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

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

Parsec-scale HI absorption structure in a low-redshift galaxy seen against a Compact Symmetric Object

We present global VLBI observations of the 21-cm transition of atomic hydrogen seen in absorption against the radio source J0855+5751. The foreground absorber (SDSS~J085519.05+575140.7) is a dwarf galaxy at $z$ = 0.026. As the background source is heavily resolved by VLBI, the data allow us to map the properties of the foreground HI gas with a spatial resolution of 2pc. The absorbing gas corresponds to a single coherent structure with an extent $>$35pc, but we also detect significant and coherent variations, including a change in the HI optical depth by a factor of five across a distance of $\leq$6pc. The large size of the structure provides support for the Heiles & Troland model of the ISM, as well as its applicability to external galaxies. The large variations in HI optical depth also suggest that caution should be applied when interpreting $T_S$ measurements from radio-detected DLAs. In addition, the distorted appearance of the background radio source is indicative of a strong jet-cloud interaction in its host galaxy. We have measured its redshift ($z$ = 0.54186) using optical spectroscopy on the William Herschel Telescope and this confirms that J0855+5751 is a FRII radio source with a physical extent of $<$1kpc and supports the previous identification of this source as a Compact Symmetric Object. These sources often show absorption associated with the host galaxy and we suggest that both HI and OH should be searched for in J0855+5751.Comment: 14 pages and 10 figures. Accepted for publication in MNRA

Functional centrality in graphs

In this paper we introduce the functional centrality as a generalization of the subgraph centrality. We propose a general method for characterizing nodes in the graph according to the number of closed walks starting and ending at the node. Closed walks are appropriately weighted according to the topological features that we need to measure