Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations

Thermal transport measurements of individual multiwalled nanotubes

The thermal conductivity and thermoelectric power of a single carbon nanotube were measured using a microfabricated suspended device. The observed thermal conductivity is more than 3000 W/K m at room temperature, which is two orders of magnitude higher than the estimation from previous experiments that used macroscopic mat samples. The temperature dependence of the thermal conductivity of nanotubes exhibits a peak at 320 K due to the onset of Umklapp phonon scattering. The measured thermoelectric power shows linear temperature dependence with a value of 80 $\mu$V/K at room temperature.Comment: 4 pages, figures include

Chemical doping of individual semiconducting carbon-nanotube ropes

We report the effects of potassium doping on the conductance of individual semiconducting single-walled carbon nanotube ropes. We are able to control the level of doping by reversibly intercalating and de-intercalating potassium. Potassium doping changes the carriers in the ropes from holes to electrons. Typical values for the carrier density are found to be ∼100–1000 electrons/μm. The effective mobility for the electrons is μeff∼20–60 cm2 V-1 s-1, a value similar to that reported for the hole effective mobility in nanotubes [R. Martel et al., Appl. Phys. Lett. 73, 2447 (1998)]

Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I

The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformation to a "string" type boundary value problem known from prior works

Two-component generalizations of the Camassa-Holm equation

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered

Bridge Hopping on Conducting Polymers in Solution

Configurational fluctuations of conducting polymers in solution can bring into proximity monomers which are distant from each other along the backbone. Electrons can hop between these monomers across the "bridges" so formed. We show how this can lead to (i) a collapse transition for metallic polymers, and (ii) to the observed dramatic efficiency of acceptor molecules for quenching fluorescence in semiconducting polymers.Comment: RevTeX 12 pages + 2 Postscript figure

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the $g=2$ case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g\geqslant 1$. The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case ($g=1$), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices