A birational mapping with a strange attractor: Post critical set and covariant curves

We consider some two-dimensional birational transformations. One of them is a birational deformation of the H\'enon map. For some of these birational mappings, the post critical set (i.e. the iterates of the critical set) is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic two-form. One may have also an infinite post critical set yielding a covariant curve which is not algebraic (transcendent). For two of the birational mappings considered, the post critical set is not infinite and we claim that there is no algebraic covariant curve and no preserved meromorphic two-form. For these two mappings with non infinite post critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.Comment: 26 pages, 11 figure

Characteristics of Copper-based Oxygen Carriers Supported on Calcium Aluminates for Chemical-Looping Combustion with Oxygen Uncoupling (CLOU)

Eight different oxygen carriers (OC) containing CuO (60 wt %) and different mass ratios of CaO to Al2O3 as the support were synthesized by wet-mixing followed by calcination at 1000 °C. The method of synthesis used involved the formation of calcium aluminum hydrate phases and ensured homogeneous mixing of the Ca2+ and Al3+ ions in the support at the molecular level. The performance of the OCs for up to 100 cycles of reduction and oxidation was evaluated in both a thermogravimetric analyzer (TGA) and a fluidized bed reactor, covering a temperature range of 800 to 950 °C. In these cycling experiments, complete conversion of the OC, from CuO to Cu and vice versa, was always achieved for all OCs. The reactivity of the materials was so high that no deactivation could be observed in the TGA, owing to mass transfer limitations. It was found that OCs prepared with a mass ratio of CaO to Al2O3 in the support >0.55 agglomerated in the fluidized bed, resulting in an apparent deactivation over 25 cycles for all temperatures investigated. High ratios of mass of CaO to Al2O3 in the support resulted in CuO interacting with CaO, forming mixed oxides that have low melting temperatures, and this explains the tendency of these materials to agglomerate. This behavior was not observed when the mass ratio of CaO to Al2O3 in the support was ≤0.55 and such materials showed excellent cyclic stability operating under redox conditions at temperatures as high as 950 °C.The authors thank Mohammad Ismail and Matthew Dunstan for helping with the XRD analysis and Alex Casabuena-Rodriguez and for helping with the SEM. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC grant EP/I010912/1).This is the final version of the article. It first appeared from ACS via http://dx.doi.org/10.1021/acs.iecr.5b0117

Scaling the state: Egypt in the third millennium BC

Discussions of the early Egyptian state suffer from a weak consideration of scale. Egyptian archaeologists derive their arguments primarily from evidence of court cemeteries, elite tombs, and monuments of royal display. The material informs the analysis of kingship, early writing, and administration but it remains obscure how the core of the early Pharaonic state was embedded in the territory it claimed to administer. This paper suggests that the relationship between centre and hinterland is key for scaling the Egyptian state of the Old Kingdom (ca. 2,700-2,200 BC). Initially, central administration imagines Egypt using models at variance with provincial practice. The end of the Old Kingdom demarcates not the collapse, but the beginning of a large-scale state characterized by the coalescence of central and local models