Edges of the Barvinok-Novik orbitope

Here we study the k^th symmetric trigonometric moment curve and its convex hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these objects and show that there is some threshold so that for two points on S^1 with arclength below this threshold, the line segment between their lifts on the curve form an edge on the Barvinok-Novik orbitope and for points with arclenth above this threshold, their lifts do not form an edge. They also give a lower bound for this threshold and conjecture that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k.Comment: 10 pages, 3 figures, corrected Lemma 4 and other minor revision

Low-temperature chemistry using the R-matrix method

Techniques for producing cold and ultracold molecules are enabling the study of chemical reactions and scattering at the quantum scattering limit, with only a few partial waves contributing to the incident channel, leading to the observation and even full control of state-to-state collisions in this regime. A new R-matrix formalism is presented for tackling problems involving low- and ultra-low energy collisions. This general formalism is particularly appropriate for slow collisions occurring on potential energy surfaces with deep wells. The many resonance states make such systems hard to treat theoretically but offer the best prospects for novel physics: resonances are already being widely used to control diatomic systems and should provide the route to steering ultracold reactions. Our R-matrix-based formalism builds on the progress made in variational calculations of molecular spectra by using these methods to provide wavefunctions for the whole system at short internuclear distances, (a regime known as the inner region). These wavefunctions are used to construct collision energy-dependent R-matrices which can then be propagated to give cross sections at each collision energy. The method is formulated for ultracold collision systems with differing numbers of atoms

Determination of the tunneling flight time as the reflected phase time

Using the time parameter in the time-dependent Schrödinger equation, we study the time of flight for a particle tunneling through a square barrier potential. Comparing the mean and variance of the energy and the flight time for transmitted and reflected particles, using both density and flux distributions, we find that, when accounting for momentum filtering, the suitably normalized transmitted and reflected distributions are identical in both the density and flux cases. In contrast to previous studies, we demonstrate that these results do not imply a vanishing tunneling time, but rather that the time it takes to tunnel through a square barrier is precisely given by the reflected phase time. For wide barriers, this becomes independent of the barrier width, as predicted independently by MacColl and Hartman. We show that these conclusions can be reached using a variety of arguments, including purely quantum mechanical ones. Analysis of the shapes of the distributions under consideration reveals that wave-packet reshaping is not an explanation for the MacColl-Hartman effect. The results presented here have direct implications for understanding recent experimental results in the study of the barrier crossing of rubidium atoms. The finite width of an incident wave packet significantly “masks” the tunneling time, and induces substantial asymmetry between the flight times of transmitted and reflected atoms

The relativistic tunneling flight time may be superluminal, but it does not imply superluminal signaling

Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the time-dependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for above-barrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that post-selected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a photon will most likely be seen first and therefore the superluminality does not imply superluminal signaling

The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations

The characteristic polynomial for a chain of dipole-dipole coupled two-level atoms with nearest-neighbour and next-nearest-neighbour interactions is developed for the study of eigenvalues and eigenvectors for single-photon excitations. We find the exact form of the polynomial in terms of the Chebyshev polynomials of the second kind that is valid for an arbitrary number of atoms and coupling strengths. We then propose a technique for expressing the roots of the polynomial as a power series in the coupling constants. The general properties of the solutions are also explored, to shed some light on the general properties that the exact, analytic form of the energy eigenvalues should have. A method for deriving the eigenvectors of the Hamiltonian is also outlined.Comment: 19 pages, 8 figures; minor correction

Learning Arbitrary Statistical Mixtures of Discrete Distributions

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, $\vartheta$, is a probability distribution over probability distributions $p$, where each such $p$ is a probability distribution over $[n] = \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$ directly, but only indirectly and in very noisy fashion, by sampling from $[n]$ repeatedly, independently $K$ times from the distribution $p$. The problem is to infer $\vartheta$ to high accuracy in transportation (earthmover) distance. We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution $\vartheta$. We bound the quality of the solution as a function of the size of the samples $K$ and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.Comment: 23 pages. Preliminary version in the Proceeding of the 47th ACM Symposium on the Theory of Computing (STOC15

Fracture of a biopolymer gel as a viscoplastic disentanglement process

We present an extensive experimental study of mode-I, steady, slow crack dynamics in gelatin gels. Taking advantage of the sensitivity of the elastic stiffness to gel composition and history we confirm and extend the model for fracture of physical hydrogels which we proposed in a previous paper (Nature Materials, doi:10.1038/nmat1666 (2006)), which attributes decohesion to the viscoplastic pull-out of the network-constituting chains. So, we propose that, in contrast with chemically cross-linked ones, reversible gels fracture without chain scission

Community calcification in Lizard Island, Great Barrier Reef: a 33 year perspective

Measurements of community calcification (G) were made during September 2008 and October 2009 on a reef flat in Lizard Island, Great Barrier Reef, Australia, 33years after the first measurements were made there by the LIMER expedition in 1975. In 2008 and 2009 we measured G=61±12 and 54±13mmolCaCOm·day, respectively. These rates are 27-49% lower than those measured during the same season in 1975-76. These rates agree well with those estimated from the measured temperature and degree of aragonite saturation using a reef calcification rate equation developed from observations in a Red Sea coral reef. Community structure surveys across the Lizard Island reef flat during our study using the same methods employed in 1978 showed that live coral coverage had not changed significantly (~8%). However, it should be noted that the uncertainty in the live coral coverage estimates in this study and in 1978 were fairly large and inherent to this methodology. Using the reef calcification rate equation while assuming that seawater above the reef was at equilibrium with atmospheric PCO and given that live coral cover had not changed G should have declined by 30±8% since the LIMER study as indeed observed. We note, however, that the error in estimated G decrease relative to the 1970's could be much larger due to the uncertainties in the coral coverage measurements. Nonetheless, the similarity between the predicted and the measured decrease in G suggests that ocean acidification may be the primary cause for the lower CaCO precipitation rate on the Lizard Island reef flat