Boundary interpolation for slice hyperholomorphic Schur functions

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad {\rm for} \quad u=1,\ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces

Nanofriction behavior of cluster-assembled carbon films

We have characterized the frictional properties of nanostructured (ns) carbon films grown by Supersonic Cluster Beam Deposition (SCBD) via an Atomic Force-Friction Force Microscope (AFM-FFM). The experimental data are discussed on the basis of a modified Amonton's law for friction, stating a linear dependence of friction on load plus an adhesive offset accounting for a finite friction force in the limit of null total applied load. Molecular Dynamics simulations of the interaction of the AFM tip with the nanostructured carbon confirm the validity of the friction model used for this system. Experimental results show that the friction coefficient is not influenced by the nanostructure of the films nor by the relative humidity. On the other hand the adhesion coefficient depends on these parameters.Comment: 22 pages, 6 figures, RevTex

Interplay between bending and stretching in carbon nanoribbons

We investigate the bending properties of carbon nanoribbons by combining continuum elasticity theory and tight-binding atomistic simulations. First, we develop a complete analysis of a given bended configuration through continuum mechanics. Then, we provide by tight-binding calculations the value of the bending rigidity in good agreement with recent literature. We discuss the emergence of a stretching field induced by the full atomic-scale relaxation of the nanoribbon architecture. We further prove that such an in-plane strain field can be decomposed into a first contribution due to the actual bending of the sheet and a second one due to edge effects.Comment: 5 pages, 6 figure

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.Comment: To appear on Boll. UMI, special volume in memory of Paolo de Bartolomei

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.Comment: 18 pages, to appear in Geometriae Dedicat

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators

Orbital dynamics of "smart dust" devices with solar radiation pressure and drag

This paper investigates how perturbations due to asymmetric solar radiation pressure, in the presence of Earth shadow, and atmospheric drag can be balanced to obtain long-lived Earth centred orbits for swarms of micro-scale 'smart dust' devices, without the use of active control. The secular variation of Keplerian elements is expressed analytically through an averaging technique. Families of solutions are then identified where Sun-synchronous apse-line precession is achieved passively to maintain asymmetric solar radiation pressure. The long-term orbit evolution is characterized by librational motion, progressively decaying due to the non-conservative effect of atmospheric drag. Long-lived orbits can then be designed through the interaction of energy gain from asymmetric solar radiation pressure and energy dissipation due to drag. In this way, the usual short drag lifetime of such high area-to-mass spacecraft can be greatly extended (and indeed selected). In addition, the effect of atmospheric drag can be exploited to ensure the rapid end-of-life decay of such devices, thus preventing long-lived orbit debris