The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.Comment: 48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phy

On the occurrence of gauge-dependent secularities in nonlinear gravitational waves

We study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this instability can not be eliminated by means of a multiscale approach, i.e. introducing suitable long variables, as it is often the case when secularities appear in a perturbative scheme. However, this is a non-physical and gauge-dependent effect that disappears in a suitable system of coordinates. In facts, we show that in a specific gauge such instability does not occur, and that it is possible to solve exactly the second order nonlinear equations of gravitational waves. Incidentally, we note that this gauge coincides with the one used by Belinski and Zakharov to find exact solitonic solutions of Einstein's equations, that is to an exactly integrable case, and this fact makes our second order nonlinear solutions less interesting. However, the important warning is that one must be aware of the existence of the instability reported in this paper, when studying nonlinear gravitational waves in the harmonic gauge

The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulas

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow to solve Dirichlet boundary value problems; then we present their Darboux transformations. Finally we construct their Lelieuvre formulas and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.Comment: 20 pages, 6 figures, submitted to Glasgow Mathematical Journal Trust for Island II proceedind

Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie on the sphere S^3, ii) the distance between any two subsequant points does not vary in time, iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves.Comment: LaTeX file, 14 pages + 4 figure

Integrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability. We present the geometric meaning of its tau-function, as the potential connecting its forward and backward data. We present the theory of transformations of the quadrilateral lattice, which is based on the discrete analogue of the theory of rectilinear congruences. In particular, we discuss the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. We also show how the sequence of Laplace transformations of a quadrilateral surface is described by the discrete Toda system. We finally show that these classical transformations are strictly related to the basic operators associated with the quantum field theoretical formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review the properties of quadrilateral hyperplane lattices, which play an interesting role in the reduction theory, when the introduction of additional geometric structures allows to establish a connection between point and hyperplane lattices. We present and fully characterize some geometrically distinguished reductions of the quadrilateral lattice, like the symmetric, circular and Egorov lattices; we review also basic geometric results of the theory of quadrilateral lattices in quadrics, and the corresponding analogue of the Ribaucour reduction of the fundamental transformation.Comment: 27 pages, 9 figures, to appear in Proceedings from the Conference "Symmetries and Integrability of Difference Equations III", Sabaudia, 199

Evidence for detrimental cross interactions between reactive oxygen and nitrogen species in Leber's hereditary optic neuropathy cells

Here we have collected evidence suggesting that chronic changes in the NO homeostasis and the rise of reactive oxygen species bioavailability can contribute to cell dysfunction in Leber’s hereditary optic neuropathy (LHON) patients.We report that peripheral blood mononuclear cells (PBMCs), derived froma female LHON patient with bilateral reduced vision and carrying the pathogenic mutation 11778/ND4, display increased levels of reactive oxygen species (ROS) and reactive nitrogen species (RNS), as revealed by flow cytometry, fluorometric measurements of nitrite/nitrate, and 3-nitrotyrosine immunodetection. Moreover, viability assays with the tetrazolium dye MTT showed that lymphoblasts from the same patient are more sensitive to prolonged NO exposure, leading to cell death. Taken together these findings suggest that oxidative and nitrosative stress cooperatively play an important role in driving LHON pathology when excess NO remains available over time in the cell environment

The nexus between forest fragmentation in Africa and Ebola virus disease outbreaks

Tropical forests are undergoing land use change in many regions of the world, including the African continent. Human populations living close to forest margins fragmented and disturbed by deforestation may be particularly exposed to zoonotic infections because of the higher likelihood for humans to be in contact with disease reservoirs. Quantitative analysis of the nexus between deforestation and the emergence of Ebola virus disease (EVD), however, is still missing. Here we use land cover change data in conjunction with EVD outbreak records to investigate the association between recent (2004-2014) outbreaks in West and Central Africa, and patterns of land use change in the region. We show how in these EVD outbreaks the index cases in humans (i.e. spillover from wildlife reservoirs) occurred mostly in hotspots of forest fragmentation

Cooperative Intersection Crossing Over 5G

IEEE Autonomous driving is a safety critical application of sensing and decision-making technologies. Communication technologies extend the awareness capabilities of vehicles, beyond what is achievable with the on-board systems only. Nonetheless, issues typically related to wireless networking must be taken into account when designing safe and reliable autonomous systems. The aim of this work is to present a control algorithm and a communication paradigm over 5G networks for negotiating traffic junctions in urban areas. The proposed control framework has been shown to converge in a finite time and the supporting communication software has been designed with the objective of minimizing communication delays. At the same time, the underlying network guarantees reliability of the communication. The proposed framework has been successfully deployed and tested, in partnership with Ericsson AB, at the AstaZero proving ground in Goteborg, Sweden. In our experiments, three heterogeneous autonomous vehicles successfully drove through a 4-way intersection of 235 square meters in an urban scenario