Symmetries and reversing symmetries of area-preserving polynomial mappings in generalised standard form

We determine the symmetries and reversing symmetries within G, the group of real planar polynomial automorphisms, of area-preserving nonlinear polynomial maps L in generalised standard form, L: x'=x+p(y), y'=y+q(x'), where p and q are polynomials. We do this by using the amalgamated free product structure of G. Our results lead to normal forms for polynomial maps in generalised standard form and to a classification of the group structures of the reversing symmetry groups for such maps.Comment: 22 page

Petrogenesis of rare-metal pegmatites in high-grade metamorphic terranes: a case study from the Lewisian Gneiss Complex of north-west Scotland

Many rare metals used today are derived from granitic pegmatites, but debate continues about the origin of these rocks. It is clear that some pegmatites represent the most highly fractionated products of a parental granite body, whilst others have formed by anatexis of local crust. However, the importance of these two processes in the formation of rare-metal pegmatites is not always evident. The Lewisian Gneiss Complex of NW Scotland comprises Archaean meta-igneous gneisses which were highly reworked during accretional and collisional events in the Palaeoproterozoic (Laxfordian orogeny). Crustal thickening and subsequent decompression led to melting and the formation of abundant granitic and pegmatitic sheets in many parts of the Lewisian Gneiss Complex. This paper presents new petrological, geochemical and age data for those pegmatites and shows that, whilst the majority are barren biotite-magnetite granitic pegmatites, a few muscovite-garnet (rare-metal) pegmatites are present. These are mainly intruded into a belt of Palaeoproterozoic metasedimentary and meta-igneous rocks known as the Harris Granulite Belt. The rare-metal pegmatites are distinct in their mineralogy, containing garnet and muscovite, with local tourmaline and a range of accessory minerals including columbite and tantalite. In contrast, the biotite-magnetite pegmatites have biotite and magnetite as their main mafic components. The rare-metal pegmatites are also distinguished by their bulk-rock and mineral chemistry, including a more peraluminous character and enrichments in Rb, Li, Cs, Be, Nb and Ta. New U-Pb ages (c. 1690–1710 Ma) suggest that these rare-metal pegmatites are within the age range of nearby biotite-magnetite pegmatites, indicating that similar genetic processes could have been responsible for their formation. The peraluminous nature of the rare-metal pegmatites strongly points towards a metasedimentary source. Notably, within the Lewisian Gneiss Complex, such pegmatites are only found in areas where a metasedimentary source is available. The evidence thus points towards all the Laxfordian pegmatites being formed by a process of crustal anatexis, with the formation of rare-metal pegmatites being largely controlled by source composition rather than solely by genetic process. This is in keeping with previous studies that have also challenged the widely accepted model that all rare-metal pegmatites are formed by fractionation from a parental granite, and raises questions about the origin of other mineralised pegmatites worldwide

Intraflagellar transport trains and motors: insights from structure

Intraflagellar transport (IFT) sculpts the proteome of cilia and flagella; the antenna-like organelles found on the surface of virtually all human cell types. By delivering proteins to the growing ciliary tip, recycling turnover products, and selectively transporting signalling molecules, IFT has critical roles in cilia biogenesis, quality control, and signal transduction. IFT involves long polymeric arrays, termed IFT trains, which move to and from the ciliary tip under the power of the microtubule-based motor proteins kinesin-II and dynein-2. Recent top-down and bottom-up structural biology approaches are converging on the molecular architecture of the IFT train machinery. Here we review these studies, with a focus on how kinesin-II and dynein-2 assemble, attach to IFT trains, and undergo precise regulation to mediate bidirectional transport

Time-distance analysis of the emerging active region NOAA 10790

We investigate the emergence of Active Region NOAA 10790 by means of time – distance helioseismology. Shallow regions of increased sound speed at the location of increased magnetic activity are observed, with regions becoming deeper at the locations of sunspot pores. We also see a long-lasting region of decreased sound speed located underneath the region of the flux emergence, possibly relating to a temperature perturbation due to magnetic quenching of eddy diffusivity, or to a dense flux tube. We detect and track an object in the subsurface layers of the Sun characterised by increased sound speed which could be related to emerging magnetic-flux and thus obtain a provisional estimate of the speed of emergence of around 1 km s−1

