  In the quantum Hall effect (QHE) the differential resistivity $r_{xx} \equiv
r$ vanishes within a range where the Hall resistivity forms a plateau. A
microscopic theory is developed, starting with a crystal lattice, setting up a
BCS-like Hamiltonian in terms of composite bosons, and using statistical
mechanical method. The main advantage of our bosonic theory is its capability
of explaning the plateau formation in the Hall resistivity, which is assumed in
the composite fermion theories. In the QHE under radiation, the resistivity
vanishes within a range with no plateau formation. This is shown in terms of
two-channels model, one channel excited by radiation where the supercurrents
run and the other (base) channel in which the normal currents run. The
transport diamonds (TD) and the zero direct current anomaly (ZCA) occur when
the resistivity $r$ is measured as a function of magnetic field and direct
current (DC). The spiral motion of an electron under a magnetic field can be
decomposed into two, the cyclotron motion with the cyclotron mass $m^*$ and the
guiding center motion with the magnetotransport $M^*$. The quantization of the
motion generates magnetic oscillations in the density of states. The
magnetoconductivity is calculated, using kinetic theory and quantum statistical
mechanics. The TR and ZCA are shown to be a breakdown of QHE. The integer QHE
minima are shown to become the Shubnikov-de Haas (SdH) maxima progressively as
the DC increases. The ZCA at low temperatures ($T=0.253$--1.2\,K) is
temperature-dependent, which is caused by the electron-optical-phonon
scattering.
