Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t-13 for position and momentum correlations and as t-23 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t-14 for position and momentum correlators and with rate t-12 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics

Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale

We report on molecular dynamics simulations of spacetime correlations of the Toda lattice in thermal equilibrium. The correlations of stretch, momentum, and energy are computed numerically over a wide range of pressure and temperature. Our numerical results are compared with the predictions from linearized generalized hydrodynamics on the Euler scale. The system size is N=3000,4000 and time t =600, at which ballistic scaling is well confirmed. With no adjustable parameters, the numerically obtained scaling functions agree with the theory within a precision of less than 3.5%.Comment: 22 pages, 5 figure

Developing a Mental Health Measurement Strategy to Capture Psychological Problems among Lower Caste Adolescent Girls in Rural, South India.

Adolescent girls vulnerable to early marriage and school dropout in rural India may be at elevated risk of psychosocial problems. However, few screening instruments have been culturally adapted and validated to measure this risk. This paper describes the process by which the Primary Health Questionnaire PHQ-9, a screening instrument for depression, was tested for cultural validity as part of the Samata evaluation - an intervention to support low caste adolescent girls in rural south India to attend and complete secondary school and to delay marriage until adulthood. Three focus groups discussions (FGDs) were held with 20 adolescent girls and six outreach workers of the Samata programme in rural north Karnataka, south India. The FGDs were used to explore local expressions of psychosocial problems and to understand the acceptability and appropriateness of PHQ-9 items. A thematic content analysis was conducted on the transcripts of the FGDs. Descriptions of local expressions of psychosocial problems generally matched the items on the PHQ-9. However, not all representations of psychological symptoms were captured by this tool. Persistent worry, loneliness and isolation, and externalised behaviours were also described by participants as common expressions of psychosocial distress. Based on the limitations of translation methods, local stakeholders must be involved in evaluating the cultural appropriateness of mental health screening tools. The current research demonstrates a strategy by which to assess the cultural validity of Western psychiatric instruments with key stakeholders in low- and middle-income settings

Ages and metallicities of circumnuclear star formation regions from Gemini IFU observations

Aims: We derive the age and metallicity of circumnuclear star formation regions (CNSFRs) located in the spiral galaxies NGC6951 and NGC1097, and investigate the cause of the very low equivalent widths of emission lines found for these regions. Methods: We used optical two-dimensional spectroscopic data obtained with Gemini GMOS-IFUs and a grid of photoionization models to derive the the metallicities and ages of CNSFRs. Results: We find star formation rates in the range 0.002-0.14 Msun/yr and oxygen abundance of 12+log(O/H)~8.8 dex, similar to those of most metal-rich nebulae located in the inner region of galactic disks. Conclusions: We conclude that the very low emission-line equivalent widths observed in CNSFRs are caused by the ``contamination'' of the continuum by (1) contribution of the underlying bulge continuum combined with (2) contribution from previous episodes of star formation at the CNSFRs.Comment: Accepted for publication in Astronomy and Astrophysic

Study of star formation in NGC 1084

We present UBVRI broad band, H$\alpha$ narrow band photometry of the star forming complexes in the infra-red bright galaxy NGC 1084. Results of medium resolution spectroscopy of some of the brighter complexes are also discussed. Spectroscopic data is used to better estimate the internal reddening within the galaxy which is found to be highly variable and to calculate metallicity which is close to the solar value. Diagnostic diagram identifies the shocked regions within this galaxy. The narrow band H$\alpha$ flux and its equivalent width are used to determine the star formation rates of the complexes and the distribution of ages. Star formation rates for a few of the complexes are found to be as high as 0.5 $M_{\odot}$/year. The star forming complexes lie in the age range 3 Myr to 6.5 Myr. U-B vs V-I colour-colour mixed population model created using the Starburst99 model colours is used to estimate the ages of the stellar populations present within these regions. Using this technique, it is found that the star formation in NGC 1084 has taken place in a series of short bursts over the last 40 Myr or so. It is proposed that the likely trigger for enhanced star formation is merger with a gas rich dwarf galaxy.Comment: 9 figures and 6 tables, Accepted in MNRAS for publicatio