Rothberger gaps in fragmented ideals

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.Comment: 28 page

Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity

We investigate mobility regimes for localized modes in the discrete nonlinear Schr\"{o}dinger (DNLS) equation with the cubic-quintic onsite terms. Using the variational approximation (VA), the largest soliton's total power admitting progressive motion of kicked discrete solitons is predicted, by comparing the effective kinetic energy with the respective Peierls-Nabarro (PN) potential barrier. The prediction is novel for the DNLS model with the cubic-only nonlinearity too, demonstrating a reasonable agreement with numerical findings. Small self-focusing quintic term quickly suppresses the mobility. In the case of the competition between the cubic self-focusing and quintic self-defocusing terms, we identify parameter regions where odd and even fundamental modes exchange their stability, involving intermediate asymmetric modes. In this case, stable solitons can be set in motion by kicking, so as to let them pass the PN barrier. Unstable solitons spontaneously start oscillatory or progressive motion, if they are located, respectively, below or above a mobility threshold. Collisions between moving discrete solitons, at the competing nonlinearities frame, are studied too.Comment: 12 pages, 15 figure

¿Serán necesarios los fondos de cesantías?

El presente trabajo, es fruto de la reflexión en torno a la validez del papel que desempeñan losfondos de cesantías en Colombia. Se trata de crear un debate académico sobre la trascendencia eimportancia de estas organizaciones, las cuales fueron creadas a partir de la ley 50 de 1990 con elpropósito de reducir la carga prestacional a las empresas que estaban afectando su viabilidad, ygenerar un ahorro obligatorio para que una vez el trabajador se retire de su empleo pueda disponerde recursos. Se parte de un análisis del papel que desempeñan los fondos de cesantías, luego serealiza un análisis basado en la simulación de los aportes que haría un trabajador durante su vidalaboral y los retornos que pueden generar estos ahorros en relación con otras alternativas deahorro. Los resultados obtenidos en la simulación permiten concluir que el papel que cumplen losfondos de cesantías no es el más apropiado y se requiere de un rediseño del sistema