Computable Diagonalizations and Turing’s Cardinality Paradox

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers.

Modelling woody material transport and deposition in alpine rivers

Recent flood events in Switzerland and Western Austria in 2005 were characterised by an increase in impacts and associated losses due to the transport of woody material. As a consequence, protection measures and bridges suffered considerable damages. Furthermore, cross-sectional obstructions due to woody material entrapment caused unexpected flood plain inundations resulting in severe damage to elements at risk. Until now, the transport of woody material is neither sufficiently taken into account nor systematically considered, leading to prediction inaccuracies during the procedure of hazard mapping. To close this gap, we propose a modelling approach that (1) allows the estimation of woody material recruitment from wood-covered banks and flood plains; (2) allows the evaluation of the disposition for woody material entrainment and transport to selected critical configurations along the stream and that (3) enables the delineation of hazard process patterns at these critical configurations. Results from a case study suggest the general applicability of the concept. This contribution to woody material transport analysis refines flood hazard assessments due to the consideration of woody material transport scenarios.

A new, fast and flexible radiative transfer method for Venus general circulation models

We present a new radiation scheme for the Oxford Planetary Unified Model System for Venus, suitable for the solar and thermal bands. This new and fast radiative parameterization uses a different approach in the two main radiative wavelength bands: solar radiation (0.1-5.5 mu m) and thermal radiation (1.7-260 mu m). The solar radiation calculation is based on the delta-Eddington approximation (two-stream-type) with an adding layer method. For the thermal radiation case, a code based on an absorptivity/emissivity formulation is used. The new radiative transfer formulation implemented is intended to be computationally light, to allow its incorporation in 3D global circulation models, but still allowing for the calculation of the effect of atmospheric conditions on radiative fluxes. This will allow us to investigate the dynamical-radiative-microphysical feedbacks. The model flexibility can be also used to explore the uncertainties in the Venus atmosphere such as the optical properties in the deep atmosphere or cloud amount. The results of radiative cooling and heating rates and the global-mean radiative-convective equilibrium temperature profiles for different atmospheric conditions are presented and discussed. This new scheme works in an atmospheric column and can be easily implemented in 3D Venus global circulation models. (C) 2014 Elsevier Ltd. All rights reserved.