Hartree-Fock-Bogoliubov theory versus local-density approximation for superfluid trapped fermionic atoms

We investigate a gas of superfluid fermionic atoms trapped in two hyperfine states by a spherical harmonic potential. We propose a new regularization method to remove the ultraviolet divergence in the Hartree-Fock-Bogoliubov equations caused by the use of a zero-range atom-atom interaction. Compared with a method used in the literature, our method is simpler and has improved convergence properties. Then we compare Hartree-Fock-Bogoliubov calculations with the semiclassical local-density approximation. We observe that for systems containing a small number of atoms shell effects, which cannot be reproduced by the semiclassical calculation, are very important. For systems with a large number of atoms at zero temperature the two calculations are in quite good agreement, which, however, is deteriorated at non-zero temperature, especially near the critical temperature. In this case the different behavior can be explained within the Ginzburg-Landau theory.Comment: 12 pages, 8 figures, revtex; v2: references and clarifying remarks adde

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 ⩾ 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

Gluon Propagator on Coarse Lattices in Laplacian Gauges

The Laplacian gauge is a nonperturbative gauge fixing that reduces to Landau gauge in the asymptotic limit. Like Landau gauge, it respects Lorentz invariance, but it is free of Gribov copies; the gauge fixing is unambiguous. In this paper we study the infrared behavior of the lattice gluon propagator in Laplacian gauge by using a variety of lattices with spacings from $a = 0.125$ to 0.35 fm, to explore finite volume and discretization effects. Three different implementations of the Laplacian gauge are defined and compared. The Laplacian gauge propagator has already been claimed to be insensitive to finite volume effects and this is tested on lattices with large volumes.Comment: RevTex 4.0, 14 pages, 9 colour figures; Correction to Reference

Changes in serum neurofilament light chain levels following narrowband ultraviolet B phototherapy in clinically isolated syndrome

Objective To determine whether serum neurofilament light chain (sNfL) levels are suppressed in patients with the clinically isolated syndrome (CIS) following narrowband ultraviolet B phototherapy (UVB-PT). Methods sNfL levels were measured using a sensitive single-molecule array assay at baseline and up to 12 months in 17 patients with CIS, 10 of whom received UVB-PT, and were compared with healthy control (HC) and early relapsing remitting multiple sclerosis (RRMS) group. sNfL levels were correlated with magnetic resonance imaging total lesion volume (LV) determined using icobrain version 4.4.1 and with clinical outcomes. Results Baseline median sNfL levels were significantly higher in the CIS (20.6 pg/mL, interquartile range [IQR] 13.7–161.4) and RRMS groups (36.6 pg/ml [IQR] 16.2–212.2) than in HC (10.7 pg/ml [IQR] 4.9–21.5) (p = .012 and p = .0002, respectively), and were strongly correlated with T2 and T1 LV at 12 months (r = .800; p = .014 and r = .833; p = .008, respectively) in the CIS group. Analysis of changes in sNfL levels over time in the CIS group showed a significant cumulative suppressive effect of UVB-PT in the first 3 months (UVB-PT −10.6% vs non-UVB-PT +58.3%; p = .04) following which the levels in the two groups converged and continued to fall. Conclusions Our findings provide the basis for further studies to determine the utility of sNfL levels as a marker of neuro-axonal damage in CIS and early MS and for assessing the efficacy of new therapeutic interventions such as UVB-PT

Integrable maps in 4D and modified Volterra lattices

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus g. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus g algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted V_g. The original motivation behind our work was the fact that, in the particular case g=2, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map V_2 yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the g=2 Volterra map V_2. We dedicate this work to a dear friend and colleague, Decio Levi

State sampling dependence of the Hopfield network inference

The fully connected Hopfield network is inferred based on observed magnetizations and pairwise correlations. We present the system in the glassy phase with low temperature and high memory load. We find that the inference error is very sensitive to the form of state sampling. When a single state is sampled to compute magnetizations and correlations, the inference error is almost indistinguishable irrespective of the sampled state. However, the error can be greatly reduced if the data is collected with state transitions. Our result holds for different disorder samples and accounts for the previously observed large fluctuations of inference error at low temperatures.Comment: 4 pages, 1 figure, further discussions added and relevant references adde