Understanding a formula on article I just read an article  "Statistical approach for figurative sentiment analysis on Social Networking Services: a case study on Twitter", which provide an algorithm to analyze tweets, and this article includes 2 formulas which I don't really understand.

Link to the article

I hope maybe someone here can help me.


The first formula is the (4) formula (Page 5) 
The second formula is the (6) formula (Page 8)


I will be very thankful if someone will help me with this. 

Examples will be most welcome! :)
Each tweet and each cluster is represented as an $m_k$-dimensional vector. The distance between a tweet $t_k$ and a cluster $\delta$ is then
$$
dis(t_k,\delta) = 1 - \frac{\langle t_k, \delta \rangle}{\|t_k\| \|\delta\|},
$$
where $\langle a,b \rangle = \sum_{i=1}^{m_k} a_i b_i$ is the inner product and $\|a\| = \sqrt{\langle a,a \rangle}$ is the norm. This explains equation (4).

The definition of $P(S_t|w)$ is given below (6):


  $P(S_t|w)$ is the
  probability that a term has a score with the given tweet score.


This is the number of tweets containing $w$ which have score $S_t$ divided by the total number of tweets containing $w$.

The rest of the formula is hopefully self-explanatory ($\times$ is just ordinary multiplication).

If you have any more questions, I suggest contacting the authors.