Far-Field Compression for Fast Kernel Summation Methods in High Dimensions

We consider fast kernel summations in high dimensions: given a large set of points in $d$ dimensions (with $d \gg 3$) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vector multiplication and scales quadratically with the number of points. Fast kernel summation algorithms reduce this cost to log-linear or linear complexity. Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by constructing approximate representations of interactions of points that are far from each other. In algebraic terms, these representations correspond to low-rank approximations of blocks of the overall interaction matrix. Existing approaches require an excessive number of kernel evaluations with increasing $d$ and number of points in the dataset. To address this issue, we use a randomized algebraic approach in which we first sample the rows of a block and then construct its approximate, low-rank interpolative decomposition. We examine the feasibility of this approach theoretically and experimentally. We provide a new theoretical result showing a tighter bound on the reconstruction error from uniformly sampling rows than the existing state-of-the-art. We demonstrate that our sampling approach is competitive with existing (but prohibitively expensive) methods from the literature. We also construct kernel matrices for the Laplacian, Gaussian, and polynomial kernels -- all commonly used in physics and data analysis. We explore the numerical properties of blocks of these matrices, and show that they are amenable to our approach. Depending on the data set, our randomized algorithm can successfully compute low rank approximations in high dimensions. We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure

Statistical Description of Hydrodynamic Processes in Ionic Melts with taking into account Polarization Effects

Statistical description of hydrodynamic processes for ionic melts is proposed with taking into account polarization effects caused by the deformation of external ionic shells. This description is carried out by means of the Zubarev nonequilibrium statistical operator method, appropriate for investigations of both strong and weak nonequilibrium processes. The nonequilibrium statistical operator and the generalized hydrodynamic equations that take into account polarization processes are received for ionic-polarization model of ionic molten salts when the nonequilibrium averaged values of densities of ions number, their momentum, dipole momentum and total energy are chosen for the reduced description parameters. A spectrum of collective excitations is investigated within the viscoelastic approximation for ion-polarization model of ionic melts.Comment: 24 pages, RevTex4.1-format, no figure

Tree-Independent Dual-Tree Algorithms

Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split: the tree, the traversal, the point-to-point base case, and the pruning rule. We provide a meta-algorithm which allows development of dual-tree algorithms in a tree-independent manner and easy extension to entirely new types of trees. Representations are provided for five common algorithms; for k-nearest neighbor search, this leads to a novel, tighter pruning bound. The meta-algorithm also allows straightforward extensions to massively parallel settings.Comment: accepted in ICML 201

Mesoscopic superpositions of Tonks-Girardeau states and the Bose-Fermi mapping

We study a one dimensional gas of repulsively interacting ultracold bosons trapped in a double-well potential as the atom-atom interactions are tuned from zero to infinity. We concentrate on the properties of the excited states which evolve from the so-called NOON states to the NOON Tonks-Girardeau states. The relation between the latter and the Bose-Fermi mapping limit is explored. We state under which conditions NOON Tonks-Girardeau states, which are not predicted by the Bose-Fermi mapping, will appear in the spectrum.Comment: 5 pages, 5 figure

Power, norms and institutional change in the European Union: the protection of the free movement of goods

How do institutions of the European Union change? Using an institutionalist approach, this article highlights the interplay between power, cognitive limits, and the normative order that underpins institutional settings and assesses their impact upon the process of institutional change. Empirical evidence from recent attempts to reinforce the protection of the free movement of goods in the EU suggests that, under conditions of uncertainty, actors with ambiguous preferences assess attempts at institutional change on the basis of the historically defined normative order which holds a given institutional structure together. Hence, path dependent and incremental change occurs even when more ambitious and functionally superior proposals are on offer

Correlations in a two-dimensional Bose gas with long range interactions

We study the correlations of two-dimensional dipolar excitons in coupled quantum wells with a dipole -- dipole repulsive interaction. We show that at low concentrations, the Bose degeneracy of the excitons is accompanied by strong multi-particle correlations and the system behaves as a Bose liquid. At high concentration the particles interaction suppresses quantum coherence and the system behaves similar to a classical liquid down to a temperature lower than typical for a Bose gas. We evaluate the interaction energy per particle and the resulting blue shift of the exciton luminescence that is a direct tool to measure the correlations. This theory can apply to other systems of bosons with extended interaction.Comment: 11 pages including 2 figure

Sharp crossover from composite fermionization to phase separation in mesoscopic mixtures of ultracold bosons

We show that a two-component mixture of a few repulsively interacting ultracold atoms in a one-dimensional trap possesses very different quantum regimes and that the crossover between them can be induced by tuning the interactions in one of the species. In the composite fermionization regime, where the interactions between both components are large, none of the species show large occupation of any natural orbital. Our results show that by increasing the interaction in one of the species, one can reach the phase-separated regime. In this regime, the weakly interacting component stays at the center of the trap and becomes almost fully phase coherent, while the strongly interacting component is displaced to the edges of the trap. The crossover is sharp, as observed in the in the energy and the in the largest occupation of a natural orbital of the weakly interacting species. Such a transition is a purely mesoscopic effect which disappears for large atom numbers.Comment: 5 pages, 3 figure

Quantum correlations and spatial localization in one-dimensional ultracold bosonic mixtures

We present the complete phase diagram for one-dimensional binary mixtures of bosonic ultracold atomic gases in a harmonic trap. We obtain exact results with direct numerical diagonalization for small number of atoms, which permits us to quantify quantum many-body correlations. The quantum Monte Carlo method is used to calculate energies and density profiles for larger system sizes. We study the system properties for a wide range of interaction parameters. For the extreme values of these parameters, different correlation limits can be identified, where the correlations are either weak or strong. We investigate in detail how the correlation evolve between the limits. For balanced mixtures in the number of atoms in each species, the transition between the different limits involves sophisticated changes in the one- and two-body correlations. Particularly, we quantify the entanglement between the two components by means of the von Neumann entropy. We show that the limits equally exist when the number of atoms is increased, for balanced mixtures. Also, the changes in the correlations along the transitions among these limits are qualitatively similar. We also show that, for imbalanced mixtures, the same limits with similar transitions exist. Finally, for strongly imbalanced systems, only two limits survive, i.e., a miscible limit and a phase-separated one, resembling those expected with a mean-field approach.Comment: 18 pages, 8 figure

Cryogenic Ion Trapping Systems with Surface-Electrode Traps

We present two simple cryogenic RF ion trap systems in which cryogenic temperatures and ultra high vacuum pressures can be reached in as little as 12 hours. The ion traps are operated either in a liquid helium bath cryostat or in a low vibration closed cycle cryostat. The fast turn around time and availability of buffer gas cooling made the systems ideal for testing surface-electrode ion traps. The vibration amplitude of the closed cycled cryostat was found to be below 106 nm. We evaluated the systems by loading surface-electrode ion traps with $^{88}$Sr$^+$ ions using laser ablation, which is compatible with the cryogenic environment. Using Doppler cooling we observed small ion crystals in which optically resolved ions have a trapped lifetime over 2500 minutes.Comment: 10 pages, 13 EPS figure