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. of gray matter volume in the left caudate and left thalamus, which would help explain why the former is more selectively responsive to mild facial inversion ([@B40]). Despite this, the lack of association between emotion processing and social network size in this study indicates that the link between these two variables is likely not driven by increased awareness of other people.

Contrary to prior findings, we found that the relationships between insula activation and social network size were not driven by awareness of social interactions. We did, however, find a relationship between activation of the ACC and social network size. Although previous studies have consistently demonstrated that activation of the ACC is linked to increased awareness of social interaction ([@B11], [@B18]), we found that the relationship was driven by increased social network size at high, but not low levels. It is noteworthy that there was no correlation between social network size and ACC activation in Study 1. It is possible that the relationship between social network size and the ACC is driven by different measures of connectivity or that it only occurs in adolescence. For example, the ACC and amygdala are directly connected in adults and connectivity between these structures has been shown to increase in adolescence ([@B25]).

The relationship between ACC activation and social network size is an important finding because studies of this region have focused largely on the perception of social interaction ([@B10]). However, the relationship between ACC activation and increased awareness of social interaction needs to be interpreted with caution. As explained in the introduction, some theories propose that the ACC plays an important role in the awareness of one’s own social environment, with activation of this region being associated with a sense of self-awareness ([@B25], [@B26]). A recent study of social pain in response to viewing one’s own facial expression did not find a relationship between ACC activation and self-reported emotion-facial recognition accuracy ([@B40]), further supporting the idea that this region is not critical in the awareness of others. However, this study did find that individuals with smaller ACC activations reported greater self-awareness of their affective state. Another recent study of social pain found that the ACC was specifically engaged by viewing an altered facial expression representing social rejection ([@B41]), which is consistent with the idea that the ACC may be active when considering one’s own social environment. It is possible that the perceived social environment in these studies does not simply provide an accurate reflection of one’s own social environment (although this is not necessarily the case), but that


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Is there any equality relation between $f(x)=x$, $g(x)=x^{2}$ and $h(x)=x^{3}$?

I would like to know if there is any equality relations among $f(x)=x$, $g(x)=x^{2}$ and $h(x)=x^{3}$


$f(x)=x$ and $g(x)=x^{2}$ are clearly not equal (unless $x=0$); by contrast $f(x)=g(x)$ is true for every real $x$.
You can show $f(x)=g(x)$ by noting that $f(x)=x$ and $g(x)=x^2$ are degree $1$ polynomials with the same leading coefficient, so they have the same leading terms; and therefore their quotient is $1$. Now the quotient of $g$ and $h$ is a polynomial of degree $2$. We then have
a polynomial with leading term $1/x^2$.
And if you want a degree $3$ polynomial with leading term $1/x^2$ then $f(x)=g(x)h(x)$ is a good choice, as

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