How is $\mathbb{R}_{+}^{\ast }$ defined? I don't know what set $\mathbb{R}_{+}^{\ast }$ usually denote? How is it defined? Does it contain zero ? Does it contain infinity?
 A: This is a somewhat common notation for $\{x \in \mathbb{R} \colon x > 0\}$. 
It does contain neither $0$ nor $+\infty$.
Further, $\mathbb{R}^{\ast} = \{x \in \mathbb{R} \colon x \neq  0\}$ and 
$\mathbb{R}_+ = \{x \in \mathbb{R} \colon x \geq  0\}$.

Update: Browsing the paper confirms what was said above. A $\sigma$ is defined with image in the set $\mathbb{R}^{\ast}_+$  and then $1/\sqrt{\sigma}$ is considered, which makes sense essentially only when it is positive and non-zero. 
The notation $\mathbb{R}_+$ is also used latter on. 
A: In general, the star symbol $\big(~^\star~\big)$ indicates the lack of $0~\big($i.e., $\mathbb N^\star,~\mathbb Z^\star,~\mathbb Q^\star,~\mathbb R^\star,~\mathbb C^\star\big)$, and the plus sign $\big(+\big)$ means $\ge0$. Similarly for the minus sign $\big(-\big),~$ which implies $\le0$. Examples include $\mathbb Z_+~,~\mathbb Q_+~,~\mathbb R_+~,~\mathbb Z_-~,~\mathbb Q_-~,~\mathbb R_-~$ All of the above exclude infinity, since $\infty$ is not a number. We can define, however, sets such as the extended real line, $\overline{\mathbb R},~$ and the extended complex plane, $\overline{\mathbb C}$.
