  A simultaneous calculation for the shape evolution and the related
spectroscopic properties of the low-lying states, and the $\beta$-decay
properties in the even- and odd-mass Ge and As nuclei in the mass
$A\approx70-80$ region, within the framework of the nuclear density functional
theory and the particle-core coupling scheme, is presented. The constrained
self-consistent mean-field calculations using a universal energy density
functional (EDF) and a pairing interaction determines the interacting-boson
Hamiltonian for the even-even core nuclei, and the essential ingredients of the
particle-boson interactions for the odd-nucleon systems, and of the
Gamow-Teller and Fermi transition operators. A rapid structural evolution from
$\gamma$-soft oblate to prolate shapes, as well as the spherical-oblate shape
coexistence around the neutron sub-shell closure $N=40$, is suggested to occur
in the even-even Ge nuclei. The predicted low-energy spectra, electromagnetic
transition rates, and $\beta$-decay $\log{ft}$ values are in a reasonable
agreement with experiment. The predicted $\log{ft}$ values reflect the
structures of the wave functions for the initial and final nuclei of $\beta$
decay, which are, to a large extent, determined by the microscopic input
provided by the underlying EDF calculation.
