Transversally Elliptic Operators

We construct certain spectral triples in the sense of A. ~Connes and H. \~Moscovici (``The local index formula in noncommutative geometry'' {\it Geom. Funct. Anal.}, 5(2):174--243, 1995) that is transversally elliptic but not necessarily elliptic. We prove that these spectral triples satisfie the conditions which ensure the Connes-Moscovici local index formula applies. We show that such a spectral triple has discrete dimensional spectrum. A notable feature of the spectral triple is that its corresponding zeta functions have multiple poles, while in the classical elliptic cases only simple poles appear for the zeta functions. We show that the multiplicities of the poles of the zeta functions have an upper bound, which is the sum of dimensions of the base manifold and the acting compact Lie group. Moreover for our spectral triple the Connes-Moscovici local index formula involves only local transverse symbol of the operator.Comment: Updated 11/25/2003 with corrected format, and in 12pt fonts Updated 5/20/2004, major reorganizatio

Confinements regulate capillary instabilities of fluid threads

We study the breakup of confined fluid threads at low flow rates to understand instability mechanisms. To determine the critical conditions between the earlier quasi-stable necking stage and the later unstable collapse stage, simulations and experiments are designed to operate at an extremely low flow rate. Critical mean radii at neck centres are identified by the stop-flow method for elementary microfluidic configurations. Analytical investigations reveal two distinct origins of capillary instabilities. One is the gradient of capillary pressure induced by the confinements of geometry and external flow, whereas the other is the competition between local capillary pressure and internal pressure determined by the confinements

Stability vs. optimality in selfish ring routing

We study the asymmetric atomic selfish routing in ring networks, which has diverse practical applications in network design and analysis. We are concerned with minimizing the maximum latency of source-destination node-pairs over links with linear latencies. We obtain the first constant upper bound on the price of anarchy and significantly improve the existing upper bounds on the price of stability. Moreover, we show that any optimal solution is a good approximate Nash equilibrium. Finally, we present better performance analysis and fast implementation of pseudo-polynomial algorithms for computing approximate Nash equilibria

Unique determination of balls and polyhedral scatterers with a single point source wave

In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in \R^N (N=2,3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a 'point-to-point' extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data incited by a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed

Anomalous Light Scattering by Pure Seawater

The latest model for light scattering by pure seawater was used to investigate the anomalous behavior of pure water. The results showed that water exhibits a minimum scattering at 24.6 °C, as compared to the previously reported values of minimum scattering at 22 °C or maximum scattering at 15 °C. The temperature corresponding to the minimum scattering also increases with the salinity, reaching 27.5 °C for S = 40 ps

Unique determination of balls and polyhedral scatterers with a single point source wave

In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in RN (N = 2, 3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a 'point-to-point extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data incited by a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed