Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical r\^ole of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non--standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.Comment: 20 pages, 13 figure

One example of general unidentifiable tensors

The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple tensors, so it is not 8-identifiable. This is the highest known example of balanced tensors of dimension 3, which are not k-identifiable, when k is smaller than the generic rank.Comment: 7 pages, one Macaulay2 script as ancillary file, two references adde

Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model

A linear elastic second gradient orthotropic two-dimensional solid that is invariant under (Formula presented.) rotation and for mirror transformation is considered. Such anisotropy is the most general for pantographic structures that are composed of two identical orthogonal families of fibers. It is well known in the literature that the corresponding strain energy depends on nine constitutive parameters: three parameters related to the first gradient part of the strain energy and six parameters related to the second gradient part of the strain energy. In this paper, analytical solutions for simple problems, which are here referred to the heavy sheet, to the nonconventional bending, and to the trapezoidal cases, are developed and presented. On the basis of such analytical solutions, gedanken experiments were developed in such a way that the whole set of the nine constitutive parameters is completely characterized in terms of the materials that the fibers are made of (i.e., of the Young’s modulus of the fiber materials), of their cross sections (i.e., of the area and of the moment of inertia of the fiber cross sections), and of the distance between the nearest pivots. On the basis of these considerations, a remarkable form of the strain energy is derived in terms of the displacement fields that closely resembles the strain energy of simple Euler beams. Numerical simulations confirm the validity of the presented results. Classic bone-shaped deformations are derived in standard bias numerical tests and the presence of a floppy mode is also made numerically evident in the present continuum model. Finally, we also show that the largeness of the boundary layer depends on the moment of inertia of the fibers

Effective criteria for specific identifiability of tensors and forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.Comment: 31 pages, 2 Macaulay2 code

Competitive reaction modelling in aqueous systems. The case of contemporary reduction of dichromates and nitrates by nZVI

In various Countries, Cr(VI) still represents one of the groundwater pollutant of major concern, mainly due to its high toxicity, furthermore enhanced by the synergic effect in presence of other contaminants. As widely reported in the recent literature, nanoscale zero valent iron particles (nZVI-p) have been proved to be particularly effective in the removal of a wide range of contaminants from polluted waters. In this work, experimental tests of hexavalent chromium reduction in polluted groundwater in the presence of nitrate by nZVI-p are presented and discussed. The effect of different nitrate amounts on Cr(VI) reduction mechanism was investigated and the obtained results were successfully interpreted by the proposed kinetic model. nZVI-p produced by the classical borohydride reduction method were added in to synthetic solutions with the initial concentration of Cr(VI) set at 93, 62 and 31 mg L-1 and different nitrate contents in the range 10-100 mg L-1. According to the experimental results, nitrate showed an adverse effect on Cr(VI) reduction, depending on the nZVI/Cr(VI) and Cr(VI)/NO3 - ratio. The proposed kinetic model soundly grasps the competitive nature of the Cr(VI) reduction process when other chemical species are present in the treated solution

An Innovative Mission Management System for Fixed-Wing UAVs

This paper presents two innovative units linked together to build the main frame of a UAV Mis- sion Management System. The first unit is a Path Planner for small UAVs able to generate optimal paths in a tridimensional environment, generat- ing flyable and safe paths with the lowest com- putational effort. The second unit is the Flight Management System based on Nonlinear Model Predictive Control, that tracks the reference path and exploits a spherical camera model to avoid unpredicted obstacles along the path. The control system solves on-line (i.e. at each sampling time) a finite horizon (state horizon) open loop optimal control problem with a Genetic Algorithm. This algorithm finds the command sequence that min- imizes the tracking error with respect to the ref- erence path, driving the aircraft far from sensed obstacles and towards the desired trajectory

Can we declare military Keynesianism dead?

This paper empirically tests the Keynesian hypothesis that government defence spending positively impacts on aggregate output, by using a long run equilibrium model for the US and the UK. Our contribution, with respect to previous works, is twofold. First, our inferences are adjusted for structural breaks exhibited by the data concerning fiscal and monetary variables. Second, we take into account different dynamics between defence spending on aggregate output, showing that the results are sensitive to sub-sample choices. Though the estimated elasticities in both countries show a lack of significance in the more recent years of the sample, defence spending priorities addressed to international security may revitalize pro-cyclical effects in the UK, by an industrial policy of defence shared with the EU members.Military spending, Output, Long run models