Fractional Iteration of Series and Transseries

We investigate compositional iteration of fractional order for transseries. For any large positive transseries $T$ of exponentiality 0, there is a family $T^{[s]}$ indexed by real numbers $s$ corresponding to teration of order $s$. It is based on Abel's Equation. We also investigate the question of whether there is a family $T^{[s]}$ all sharing a single support set. A subset of the transseries of exponentiality 0 is divided into three classes ("shallow", "moderate" and "deep") with different properties related to fractional iteration.Comment: 17 pages; 1 figur

Type O pure radiation metrics with a cosmological constant

In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant $\Lambda$, and $\tau \ne 0$, by considering the case $\Lambda +\tau\bar\tau \ne 0$. This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction, (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case $\Lambda +\tau\bar\tau = 0$) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics, with zero cosmological constant

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we are able to obtain complete agreement with the results obtained in intrinsic coordinates. In particular, we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.Comment: Extended version of GRG publication, with some typos etc correcte

Dimensionally Dependent Tensor Identities by Double Antisymmetrisation

Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p >= n$. We generalise Lovelock's results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrising over n+1 indices, we establish a very general 'master' identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner.Comment: 17 pages, 2 figure

Reconstruction of eolian bed forms and paleocurrents from cross-bedded strata at Victoria Crater, Meridiani Planum, Mars

Outcrop exposures imaged by the Opportunity rover at Victoria Crater, a 750 m diameter crater in Meridiani Planum, are used to delineate sedimentary structures and further develop a dune-interdune depositional model for the region. The stratigraphy at Victoria Crater, observed during Opportunity's partial traverse of its rim, includes the best examples of meter-scale eolian cross bedding observed on Mars to date. The Cape St. Mary promontory, located at the southern end of the rim traverse, is characterized by meter-scale sets of trough cross bedding, suggesting northward migrating sinuous-crested bed forms. Cape St. Vincent, which is located at the opposite end of the traverse, shows tabular-planar stratification indicative of climbing bed forms with meter- to decameter-scale dune heights migrating southward. Promontories located between Cape St. Mary and Cape St. Vincent contain superposed stratigraphic units with northward and southward dipping beds separated by outcrop-scale bounding surfaces. These bounding surfaces are interpreted to be either reactivation and/or superposition surfaces in a complex erg sea. Any depositional model used to explain the bedding must conform to reversing northward and southward paleomigration directions and include multiple scales of bed forms. In addition to stratified outcrop, a bright diagenetic band is observed to overprint bedding and to lie on an equipotential parallel to the preimpact surface. Meter-scale cross bedding at Victoria Crater is similar to terrestrial eolian deposits and is interpreted as a dry dune field, comparable to Jurassic age eolian deposits in the western United States