On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets

Let $\Omega$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_{\Omega}$ be the torsion function for $\Omega$, i.e. the solution of $-\Delta v=1, v\in H_0^1(\Omega)$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge \frac{\pi^2}{24}$, and that this bound is sharp.Comment: 12 pages, 4 figure

Superconducting Fluctuation Corrections to the Thermal Current in Granular Metals

The first-order superconducting fluctuation corrections to the thermal conductivity of a granular metal are calculated. A suppression of thermal conductivity proportional to $T_c/(T-T_c)$ is observed in a region not too close to the critical temperature $T_c$. As $T\simeq T_c$, a saturation of the correction is found, and its sign depends on the ratio between the barrier transparency and the critical temperature. In both regimes, the Wiedemann-Franz law is violated.Comment: 9 pages, 7 figures. Replaced with published version. Important change

Protein Gaussian Image (PGI): A protein structural representation based on the spatial attitude of secondary structure

A well-known shape representation usually applied for 3D object recognition is the Extended Gaussian Image (EGI) which maps the histogram of the orientations of the object surface on the unitary sphere. We propose to adopt an analogous “abstract” data-structure named Protein Gaussian Image (PNM) for representing the orientation of the protein secondary structures (e.g. helices or strands) which combines the characteristics of the EGI and the ones of needle maps. The “concrete” data structures is the same as for the EGI, with a hierarchy that starting with a discretization corresponding to the 20 orientations of the icosahedron facets, it is iteratively refined with a factor 4 at each new level (80, 320, 1280, . . . ) up to the maximum precision required. However, in this case to each orientation does not correspond the area of the patches having that orientation but the features of the protein secondary structures having that direction. Among the features we may include the versus (origin versus surface or vice versa), the length of the structure (e.g. the number of amino acids), biochemical properties, and even the sequence of the amino acids (stored as a list). We consider this representation very effective for a preliminary screening when looking in a protein data base for retrieval of a given structural block, or a domain, or even an entire protein. In fact, on this structure it is possible to identify the presence of a given motif, or also sheets (note that parallel or anti-parallel β-sheets are characterized by common or opposite directions of ladders). Herewith some known proteins are described with common typical motifs easily marked in the PGI

High-temperature excess current and quantum suppression of electronic backscattering in a 1-D system

We consider the electronic current through a one-dimensional conductor in the ballistic transport regime and show that the quantum oscillations of a weakly pinned single scattering target results in a temperature- and bias-voltage independent excess current at large bias voltages. This is a genuine effect on transport that derives from an exponential reduction of electronic backscattering in the elastic channel due to quantum delocalization of the scatterer and from suppression of low-energy electron backscattering in the inelastic channels caused by the Pauli exclusion principle. We show that both the mass of the target and the frequency of its quantum vibrations can be measured by studying the differential conductance and the excess current. We apply our analysis to the particular case of a weakly pinned C60 molecule encapsulated by a single-wall carbon nanotube and find that the discussed phenomena are experimentally observable.Comment: 4 pages, 4 figure

Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.Comment: To appear in Comm. PD