We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point (x, ax + b), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.
We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if p ≥ 1 is a computable real, and if Ω is a nonzero, non-atomic, and separable measure space, then every computable presentation of L p (Ω) is computably linearly isometric to the standard computable presentation of L p [0, 1]; in particular, L p [0, 1] is computably categorical. We also show that there is a measure space Ω that does not have a computable presentation even though L p (Ω) does for every computable real p ≥ 1.Laboratoire lorrain de recherche en informatique et ses applications,
Abstract-DNA nanotechnology uses the information processing capabilities of nucleic acids to design self-assembling, programmable structures and devices at the nanoscale. Devices developed to date have been programmed to implement logic circuits and neural networks, capture or release specific molecules, and traverse molecular tracks and mazes.Here we investigate the use of requirements engineering methods to make DNA nanotechnology more productive, predictable, and safe. We use goal-oriented requirements modeling to identify, specify, and analyze a product family of DNA nanodevices, and we use PRISM model checking to verify both common properties across the family and properties that are specific to individual products. Challenges to doing requirements engineering in this domain include the errorprone nature of nanodevices carrying out their tasks in the probabilistic world of chemical kinetics, the fact that roughly a nanomole (a 1 followed by 14 0s) of devices are typically deployed at once, and the difficulty of specifying and achieving modularity in a realm where devices have many opportunities to interfere with each other. Nevertheless, our results show that requirements engineering is useful in DNA nanotechnology and that leveraging the similarities among nanodevices in the product family improves the modeling and analysis by supporting reuse.
We investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several characterizations. We show that some natural effectivizations of quasi-metric spaces are strictly stronger.Proposition 6.1. The space [α, 1] < is an effective quasi-Polish space iff α is left-c.e. relative to the halting set.
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