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

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

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

Chebyshev matrix product state approach for spectral functions

We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebychev matrix product state (CheMPS) approach are: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model's many-body bandwidth; (iii) it offers a well-controlled broadening scheme; (iv) it is based on a succession of Chebychev vectors |t_n>, (v) whose entanglement entropies were found to remain bounded with increasing recursion order n for all cases analyzed here; (vi) it distributes the total entanglement entropy that accumulates with increasing n over the set of Chebyshev vectors |t_n>. We present zero-temperature CheMPS results for the structure factor of spin-1/2 antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (1) yields results comparable in quality to those of correction vector DMRG, at dramatically reduced numerical cost; (2) agrees well with Bethe Ansatz results for an infinite system, within the limitations expected for numerics on finite systems; (3) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.Comment: 22 pages, 13 figure

The unfolded protein response affects readthrough of premature termination codons

One-third of monogenic inherited diseases result from premature termination codons (PTCs). Readthrough of in-frame PTCs enables synthesis of full-length functional proteins. However, extended variability in the response to readthrough treatment is found among patients, which correlates with the level of nonsense transcripts. Here, we aimed to reveal cellular pathways affecting this inter-patient variability. We show that activation of the unfolded protein response (UPR) governs the response to readthrough treatment by regulating the levels of transcripts carrying PTCs. Quantitative proteomic analyses showed substantial differences in UPR activation between patients carrying PTCs, correlating with their response. We further found a significant inverse correlation between the UPR and nonsense-mediated mRNA decay (NMD), suggesting a feedback loop between these homeostatic pathways. We uncovered and characterized the mechanism underlying this NMD-UPR feedback loop, which augments both UPR activation and NMD attenuation. Importantly, this feedback loop enhances the response to readthrough treatment, highlighting its clinical importance. Altogether, our study demonstrates the importance of the UPR and its regulatory network for genetic diseases caused by PTCs and for cell homeostasis under normal conditions

Deformation of the three-term recursion relation and the generation of new orthogonal polynomials

We find solutions for a linear deformation of the symmetric three-term recursion relation. The orthogonal polynomials of the first and second kind associated with the deformed relation are obtained. The new density (weight) function is written in terms of the original one and the deformation parameters.Comment: 12 pages, 3 figure

Preserving Differential Privacy in Convolutional Deep Belief Networks

The remarkable development of deep learning in medicine and healthcare domain presents obvious privacy issues, when deep neural networks are built on users' personal and highly sensitive data, e.g., clinical records, user profiles, biomedical images, etc. However, only a few scientific studies on preserving privacy in deep learning have been conducted. In this paper, we focus on developing a private convolutional deep belief network (pCDBN), which essentially is a convolutional deep belief network (CDBN) under differential privacy. Our main idea of enforcing epsilon-differential privacy is to leverage the functional mechanism to perturb the energy-based objective functions of traditional CDBNs, rather than their results. One key contribution of this work is that we propose the use of Chebyshev expansion to derive the approximate polynomial representation of objective functions. Our theoretical analysis shows that we can further derive the sensitivity and error bounds of the approximate polynomial representation. As a result, preserving differential privacy in CDBNs is feasible. We applied our model in a health social network, i.e., YesiWell data, and in a handwriting digit dataset, i.e., MNIST data, for human behavior prediction, human behavior classification, and handwriting digit recognition tasks. Theoretical analysis and rigorous experimental evaluations show that the pCDBN is highly effective. It significantly outperforms existing solutions