Symmetries on almost symmetric numerical semigroups

The notion of almost symmetric numerical semigroup was given by V. Barucci and R. Fr\"oberg. We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for $H^*$ (the dual of $M$) to be almost symmetric numerical semigroup. Using these results we give a formula for multiplicity of an opened modular numerical semigroups. Finally, we show that if $H_1$ or $H_2$ is not symmetric, then the gluing of $H_1$ and $H_2$ is not almost symmetric.