We investigate the occurrence of topologically protected waves in classical fluids confined on curved surfaces. Using a combination of topological band theory and real space analysis, we demonstrate the existence of a system-independent mechanism beh
ind topological protection in two-dimensional passive and active fluids. This allows us to formulate an index theorem linking the number of modes, determined by the topology of Fourier space, to the real space topology of the surface on which they are hosted. With this framework in hand, we review two examples of topological waves in two-dimensional fluids, namely oceanic shallow-water waves propagating on the Earths rotating surface and momentum waves in active polar fluids spontaneously flocking on substrates endowed with a ${rm U}(1)$ isometry (e.g. surfaces of revolution). Our work suggests some simple rules to engineer topological modes on surfaces in passive and active soft matter systems.
A spectral principal component analysis (SPCA) of a sample of 87 PG QSOs at $z < 0.5$ is presented for their mid-infrared spectra from Spitzer Space Telescope. We have derived the first five eigenspectra, which account for 85.2% of the mid-infrared s
pectral variation. It is found that the first eigenspectrum represents the mid-infrared slope, forbidden emission line strength and $9.7~mu m$ silicate feature, the 3rd and 4th eigenspectra represent the silicate features at $18~ mu m$ and $9.7~mu m$, respectively. With the principal components (PC) from optical PCA, we find that there is a medium strong correlation between spectral SPC1 and PC2 (accretion rate). It suggests that more nuclear contribution to the near-IR spectrum leads to the change of mid-IR slope. We find mid-IR forbidden lines are suppressed with higher accretion rate. A medium strong correlation between SPC3 and PC1 (Eddington ratio) suggests a connection between the silicate feature at $18~mu m$ and the Eddington ratio. For the ratio of the silicate strength at 9.7 $mu m$ to that at 18 $mu m$, we find a strong correlation with PC2 (accretion rate or QSO luminosity). We also find that there is a medium strong correlation between the star formation rate (SFR) and PC2. It implies a correlation between star formation rate and the central accretion rate in PG QSOs.
We present N-body simulations of resonant planets with inclined orbits that show chaotically evolving eccentricities and inclinations that can persist for at least 10 Gyr. A wide range of behavior is possible, from fast, low amplitude variations to s
ystems in which eccentricities reach 0.9999 and inclinations 179.9 degrees. While the orbital elements evolve chaotically, at least one resonant argument always librates. We show that the HD 73526, HD 45364 and HD 60532 systems may be in chaotically-evolving resonances. Chaotic evolution is apparent in the 2:1, 3:1 and 3:2 resonances, and for planetary masses from lunar- to Jupiter-mass. In some cases, orbital disruption occurs after several Gyr, implying the mechanism is not rigorously stable, just long-lived relative to the main sequence lifetimes of solar-type stars. Planet-planet scattering appears to yield planets in inclined resonances that evolve chaotically in about 0.5% of cases. These results suggest that 1) approximate methods for identifying unstable orbital architectures may have limited applicability, 2) the observed close-in exoplanets may be produced during the high eccentricity phases induced by inclined resonances, 3) those exoplanets orbital planes may be misaligned with the host stars spin axis, 4) systems with resonances may be systematically younger than those without, 5) the distribution of period ratios of adjacent planets detected via transit may be skewed due to inclined resonances, and 6) potentially habitable planets in resonances may have dramatically different climatic evolution than the Earth. The GAIA spacecraft is capable of discovering giant planets in these types of orbits.
We evaluate the orbital evolution and several plausible origins scenarios for the mutually inclined orbits of Upsilon Andromedae c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boun
dary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, Upsilon And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of ~30 deg. However, it could incite ejections or collisions between planetary companions that subsequently raise the mutual inclination to >30 deg. Our simulated systems with large mutual inclinations tend to be further from the stability boundary than Upsilon And, but we are able to produce similar systems. We conclude that scattering is a plausible mechanism to explain the observed orbits of Upsilon And c and d, but we cannot determine whether the scattering was caused by instabilities among the planets themselves or by perturbations from Upsilon And B. We also develop a procedure to quantitatively compare numerous properties of the observed system to our numerical models. Although we only implement this procedure to Upsilon And, it may be applied to any exoplanetary system.
