Real null coframes in general relativity and GPS type coordinates

Based on work of Derrick, Coll, and Morales, we define a `symmetric' null coframe with {\it four real null covectors}. We show that this coframe is closely related to the GPS type coordinates recently introduced by Rovelli.Comment: Latex script, 9 pages, 4 figures; references added to work of Derrick, Coll, and Morales, 1 new figur

Relaxation Properties of Small-World Networks

Recently, Watts and Strogatz introduced the so-called small-world networks in order to describe systems which combine simultaneously properties of regular and of random lattices. In this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin. The results are intermediate between the corresponding ones for fractals and for Cayley trees.Comment: 16 pages, 6 figure

Lyapunov exponents for products of complex Gaussian random matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.Comment: 15 page

Instream and riparian implications of weed cutting in a chalk river

Macrophyte growth is extensive in the iconic chalk streams that are concentrated in southern and eastern England. Widespread and frequent weed cutting is undertaken to maintain their key functions (e.g. flood water conveyance and maintenance of viable fisheries). In this study, a multidisciplinary approach was adopted to quantify coincident physico-chemical responses (instream and riparian) that result from weed cutting and to discuss their potential implications. Three weed cuts were monitored at a site on the River Lambourn (The CEH River Lambourn Observatory) and major instream and riparian impacts were observed. Measurements clearly demonstrated how weed cutting enhanced flood flow conveyance, reduced water levels (river and wetland), increased river velocities, and mobilised suspended sediment (with associated chemicals) and reduced the capacity for its retention within the river channel. Potential implications in relation to flood risk, water resources, downstream water quality, instream and riparian ecology, amenity value of the river, and wetland greenhouse gas emissions were considered. Provided the major influence of macrophytes on instream and riparian environments is fully understood then the manipulation of macrophytes represents an effective management tool that demonstrates the great potential of working with nature

Dynamics of systems with isotropic competing interactions in an external field: a Langevin approach

We study the Langevin dynamics of a ferromagnetic Ginzburg-Landau Hamiltonian with a competing long-range repulsive term in the presence of an external magnetic field. The model is analytically solved within the self consistent Hartree approximation for two different initial conditions: disordered or zero field cooled (ZFC), and fully magnetized or field cooled (FC). To test the predictions of the approximation we develop a suitable numerical scheme to ensure the isotropic nature of the interactions. Both the analytical approach and the numerical simulations of two-dimensional finite systems confirm a simple aging scenario at zero temperature and zero field. At zero temperature a critical field $h_c$ is found below which the initial conditions are relevant for the long time dynamics of the system. For $h < h_c$ a logarithmic growth of modulated domains is found in the numerical simulations but this behavior is not captured by the analytical approach which predicts a $t^1/2$ growth law at $T = 0$

Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models

We compute exactly the distribution of the occupation time in a discrete {\em non-Markovian} toy sequence which appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function which is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting new question for a generic finite sized spin glass model: at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included

Glassiness and constrained dynamics of a short-range non-disordered spin model

We study the low temperature dynamics of a two dimensional short-range spin system with uniform ferromagnetic interactions, which displays glassiness at low temperatures despite the absence of disorder or frustration. The model has a dual description in terms of free defects subject to dynamical constraints, and is an explicit realization of the ``hierarchically constrained dynamics'' scenario for glassy systems. We give a number of exact results for the statics of the model, and study in detail the dynamical behaviour of one-time and two-time quantities. We also consider the role played by the configurational entropy, which can be computed exactly, in the relation between fluctuations and response.Comment: 10 pages, 9 figures; minor changes, references adde

Slow Relaxation in a Constrained Ising Spin Chain: a Toy Model for Granular Compaction

We present detailed analytical studies on the zero temperature coarsening dynamics in an Ising spin chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m=-1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compaction. This toy model then predicts analytically an inverse logarithmic law for the growth of density of granular particles, as seen in recent experiments and thereby provides a new mechanism for the inverse logarithmic relaxation. Our analysis utilizes an independent interval approximation for the particle and the hole clusters and is argued to be exact at late times (supported also by numerical simulations).Comment: 9 pages RevTeX, 1 figures (.eps

Understanding the controls on deposited fine sediment in the streams of agricultural catchments

Excessive sediment pressure on aquatic habitats is of global concern. A unique dataset, comprising instantaneous measurements of deposited fine sediment in 230 agricultural streams across England and Wales, was analysed in relation to 20 potential explanatory catchment and channel variables. The most effective explanatory variable for the amount of deposited sediment was found to be stream power, calculated for bankfull flow and used to index the capacity of the stream to transport sediment. Both stream power and velocity category were highly significant (p<<0.001), explaining some 57% variation in total fine sediment mass. Modelled sediment pressure, predominantly from agriculture, was marginally significant (p<0.05) and explained a further 1% variation. The relationship was slightly stronger for erosional zones, providing 62% explanation overall. In the case of the deposited surface drape, stream power was again found to be the most effective explanatory variable (p<0.001) but velocity category, baseflow index and modelled sediment pressure were all significant (p<0.01); each provided an additional 2% explanation to an overall 50%. It is suggested that, in general, the study sites were transport-limited and the majority of stream beds were saturated by fine sediment. For sites below saturation, the upper envelope of measured fine sediment mass increased with modelled sediment pressure. The practical implications of these findings are that (i) targets for fine sediment loads need to take into account the ability of streams to transport/retain fine sediment, and (ii) where agricultural mitigation measures are implemented to reduce delivery of sediment, river management to mobilise/remove fines may also be needed in order to effect an improvement in ecological status in cases where streams are already saturated with fines and unlikely to self-cleanse

Glacial to Holocene terrigenous organic matter input to sediments from Orca Basin, Gulf of Mexico — A combined optical and biomarker approach

In this study we assessed changes in the contribution of terrigenous organic matter (OM) to the Gulf of Mexico over the course of the last deglaciation (the last 25 kyr). To this end, we combined optical kerogen analyses with bulk sedimentary, biomarker, and compound-specific carbon isotope analyses. Samples were obtained from core MD02-2550 from Orca Basin (2249 m water depth at 26°56.77N, 91°20.74W) with temporal resolution ranging from multi-decadal to millennial-scale, depending on the proxy. All proxies confirmed larger terrigenous input during glacial times compared to the Holocene. In addition, the kerogen analyses suggest that much of the glacial OM is reworked (at least 50% of spores and pollen grains and 40% of dinoflagellate cysts). The Holocene sediments, in contrast, contain mainly marine OM, which is exceptionally well preserved. During the deglaciation, terrigenous input was generally high due to large meltwater fluxes, whereby discrepancies between different proxies call for additional influences, such as the change in distance to the river mouth, local productivity changes, and hydrodynamic particle sorting. It is possible that kerogen particles and the terrigenous biomarkers studied here represent distinct pools of land-derived OM with inputs varying independently