Box splines and the equivariant index theorem

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra. The morphism from K-theory to cohomology is analyzed and the multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semidiscrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol.Comment: 44 page

Quasi-T\"oplitz functions in KAM theorem

We define and describe the class of Quasi-T\"oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr\"odinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment.Comment: 34 pages, 1 figur

On the Ado Theorem for finite Lie conformal algebras with Levi decomposition

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.Comment: 11 page

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

We study stability times for a family of parameter dependent nonlinear Schrodinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively

Widespread abiotic methane in chromitites

Recurring discoveries of abiotic methane in gas seeps and springs in ophiolites and peridotite massifs worldwide raised the question of where, in which rocks, methane was generated. Answers will impact the theories on life origin related to serpentinization of ultramafic rocks, and the origin of methane on rocky planets. Here we document, through molecular and isotopic analyses of gas liberated by rock crushing, that among the several mafic and ultramafic rocks composing classic ophiolites in Greece, i.e., serpentinite, peridotite, chromitite, gabbro, rodingite and basalt, only chromitites, characterized by high concentrations of chromium and ruthenium, host considerable amounts of 13C-enriched methane, hydrogen and heavier hydrocarbons with inverse isotopic trend, which is typical of abiotic gas origin. Raman analyses are consistent with methane being occluded in widespread microfractures and porous serpentine- or chlorite-filled veins. Chromium and ruthenium may be key metal catalysts for methane production via Sabatier reaction. Chromitites may represent source rocks of abiotic methane on Earth and, potentially, on Mars

Quasi-periodic solutions of completely resonant forced wave equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur

Corrigendum to 'Reducibility of first order linear operators on tori via Moser's theorem' [Journal of Functional Analysis 276 (3) (2019) 932–970]

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Normal Form for the Schr\"odinger equation with analytic non--linearities

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and applications to quasi--periodic solutions will appear in a forthcoming article. This paper replaces a previous version correcting some mistakes.Comment: 52 pages, 2 figure

Development of an Italian catalogue of potential CO2storage sites: an approach from deep wells data

Stabilize and reduce the atmospheric concentration of anthropogenic greenhouse gases is one of the principal goal that have to be accomplished in short time, in order to reduce the climate changes and the global warming, following the World Energy Outlook 2007 program by IEA. The most promising remedy, proposed for large CO2 sources like thermoelectric power plants, refineries and cement industries, is to separate the flue gas capturing the CO2 and to store it into deep sub-surface geological reservoirs, such as deep saline aquifers, depleted oil and gas fields and unminable coal beds. Among these options, deep saline aquifers are considered the reservoirs with the larger storage potentiality, as a consequence of a wide availability with respect to deep coal seems, depleted oil fields and gas reservoirs. The identification of a possible storage site necessarily passes through the demonstration that CO2 can be injected in extremely safe conditions into geological deep formations, with impermeable caprock above the aquifer/s, which physic-chemical-mineralogical conditions are useful to a better mineral and solubility trapping as well as the hydrodynamic or physical/ structural ones. In order to support the identification of potential storage reservoirs in Italy, INGV jointly with CESI RICERCA S.p.A. accomplished a detailed reworking of available geological, geophysical, geochemical and seismological data, in order to support the existing European GESTCO as well as the CO2GeoCapacity projects. Aim of this work is to establish some site selection criteria to demonstrate the possibility of the geological storage of CO2 in Italy, even if it is located in an active geodynamical domain. This research started from the study of 7575 wells drilled on Italian territory during the last 50 years for gas/oil and geothermal exploration. Among this data-set as a whole, only 1700 wells (deeper than 800 m) have been selected. Only 1290 of these wells have a public-available composite log and fit with the basic prerequisites for CO2 storage potential, mostly as deep saline aquifer/s presence. Wells data have been organized into a geodatabase containing information about the nature and the thickness of geological formations, the presence of fresh, saline or brackish water, brine, gas and oil, the underground temperature, the permeability, porosity and geochemical characteristics of the caprock and the reservoirs lithologies. Available maps, seismic and geological profiles containing or closer to the analyzed wells have been catalogued too. In order to constrain the supercritical behaviour of the CO2 and to prevent the escape of gaseous CO2 to the surface, a first evaluation of the caprock presence and quality has been done on these selected wells. Using a numerical parameterization of the caprock lithologies, a “Caprock Quality Factor” (Fbp) has been defined, which clustered the wells into 5 different classes of caprock impermeability (ranging between the lowest 1 to highest 5). The analysis shows that more than 50% of the selected wells have an Fbp Factor between 4 and 5 (good and optimal quality of caprock), and are mostly located in foredeep basins of the Alps-Apenninic Chain. The geodatabase also includes: i) the seismogenetic sources (INGV DISS 3.0.4 Database of Individual Seismogenetic Sources), ii) an elaboration of seismic events catalogues (INGV CFTI, CPTI04, NT4.1), iii) the Diffuse Degassing Structures (DDS), as part of the INGV project V5 diffuse degassing in Italy geodatabase, considered as “CO2 analogue” field-tests, iv) the distribution of the thermal anomalies on the Italian Territory, linked to the presence of volcanic CO2 emissions, in order to consider the CO2 diffuse degassing risk assessment on the Italian territory Successively it has been created a geodatabase on the nature and quality of deep aquifers for the high-ranking wells sub-dataset (where the aquifers data are available), containing the following parameters: i) presence of one or more aquifers deeper than 800 meters; ii) thickness of the aquifer/s; iii) lithology of the reservoir/s; iv) available chemical analysis; v) distance from closer power plants or other anthropogenic CO2 sources.The final aim of these work is to help to find potential areas in Italy where CO2 storage feasibility studies can be done. In these cases it is necessary to implement the knowledge by: i) better evaluation of saline aquifer quality; ii) estimation of CO2 storage capacity by 3D-modeling of deep crustal structures; iii) fluid-dynamic and geochemical modelling of water-rock-CO2 interaction paths