Operator pencils on the algebra of densities

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators, and the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO-pencil lifting to describe all regular proj-equivariant pencil liftings. In particular the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian. Within this paper the question of whether the pencil lifting factors through a full symbol map naturally arises.Comment: 23 pages, LaTeX file Small corrections are mad

Vegetation-soil relations in a highly sodic landscape, Yelarbon, southern Queensland

Soil and vegetation data were collected from a sodic-scald near Yelarbon in southern Queensland. The surface of the landscape includes relatively light textured pedestals of the A-horizon with slightly alkaline pH and slopes leading down to scalded basement representing the surface of the strongly alkaline B-horizon. The strongest gradient within the floristic patterns was associated with wetland vegetation in drainage lines, but a secondary and orthogonal gradient was related to soil pH, which was probably a function of lower alkalinity on the more stable and weathered A-horizons. There were few significant differences between soil or vegetation characteristics from plot data comparing parts of the landscape with differing historical grazing regimes. Sites included stock routes heavily grazed between the 1920s and 1970s, and subsequently almost ungrazed; and grazed paddocks that have had moderate use throughout this period. There is clear evidence that the area is naturally active in terms of erosion and deposition during flooding regardless of grazing

Unevenly-sampled signals: a general formalism of the Lomb-Scargle periodogram

The periodogram is a popular tool that tests whether a signal consists only of noise or if it also includes other components. The main issue of this method is to define a critical detection threshold that allows identification of a component other than noise, when a peak in the periodogram exceeds it. In the case of signals sampled on a regular time grid, determination of such a threshold is relatively simple. When the sampling is uneven, however, things are more complicated. The most popular solution in this case is to use the "Lomb-Scargle" periodogram, but this method can be used only when the noise is the realization of a zero-mean, white (i.e. flat-spectrum) random process. In this paper, we present a general formalism based on matrix algebra, which permits analysis of the statistical properties of a periodogram independently of the characteristics of noise (e.g. colored and/or non-stationary), as well as the characteristics of sampling.Comment: 10 pages, 11 figures, Astronomy and Astrophysics, in pres

Assessing the importance of a self-generated detachment process in river biofilm models

1. Epilithic biofilm biomass was measured for 14 months in two sites, located up- and downstream of the city of Toulouse in the Garonne River (south-west France). Periodical sampling provided a biomass data set to compare with simulations from the model of Uehlinger, Bürher and Reichert (1996: Freshwater Biology, 36, 249–263.), in order to evaluate the impact of hydraulic disturbance. 2. Despite differences in application conditions (e.g. river size, discharge, frequency of disturbance), the base equation satisfactorily predicted biomass between low and high water periods of the year, suggesting that the flood disturbance regime may be considered a universal mechanism controlling periphyton biomass. 3. However modelling gave no agreement with biomass dynamics during the 7-month long low water period that the river experienced. The influence of other biomass-regulating factors (temperature, light and soluble reactive phosphorus) on temporal biomass dynamics was weak. 4. Implementing a supplementary mechanism corresponding to a temperature-dependent self-generated loss because of heterotrophic processes allowed us to accurately reproduce the observed pattern: a succession of two peaks. This case study suggests that during typical summer low water periods (flow stability and favourable temperature) river biofilm modelling requires self-generated detachment to be considered

Operator pencil passing through a given operator

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight \lo on a manifold $M$. One can consider a pencil of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator $\Delta$ such that any \Delta_\l is a linear differential operator acting on densities of weight \l. This pencil can be identified with a linear differential operator \hD acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of \diff(M)-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil