On the behavior of the numerical abscissa in the triangular matrices case

L. Trefethen gave a conjecture concerning a phenomenon linked to the initial transient and asymptotic behavior of ∥ exp(tA)∥ as a function of t at time t = 0, ( t → ∞ is a familiar limit). Motivated by this interesting phenomenon, we propose an answer for it in this particular case and in a general one as well. Also some results concerning particular quantities which are important in the study of time-dependent dynamical systems are given.