On the composition of convex envelopes for quadrilinear terms

International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting

A holographic model for the fractional quantum Hall effect

Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a modular symmetry group commuting with the renormalization group flow and hence mapping different phases of two-dimensional electron gases into each other. Based on this insight, we construct a phenomenological holographic model which captures many features of the fractional quantum Hall effect. Using an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton theory capturing the important modular transformation properties of quantum Hall physics, we find dyonic diatonic black hole solutions which are gapped and have a Hall conductivity equal to the filling fraction, as expected for quantum Hall states. We also provide several technical results on the general behavior of the gauge field fluctuations around these dyonic dilatonic black hole solutions: We specify a sufficient criterion for IR normalizability of the fluctuations, demonstrate the preservation of the gap under the SL(2,Z) action, and prove that the singularity of the fluctuation problem in the presence of a magnetic field is an accessory singularity. We finish with a preliminary investigation of the possible IR scaling solutions of our model and some speculations on how they could be important for the observed universality of quantum Hall transitions.Comment: 86 pages, 16 figures; v.2 references added, typos fixed, improved discussion of ref. [39]; v.3 more references added and typos fixed, several statements clarified, v.4 version accepted for publication in JHE

Inverse spin-s portrait and representation of qudit states by single probability vectors

Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j+1)(4j+1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits (j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and (2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s portraits. We also address an auxiliary problem of the optimum reconstruction of qudit states, where the optimality implies a minimum relative error of the density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian Laser Researc

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques

Widespread forest vertebrate extinctions induced by a mega hydroelectric dam in lowland Amazonia

Mega hydropower projects in tropical forests pose a major emergent threat to terrestrial and freshwater biodiversity worldwide. Despite the unprecedented number of existing, underconstruction and planned hydroelectric dams in lowland tropical forests, long-term effects on biodiversity have yet to be evaluated. We examine how medium and large-bodied assemblages of terrestrial and arboreal vertebrates (including 35 mammal, bird and tortoise species) responded to the drastic 26-year post-isolation history of archipelagic alteration in landscape structure and habitat quality in a major hydroelectric reservoir of Central Amazonia. The Balbina Hydroelectric Dam inundated 3,129 km2 of primary forests, simultaneously isolating 3,546 land-bridge islands. We conducted intensive biodiversity surveys at 37 of those islands and three adjacent continuous forests using a combination of four survey techniques, and detected strong forest habitat area effects in explaining patterns of vertebrate extinction. Beyond clear area effects, edge-mediated surface fire disturbance was the most important additional driver of species loss, particularly in islands smaller than 10 ha. Based on species-area models, we predict that only 0.7% of all islands now harbor a species-rich vertebrate assemblage consisting of ≥80% of all species. We highlight the colossal erosion in vertebrate diversity driven by a man-made dam and show that the biodiversity impacts of mega dams in lowland tropical forest regions have been severely overlooked. The geopolitical strategy to deploy many more large hydropower infrastructure projects in regions like lowland Amazonia should be urgently reassessed, and we strongly advise that long-term biodiversity impacts should be explicitly included in pre-approval environmental impact assessments

Super-resolving phase measurements with a multi-photon entangled state

Using a linear optical elements and post-selection, we construct an entangled polarization state of three photons in the same spatial mode. This state is analogous to a ``photon-number path entangled state'' and can be used for super-resolving interferometry. Measuring a birefringent phase shift, we demonstrate two- and three-fold improvements in phase resolution.Comment: 4 pages, 3 figure

Designing libraries of chimeric proteins using SCHEMA recombination and RASPP

SCHEMA is a method for designing libraries of novel proteins by recombination of homologous sequences. The goal is to maximize the number of folded proteins while simultaneously generating significant sequence diversity. Here, we use the RASPP algorithm to identify optimal SCHEMA designs for shuffling contiguous elements of sequence. To exemplify the method, SCHEMA is used to recombine five fungal cellobiohydrolases (CBH1s) to produce a library of more than 390,000 novel CBH1 sequences

How to Specify It!: A Guide to Writing Properties of Pure Functions

Property-based testing tools test software against a\ua0specification, rather than a set of examples. This tutorial paper presents five generic approaches to writing such specifications (for purely functional code). We discuss the costs, benefits, and bug-finding power of each approach, with reference to a simple example with eight buggy variants. The lessons learned should help the reader to develop effective property-based tests in the future

Experimental investigation of classical and quantum correlations under decoherence

It is well known that many operations in quantum information processing depend largely on a special kind of quantum correlation, that is, entanglement. However, there are also quantum tasks that display the quantum advantage without entanglement. Distinguishing classical and quantum correlations in quantum systems is therefore of both fundamental and practical importance. In consideration of the unavoidable interaction between correlated systems and the environment, understanding the dynamics of correlations would stimulate great interest. In this study, we investigate the dynamics of different kinds of bipartite correlations in an all-optical experimental setup. The sudden change in behaviour in the decay rates of correlations and their immunity against certain decoherences are shown. Moreover, quantum correlation is observed to be larger than classical correlation, which disproves the early conjecture that classical correlation is always greater than quantum correlation. Our observations may be important for quantum information processing.Comment: 7 pages, 4 figures, to appear in Nature Communication