This study aims to investigate serum makorin ring finger protein 3 (MKRN3) levels in girls with idiopathic central precocious puberty (ICPP) and premature thelarche (PT), in order to determine whether circulating MKRN3 level is associated with ICPP and PT. A total of 90 girls were enrolled in the study. 30 age-matched girls were allocated for each group (ICPP, PT and healthy controls [HC], respectively). The base LH (B-LH) and E2 levels were higher in ICPP girls than those in HC and PT girls. The peak LH (P-LH) levels and P-LH/P-FSH values were obviously higher in ICPP girls than those in PT girls, while higher peak FSH (P-FSH) levels were detected in PT girls when compared to those in ICPP girls. Kisspeptin levels were lower in HC girls than those in ICPP and PT girls. MKRN3 levels were the highest in HC girls among the three groups. There were relatively strong negative correlations among MKRN3, kisspeptin and P-LH/P-FSH. Circulating MKRN3 can have an important role in the onset of ICPP and PT. However, this should not be used as an independent diagnostic criterion for diagnosing ICPP or differentiating ICPP from PT, but should be used only as an adjunctive diagnostic biomarker.
Let $\Vert {\cdot }\Vert $ be a norm on the algebra ${\mathcal M}_n$ of all $n\times n$ matrices over ${\mathbb{C}}$. An interesting problem in matrix theory is that “Are there two norms $\Vert {\cdot }\Vert _1$ and $\Vert {\cdot }\Vert _2$ on ${\mathbb{C}}^n$ such that $\Vert A\Vert =\max \lbrace \Vert Ax\Vert _{2}\: \Vert x\Vert _{1}=1\rbrace $ for all $A\in {\mathcal M}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which $\Vert {\cdot }\Vert _1=\Vert {\cdot }\Vert _2$.