Understanding a formula on article
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.