Toward a new theory of the fractional quantum Hall effect

The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx = e 2/3h, in the vicinity of the Landau level filling factor ν = 1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N ≤ 7 electrons continuously as a function of magnetic field in the range 1/4 ≲ ν < 1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν = 1/3 Laughlin state for N ≤ 8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect

Periodic Solutions in Rn\mathbb R^n for Stationary Anisotropic Stokes and Navier-Stokes Systems

First, the solution uniqueness and existence of a stationary anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework on $n$-dimensional flat torus are analysed in a range of periodic Sobolev (Bessel-potential) spaces. By employing the Leray-Schauder fixed point theorem, the linear results are employed to show existence of solution to the stationary anisotropic (non-linear) Navier-Stokes incompressible system on torus in a periodic Sobolev space. Then the solution regularity results for stationary anisotropic Navier-Stokes system on torus are established.Comment: 15 pages, correcte

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2011 ElsevierFor functions from the Sobolev space H^s(\Omega­), 1/2 < s < 3/2 , definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain­, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although de¯ned on a more narrow function class than the generalized ones, are continuous extensions of the classical co-norma derivatives. Some new results about trace operator estimates and Sobolev spaces haracterizations, are also presented

CAA Modeling of Helicopter Main Rotor in Hover

In this work rotor aeroacoustics in hover is considered. Farfield observers are used and the nearfield flow parameters are obtained using the in house HMB and commercial Fluent CFD codes (identical hexa-grids are used for both solvers). Farfield noise at a remote observer position is calculated at post processing stage using FW–H solver implemented in Fluent and HMB. The main rotor of the UH-1H helicopter is considered as a test case for comparison to experimental data. The sound pressure level is estimated for different rotor blade collectives and observation angles