Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. L\'{e}vy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the L\'{e}vy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strongly interacting neutrinos as the highest energy cosmic rays

We show that all features of the ultrahigh energy cosmic ray spectrum from 10^{17} eV to 10^{21} eV can be described with a simple power-like injection spectrum of protons under the assumption that the neutrino-nucleon cross-section is significantly enhanced at center of mass energies above \approx 100 TeV. In our scenario, the cosmogenic neutrinos produced during the propagation of protons through the cosmic microwave background initiate air showers in the atmosphere, just as the protons. The total air shower spectrum induced by protons and neutrinos shows excellent agreement with the observations. A particular possibility for a large neutrino-nucleon cross-section exists within the Standard Model through electroweak instanton-induced processes.Comment: 8 pages, 4 figures, talk given at Beyond the Desert '03, Castle Ringberg, 9-14 June, 200

Domain structure in the monoclinic Pm phase of Pb(Mg1/3Nb2/3)O3 - PbTiO3 single crystals

The domain structure of (1-x)Pb(Mg1/3Nb2/3)O3- xPbTiO3 single crystals with composition x ~ 0.33 in the range of the morphotropic phase boundary (MPB) was studied. Based on the analysis of spontaneous strain compatibility and charge of domain walls, we have established the permissible domain arrangements for the ferroelectric phases of different symmetry, which are expected to occur in the range of the MPB. Examination of (001)-oriented unpoled and electrically poled (along the [001] direction) crystal plates in polarizing microscope reveals a monophase state with the domain structure compatible with the structure theoretically predicted for the Mc monoclinic phase (space group Pm), which was recently discovered in the compositions close to the MPB by x-ray and neutron diffraction studies. In the unpoled crystal the 180 deg. walls between the domains whose spontaneous polarization vectors are parallel to the plane of the crystal plate (i.e. a-domains) are observed. The domain structure of the poled crystal is predominantly composed of crystallographically prominent Wf walls parallel to (001) (i.e. the plane of the crystal plate) and inclined S walls parallel to [110] or [1-10] direction. In poled and unpoled samples the optical rotatory polarization effect is found, which is related to the inhomogeneity of the sample resulting from the mixture of domains.Comment: PDF file, 30 pages, 15 figures (3 in color) collected on pp.18-3

Maximized string order parameters in the valence bond solid states of quantum integer spin chains

We propose a set of maximized string order parameters to describe the hidden topological order in the valence bond solid states of quantum integer spin-S chains. These optimized string order parameters involve spin-twist angles corresponding to $Z_{S+1}$ rotations around $z$ or $x$-axes, suggesting a hidden $Z_{S+1}\times Z_{S+1}$ symmetry. Our results also suggest that a local triplet excitation in the valence bond solid states carries a $Z_{S+1}$ topological charge measured by these maximized string order parameters.Comment: 5 pages, 1 figur