  We consider a quantum dot coupled to both superconducting and spin-polarized
electrodes, and study the triad interplay of the Kondo effect,
superconductivity, and ferromagnetism, any pair of which compete with and
suppress each other. We find that the interplay leads to a mixed-valence
quantum phase transition, which for other typical sysmstems is merely a
crossover rather than a true transition. At the transition, the system changes
from the spin doublet to singlet state. The singlet phase is adiabatically
connected (through crossovers) to the so-called 'charge Kondo state' and to the
superconducting state. We analyze in detail the physical characteristics of
different states and propose that the measurement of the cross-current
correlation and the charge relaxation resistance can clearly distinguish
between them.
