation of the zero temperature limit free energy becomes the *actual*-softmin.
 **Examples:**
 We will now revisit examples from the previous practicum and see the effects from applying the relaxed version.
 
@@ -115,7 +115,7 @@ Objective -  Finding a well behaved energy function
 A loss functional, minimized during learning, is used to measure the quality of the available energy functions. In simple terms, loss functional is a scalar function that tells us how good our energy function is. A distinction should be made between the energy function, which is minimized by the inference process, and the loss functional (introduced in Section 2), which is minimized by the learning process.
 
 
-$$\mathcal{L}(F(.),Y) = \frac{1}{N} \sum_{n=1}{N} l(F(.),y^{(n)}) \in R$$
+$$\mathcal{L}(F(.),Y) = \frac{1}{N} \sum_{n=1}^{N} l(F(.),y^{(n)}) \in \R$$
 
 
 $\mathcal{L}$ is the loss function of the whole dataset that can be expressed as the average of these per sample loss function functionals.