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We show that the Hausdorff dimension of the singular set of perimeter minimizers in non-collapsed Ricci limit spaces with a two-sided Ricci curvature bound is at most $N-5$, where $N$ is the dimension of the ambient space. The estimate is sharp.

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non collapsed RCD spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth., Comment: 37 pages. Final version, accepted for publication in Commentarii Mathematici Helvetici

The goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbour approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbour approximations are not generally appropriate., Comment: 25 pages, 2 figures

Laser induced fluorescence (LIF) is a powerful remote and non-invasive analysis technique that has been successfully applied to the real-time diagnosis of historical artworks. Hyperspectral images collection on fresco' s and their false colours processing allowed to reveal features invisible to the naked eye and to obtain specific information on pigments composition and consolidants utilization, the latter also related to former restorations. This report presents the results obtained by ENEA LIF scanning system during a field campaign conducted in June 2010 on fresco's by Giusto de' Menabuoi in the Padua Baptistery. The data collected by LidArt allowed the detection of Paraloid B72 and Movilith/Primal AC33, guiding the restorers in their conservation actions.

We establish a version of the first Noether Theorem, according to which the (equivalence classes of) conserved quantities of given Euler-Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action, from which the Euler-Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such correspondence between symmetries and conservation laws is built on an explicit linear map f_A from the vector fields satisfying (a) and (b) into the conserved differential operators, and not into their divergences as it occurs in other proofs of Noether Theorem. This map f_A is not new: it is the map determined by contracting symmetries with a form of Poincare'-Cartan type A and it is essentially the same considered for instance in a paper by Kupershmidt. There it was shown that f_A determines a bijection between symmetries and conservation laws in a special form. Here we show that, if appropriate regularity assumptions are satisfied, any conservation law is equivalent to one that belongs to the image of f_A, proving that the corresponding induced map F_A between equivalence classes of symmetries and equivalence classes of conservation laws is actually a bijection. All results are given coordinate-free formulations and rely just on basic differential geometric properties of finite-dimensional manifolds., Comment: 33 pages; in v2, minor changes in the Abstract, in the Introduction and in the last section; a few references added

We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler-Lagrange equations are derived., Comment: 32 pages; to appear on Journal of Geometry and Physics

We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville-Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko-Fomenko theorem on noncommutative integrability, and for each of them we give a version suitable for the noncompact case. We give a possible global version of the previous local results, under certain topological hypotheses on the base space. It turns out that locally affine structures arise naturally in this setting., Comment: It will appear on International Journal of Geometric Methods in Modern Physics vol.5 n.3 (May 2008) issue

The obstruction to the existence of global action-angle coordinates of Abelian and noncommutative (non-Abelian) completely integrable systems with compact invariant submanifolds has been studied. We extend this analysis to the case of noncompact invariant submanifolds., Comment: 13 pages, to be published in J. Math. Phys. (2007)

The Mishchenko-Fomenko theorem on noncommutative integrability of Hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds., Comment: 11 pages, accepted for publication in J.Phys. A

Under ceratin conditions, generalized action-angle coordinates can be introduced near non-compact invariant manifolds of completely and partially integrable Hamiltonian systems., Comment: 8 pages

A trivial bundle of regular connected invariant manifolds of a completely integrable Hamiltonian system can be provided with action-angle coordinates., Comment: 3 pages

A time-dependent completely integrable Hamiltonian system is quantized with respect to time-dependent action-angle variables near an instantly compact regular invariant manifold. Its Hamiltonian depends only on action variables, and has a time-independent countable energy spectrum., Comment: 16 pages

In this paper we report magnetic relaxation data for antiferromagnetic alpha-Fe2O3 particles of 5 nm mean diameter in the temperature range 0.1 K to 25 K. The average spin value of these particles S=124 and the uniaxial anisotropy constant D=1.6x10^-2 K have been estimated from the experimental values of the blocking temperature and anisotropy field. The observed plateau in the magnetic viscosity from 3 K down to 100 mK agrees with the occurrence of spin tunneling from the ground state Sz = S. However, the scaling M vs Tln(nu t) is broken below 5 K, suggesting the occurrence of tunneling from excited states below this temperature., Comment: 4 pages (two columns), 4 figures

The issue of the approximate isotropy and homogeneity of the observable universe is one of the major topics in modern Cosmology: the common use of the Friedmann-Robertson-Walker [FWR] metric relies on these assumptions. Therefore, results conflicting with the ``canonical'' picture would be of the utmost importance. In a number of recent papers it has been suggested that strong evidence of a fractal distribution with dimension D~2 exists in several samples, including Abell clusters [ACO] and galaxies from the ESO Slice Project redshift survey [ESP].Here we report the results of an independent analysis of the radial density run,N(