Dirac operators and spectral triples for some fractal sets built on curves

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension $\log 3/ \log 2$.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv. Mat

From Monomials to Words to graphs

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal generated by \sigma(I) in the free monoid is finitely generated. Whether there exists an ordering such that is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.Comment: 27 pages, 2 postscript figures, uses gastex.st

Contractile stresses in cohesive cell layers on finite-thickness substrates

Using a minimal model of cells or cohesive cell layers as continuum active elastic media, we examine the effect of substrate thickness and stiffness on traction forces exerted by strongly adhering cells. We obtain a simple expression for the length scale controlling the spatial variation of stresses in terms of cell and substrate parameters that describes the crossover between the thin and thick substrate limits. Our model is an important step towards a unified theoretical description of the dependence of traction forces on cell or colony size, acto-myosin contractility, substrate depth and stiffness, and strength of focal adhesions, and makes experimentally testable predictions.Comment: 5 pages, 3 figure

Oxygen content variation and cation doping dependence of (La)1.4(Sr1-yCay)1.6Mn2O7 (y = 0, 0.25, 0.5) bilayered manganites properties

The results of the synthesis and characterization of the optimally doped (La)1.4(Sr1-yCay)1.6Mn2O7 solid solution with y=0, 0.25 and 0.5 are reported. By progressively replacing the Sr with the smaller Ca, while keeping fixed the hole-concentration due to the divalent dopant, the 'size effect' of the cation itself on the structural, transport and magnetic properties of the bilayered manganite has been analysed. Two different annealing treatments of the solid solution, in pure oxygen and in pure argon, allowed also to study the effect of the oxygen content variation. Structure and electronic properties of the samples have been investigated by means of X-ray powder diffraction and X-ray absorption spectroscopy measurements. Magnetoresistivity and static magnetization measurements have been carried out to complete the samples characterization. Oxygen annealing of the solid solution, that showed a limit for about y=0.5, induces an increase of the Mn average valence state and a transition of the crystal structure from tetragonal to orthorhombic while the argon annealing induces an oxygen under-stoichiometry and, in turn, a reduction of the Mn average valence state. Along with the Ca substitution, the Jahn-Teller distortion of the MnO6 octahedra is reduced. This has been directly connected to a general enhancement of the transport properties induced by the Ca-doping. For the same cation composition, oxygen over-stoichiometry leads to higher metal-insulator transition temperatures and lower resistivity values. Curie temperatures (TC) reduce by increasing the Ca-doping. The lower TC for all the annealed samples with respect to the 'as prepared' ones are connected to the strong influence on the magnetic interaction of the point defects due to the oxygen content variation.Comment: 49 pages, 13 figure

The effects of non-abelian statistics on two-terminal shot noise in a quantum Hall liquid in the Pfaffian state

We study non-equilibrium noise in the tunnelling current between the edges of a quantum Hall liquid in the Pfaffian state, which is a strong candidate for the plateau at $\nu=5/2$. To first non-vanishing order in perturbation theory (in the tunneling amplitude) we find that one can extract the value of the fractional charge of the tunnelling quasiparticles. We note however that no direct information about non-abelian statistics can be retrieved at this level. If we go to higher-order in the perturbative calculation of the non-equilibrium shot noise, we find effects due to non-Abelian statistics. They are subtle, but eventually may have an experimental signature on the frequency dependent shot noise. We suggest how multi-terminal noise measurements might yield a more dramatic signature of non-Abelian statistics and develop some of the relevant formalism.Comment: 13 pages, 8 figures, a few change

Emission and absorption noise in the fractional quantum Hall effect

We compute the high-frequency emission and absorption noise in a fractional quantum Hall effect (FQHE) sample at arbitrary temperature. We model the edges of the FQHE as chiral Luttinger liquids (LL) and we use the non-equilibrium perturbative Keldysh formalism. We find that the non-symmetrized high frequency noise contains important signatures of the electron-electron interactions that can be used to test the Luttinger liquid physics, not only in FQHE edge states, but possibly also in other one-dimensional systems such as carbon nanotubes. In particular we find that the emission and absorption components of the excess noise (defined as the difference between the noise at finite voltage and at zero voltage) are different in an interacting system, as opposed to the non-interacting case when they are identical. We study the resonance features which appear in the noise at the Josephson frequency (proportional to the applied voltage), and we also analyze the effect of the distance between the measurement point and the backscattering site. Most of our analysis is performed in the weak backscattering limit, but we also compute and discuss briefly the high-frequency noise in the tunneling regime.Comment: 26 pages, 11 figure