Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of {\em parabolic molecules}, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of {\em sparsity equivalence} is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets

Re-entrant pinning of Wigner molecules in a magnetic field due to a Coulomb impurity

Pinning of magnetic-field induced Wigner molecules (WMs) confined in parabolic two-dimensional quantum dots by a charged defect is studied by an exact diagonalization approach. We found a re-entrant pinning of the WMs as function of the magnetic field, a magnetic field induced re-orientation of the WMs and a qualitatively different pinning behaviour in the presence of a positive and negative Coulomb impurity

Far-infrared spectra of lateral quantum dot molecules

We study effects of electron-electron interactions and confinement potential on the magneto-optical absorption spectrum in the far-infrared range of lateral quantum dot molecules. We calculate far-infrared (FIR) spectra for three different quantum dot molecule confinement potentials. We use accurate exact diagonalization technique for two interacting electrons and calculate dipole-transitions between two-body levels with perturbation theory. We conclude that the two-electron FIR spectra directly reflect the symmetry of the confinement potential and interactions cause only small shifts in the spectra. These predictions could be tested in experiments with nonparabolic quantum dots by changing the number of confined electrons. We also calculate FIR spectra for up to six noninteracting electrons and observe some additional features in the spectrum.Comment: For better quality Figs download manuscript from http://www.fyslab.hut.fi/~mma/FIR/Helle_qdmfir.ps.g

Two-electron lateral quantum-dot molecules in a magnetic field

Laterally coupled quantum dot molecules are studied using exact diagonalization techniques. We examine the two-electron singlet-triplet energy difference as a function of magnetic field strength and investigate the magnetization and vortex formation of two- and four-minima lateral quantum dot molecules. Special attention is paid to the analysis of how the distorted symmetry affects the properties of quantum-dot molecules.Comment: 18 pages, 26 figure

Topological characterization of crystalline ice structures from coordination sequences

Topological properties of crystalline ice structures are studied by considering ring statistics, coordination sequences, and topological density of different ice phases. The coordination sequences (number of sites at topological distance k from a reference site) have been obtained by direct enumeration until at least 40 coordination spheres for different ice polymorphs. This allows us to study the asymptotic behavior of the mean number of sites in the k-th shell, M_k, for high values of k: M_k ~ a k^2, a being a structure-dependent parameter. Small departures from a strict parabolic dependence have been studied by considering first and second differences of the series {M_k} for each structure. The parameter a ranges from 2.00 for ice VI to 4.27 for ice XII, and is used to define a topological density for these solid phases of water. Correlations between such topological density and the actual volume of ice phases are discussed. Ices Ih and Ic are found to depart from the general trend in this correlation due to the large void space in their structures.Comment: 10 pages, 7 figures, 3 table

Molecular dynamics simulation of plane poiseuille flow in nanochannels

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.This paper presents new techniques and results of simulating microflows in plane channels by the molecular dynamics (MD) method. Mass forces and thermostat are not used in these techniques. The flows are simulated by both hard-sphere molecules and molecules with the Lennard-Jones intermolecular potential. Flow at a given fluid flow rate is implemented. In this case, the initial shock profile is transformed to a parabolic type profile. However, unlike in ordinary Poiseuille flows, a slip effect is recorded on the channel walls. It is shown that, in a nanochannel, a linear pressure gradient occurs. Fluid structuring is studied. The effects of fluid density, accommodation coefficients, and channel dimensions on flow properties are investigated.This work was supported in part by the Russian Foundation for Basic Researches (grant No. 07-08-00164) and by the grant of the President of the Russian Federation for Support of Leading Scientific Schools (project no. NSh-454.2008.1)

Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets