Symplectic Maps from Cluster Algebras

We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map

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

Backlund transformations for many-body systems related to KdV

We present Backlund transformations (BTs) with parameter for certain classical integrable n-body systems, namely the many-body generalised Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Backlund parameter.Comment: LaTeX2e, 8 page

Graphene field-effect transistors based on boron nitride gate dielectrics

Graphene field-effect transistors are fabricated utilizing single-crystal hexagonal boron nitride (h-BN), an insulating isomorph of graphene, as the gate dielectric. The devices exhibit mobility values exceeding 10,000 cm2/V-sec and current saturation down to 500 nm channel lengths with intrinsic transconductance values above 400 mS/mm. The work demonstrates the favorable properties of using h-BN as a gate dielectric for graphene FETs.Comment: 4 pages, 8 figure

Discrete Painlevé equations from Y-systems

We consider T-systems and Y-systems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficient-free T-systems) were described in the work of Fordy and Marsh, who completely classified all such quivers in the case of period 1, and characterized them in terms of the skew-symmetric exchange matrix B that defines the quiver. A broader notion of periodicity in general cluster algebras was introduced by Nakanishi, who also described the corresponding Y-systems, and T-systems with coefficients. A result of Fomin and Zelevinsky says that the coefficient-free T-system provides a solution of the Y-system. In this paper, we show that in general there is a discrepancy between these two systems, in the sense that the solution of the former does not correspond to the general solution of the latter. This discrepancy is removed by introducing additional non-autonomous coefficients into the T-system. In particular, we focus on the period 1 case and show that, when the exchange matrix B is degenerate, discrete Painlev\'e equations can arise from this construction

User interface design for mobile-based sexual health interventions for young people: Design recommendations from a qualitative study on an online Chlamydia clinical care pathway

Background: The increasing pervasiveness of mobile technologies has given potential to transform healthcare by facilitating clinical management using software applications. These technologies may provide valuable tools in sexual health care and potentially overcome existing practical and cultural barriers to routine testing for sexually transmitted infections. In order to inform the design of a mobile health application for STIs that supports self-testing and self-management by linking diagnosis with online care pathways, we aimed to identify the dimensions and range of preferences for user interface design features among young people. Methods: Nine focus group discussions were conducted (n=49) with two age-stratified samples (16 to 18 and 19 to 24 year olds) of young people from Further Education colleges and Higher Education establishments. Discussions explored young people's views with regard to: the software interface; the presentation of information; and the ordering of interaction steps. Discussions were audio recorded and transcribed verbatim. Interview transcripts were analysed using thematic analysis. Results: Four over-arching themes emerged: privacy and security; credibility; user journey support; and the task-technology-context fit. From these themes, 20 user interface design recommendations for mobile health applications are proposed. For participants, although privacy was a major concern, security was not perceived as a major potential barrier as participants were generally unaware of potential security threats and inherently trusted new technology. Customisation also emerged as a key design preference to increase attractiveness and acceptability. Conclusions: Considerable effort should be focused on designing healthcare applications from the patient's perspective to maximise acceptability. The design recommendations proposed in this paper provide a valuable point of reference for the health design community to inform development of mobile-based health interventions for the diagnosis and treatment of a number of other conditions for this target group, while stimulating conversation across multidisciplinary communities

A lattice model of hydrophobic interactions

Hydrogen bonding is modeled in terms of virtual exchange of protons between water molecules. A simple lattice model is analyzed, using ideas and techniques from the theory of correlated electrons in metals. Reasonable parameters reproduce observed magnitudes and temperature dependence of the hydrophobic interaction between substitutional impurities and water within this lattice.Comment: 7 pages, 3 figures. To appear in Europhysics Letter

Cluster Algebras and Discrete Integrability

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the context of cluster mutation. In particular, we give examples of birational maps that are integrable in the Liouville sense and arise from cluster algebras with periodicity, as well as examples of discrete Painleve equations that are derived from Y-systems

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