  We present the results of our ALMA HCN J=3-2 and HCO+ J=3-2 line observations
of a uniformly selected sample (>25) of nearby ultraluminous infrared galaxies
(ULIRGs) at z < 0.15. The emission of these dense molecular gas tracers and
continuum are spatially resolved in the majority of observed ULIRGs for the
first time with achieved synthesized beam sizes of ~0.2 arcsec or ~500 pc. In
most ULIRGs, the HCN-to-HCO+ J=3-2 flux ratios in the nuclear regions within
the beam size are systematically higher than those in the spatially extended
regions. The elevated nuclear HCN J=3-2 emission could be related to (a)
luminous buried active galactic nuclei, (b) the high molecular gas density and
temperature in ULIRG's nuclei, and/or (c) mechanical heating by spatially
compact nuclear outflows. A small fraction of the observed ULIRGs display
higher HCN-to-HCO+ J=3-2 flux ratios in localized off-nuclear regions than
those of the nuclei, which may be due to mechanical heating by spatially
extended outflows. The observed nearby ULIRGs are generally rich in dense
(>10^5 cm^-3) molecular gas, with an estimated mass of >10^9 Msun within the
nuclear (a few kpc) regions, and dense gas can dominate the total molecular
mass there. We find a low detection rate (<20%) regarding the possible
signature of a vibrationally excited (v2=1f) HCN J=3-2 emission line in the
vicinity of the bright HCO+ J=3-2 line that may be due, in part, to the large
molecular line widths of ULIRGs.
