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linear algebra - matrix exponential and Spectral abscissa - Mathematics . . . $\lim_{t \rightarrow \infty} \|e^{tA}\| = 0$ if and only if $\alpha(A) < 0 $, where $\alpha$ is the Spectral abscissa, defined as $\max{Re(\lambda_i)}$ I tried to approach this problem by proving that $\|e^{tA}\| \leq \|e^{t\alpha(A)}\|$ but didn't succeed The hints given is First show what happens when you exponentiate a triangular matrix
Abscissa of convergence for a Dirichlet series Since $|f(n)| = 1$, it follows that $\sigma_a = 1$ (where $\sigma_a$ is the abscissa of absolute convergence), and we may conclude from general theory of Dirichlet series that $\sigma_c \in [0,1]$ My feeling is that there should be a bit of cancellation, resulting in $\sigma_c$ being smaller than 1, though I haven't been able to quantify this
coordinate systems - Abscissa, Ordinate, and Applicate -- Origins . . . Abscissa is found in Latin in 1656 in Exercitationum Mathematicarum by Frans van Schooten See page 285 [Bill Stockich] According to Cajori (1906, page 185), “The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome ”
Abscissa of absolute convergence of a Dirichlet series I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute