A Galois connection between Turing jumps and limits

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis

¿Son las perspectivas socioculturales de las matemáticas y de las pedagogías de las matemáticas incompatibles con las perspectivas cognitivas?

En esta conferencia se presenta una reflexión sobre la relación entre las perspectivas congitivas que se han servido de marco conceptual para diversas investigaciones en Educación Matemática y las nuevas perspectivas socioculturales. Se concluye que cada uno de estos programas, con sus sucesivos o paralelos modelos y teorías ha mostrado aspectos que permanecían ocultos para los paradigmas previos, y tienen aún mucho que aportar. La única manera de saber cuánto aportará cada uno es practicarlos, impulsarlos, revisar y criticar sus resultados y contrastarlos con otros

Completion of Choice

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal role in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete.Comment: 30 page

Weihrauch goes Brouwerian

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations that are required to describe these concepts. In order to show that the parallelized total Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized total Weihrauch lattice forms a Brouwer algebra with this implication, the total Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms. Finally, we show that the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which also implies that the theory of our Brouwer algebra is Jankov logic.Comment: 36 page

An Unsupervised Method for Estimating the Global Horizontal Irradiance from Photovoltaic Power Measurements

In this paper, we present a method to determine the global horizontal irradiance (GHI) from the power measurements of one or more PV systems, located in the same neighborhood. The method is completely unsupervised and is based on a physical model of a PV plant. The precise assessment of solar irradiance is pivotal for the forecast of the electric power generated by photovoltaic (PV) plants. However, on-ground measurements are expensive and are generally not performed for small and medium-sized PV plants. Satellite-based services represent a valid alternative to on site measurements, but their space-time resolution is limited. Results from two case studies located in Switzerland are presented. The performance of the proposed method at assessing GHI is compared with that of free and commercial satellite services. Our results show that the presented method is generally better than satellite-based services, especially at high temporal resolutions