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Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set $$ S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}. $$ Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce $$ ...