  We investigate criticality near the jamming transition in both quiescent
systems and those under shear by considering the effect of mechanical training
on the jamming transition and nonlinear rheology. We simulate frictionless soft
particles undergoing athermal quasi-static shear using initial configurations
trained with athermal quasi-static cyclic volume deformations. The jamming
transition density of the initial configuration $\varphi_{\rm J0}$ is
systematically altered by tuning the ``depth'' of mechanical training. We exert
a steady shear on these configurations and observe either shear jamming (gain
of stiffness due to shear) or shear melting (loss of stiffness due to shear),
depending on the depth of training and proximity to the jamming transition
density. We also observe that the characteristic strains, at which shear
jamming or melting occur, diverge at a unique density $\varphi_{\rm JS}$. This
is due to the shift of the jamming transition density from $\varphi_{\rm J0}$
to $\varphi_{\rm JS}$ under shear, associated with loss of memory of the
initial configuration. Finally, we thoroughly investigate nonlinear rheology
near the jamming transition density, and contrary to previous works, we find a
nonlinear ``softening'' takes place below as well as above the jamming
transition density.

Soft Condensed Matter;Disordered Systems and Neural Networks