Polarizations and Nullcone of Representations of Reductive Groups

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f_1,f_2,...,f_n be homogeneous invariant functions. Then the polarizations of f_1,f_2,...,f_n define the nullcone of k 0} h(t) x = 0 for all x in L. This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of $SL_2$ on the n-fold tensor product C^2 otimes C^2 otimes ... otimes C^2. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n <= 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. {bf 224} (2006), 119--140.

Image interpolation using Shearlet based iterative refinement

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images

Evidence of sympathetic cooling of Na+ ions by a Na MOT in a hybrid trap

A hybrid ion-neutral trap provides an ideal system to study collisional dynamics between ions and neutrals. This system provides a general cooling method that can be applied to optically inaccessible species and can also potentially cool internal degrees of freedom. The long range polarization potentials ($V\propto-\alpha/r^4$) between ions and neutrals result in large scattering cross sections at cold temperatures, making the hybrid trap a favorable system for efficient sympathetic cooling of ions by collisions with neutral atoms. We present experimental evidence of sympathetic cooling in a hybrid trap of \ce{Na+} ions, which are closed shell and therefore do not have a laser induced atomic transition, by equal mass cold Na atoms in a magneto-optical trap (MOT).Comment: 7 figure

Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata

Functions which are equivariant or invariant under the transformations of a compact linear group $G$ acting in an euclidean space $\real^n$, can profitably be studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same orbit-type. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of equivariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau theory), in equivariant bifurcation theory, in crystal field theory and in most areas where use is made of symmetry adapted functions. A physically significant example of utilization of the recipe is given, related to spontaneous polarization in chiral biaxial liquid crystals, where the advantages with respect to previous heuristic approaches are shown.Comment: Figures generated through texdraw package; revised version appearing in J. Phys. A: Math. Ge

Genus Topology of the Cosmic Microwave Background from the WMAP 3-Year Data

We have independently measured the genus topology of the temperature fluctuations in the cosmic microwave background seen in the Wilkinson Microwave Anisotropy Probe (WMAP) 3-year data. A genus analysis of the WMAP data indicates consistency with Gaussian random-phase initial conditions, as predicted by standard inflation. We set 95% confidence limits on non-linearities of -101 < f_{nl} < 107. We also find that the observed low l (l <= 8) modes show a slight anti-correlation with the Galactic foreground, but not exceeding 95% confidence, and that the topology defined by these modes is consistent with that of a Gaussian random-phase distribution (within 95% confidence).Comment: MNRAS LaTeX style (mn2e.cls), EPS and JPEG figure

Ion-neutral sympathetic cooling in a hybrid linear rf Paul and magneto-optical trap

Long range polarization forces between ions and neutral atoms result in large elastic scattering cross sections, e.g., 10^6 a.u. for Na+ on Na or Ca+ on Na at cold and ultracold temperatures. This suggests that a hybrid ion-neutral trap should offer a general means for significant sympathetic cooling of atomic or molecular ions. We present SIMION 7.0 simulation results concerning the advantages and limitations of sympathetic cooling within a hybrid trap apparatus, consisting of a linear rf Paul trap concentric with a Na magneto-optical trap (MOT). This paper explores the impact of various heating mechanisms on the hybrid system and how parameters related to the MOT, Paul trap, number of ions, and ion species affect the efficiency of the sympathetic cooling

Dynamics of the DBI Spike Soliton

We compare oscillations of a fundamental string ending on a D3-brane in two different settings: (1) a test-string radially threading the horizon of an extremal black D3-brane and (2) the spike soliton of the DBI effective action for a D3-brane. Previous work has shown that overall transverse modes of the test-string appear as l=0 modes of the transverse scalar fields of the DBI system. We identify DBI world-volume degrees of freedom that have dynamics matching those of the test-string relative transverse modes. We show that there is a map, resembling T-duality, between relative and overall transverse modes for the test-string that interchanges Neumann and Dirichlet boundary conditions and implies equality of the absorption coefficients for both modes. We give general solutions to the overall and relative transverse parts of the DBI coupled gauge and scalar system and calculate absorption coefficients for the higher angular momentum modes in the low frequency limit. We find that there is a nonzero amplitude for l>0 modes to travel out to infinity along the spike, demonstrating that the spike remains effectively 3+1-dimensional.Comment: 15 pages, 1 figur

Coherent laminar and turbulent motion of toroidal vortex bundles

Motivated by experiments performed in superfluid helium, we study numerically the motion of toroidal bundles of vortex filaments in an inviscid fluid. We find that the evolution of these large-scale vortex structures involves the generalised leapfrogging of the constituent vortex rings. Despite three dimensional perturbations in the form of Kelvin waves and vortex reconnections, toroidal vortex bundles retain their coherence over a relatively large distance (compared to their size), in agreement with experimental observations.Comment: 22 pages, 12 figure

Relativistic J-matrix method

The relativistic version of the J-matrix method for a scattering problem on the potential vanishing faster than the Coulomb one is formulated. As in the non-relativistic case it leads to a finite algebraic eigenvalue problem. The derived expression for the tangent of phase shift is simply related to the non-relativistic case formula and gives the latter as a limit case. It is due to the fact that the used basis set satisfies the ``kinetic balance condition''.Comment: 21 pages, RevTeX, accepted for publication in Phys. Rev.

Classical and quantum regimes of the superfluid turbulence

We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with 1/q playing the same role as the Reynolds number Re=UR/\nu in classical hydrodynamics. It marks the transition between the "laminar" and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re >> 1 in classical hydrodynamics, and q << 1 in the superfluid hydrodynamics. Another parameter of the superfluid turbulence is the superfluid Reynolds number Re_s=UR/\kappa, which contains the circulation quantum \kappa characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; and (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. The phase diagram of the dynamical vortex states is suggested.Comment: 12 pages, 1 figure, version accepted in JETP Letter