On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces

Computably regular topological spaces

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces

Products of effective topological spaces and a uniformly computable Tychonoff Theorem

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we introduce natural multi-representations of the class of all effective topological spaces, of their points, of their subsets and of their compact subsets. We show that the binary, finite and countable product operations on effective topological spaces are computable. For spaces with non-empty base sets the factors can be retrieved from the products. We study computability of the product operations on points, on arbitrary subsets and on compact subsets. For the case of compact sets the results are uniformly computable versions of Tychonoff's Theorem (stating that every Cartesian product of compact spaces is compact) for both, the cover multi-representation and the "minimal cover" multi-representation

Computable Jordan Decomposition of Linear Continuous Functionals on C[0;1]C[0;1]

By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures on the Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition into non-negative or non-decreasing objects. Using the representation approach to computable analysis, a computable version of the Riesz representation theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we extend this result. We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm. We introduce natural representations for defining computability. We prove that the canonical linear bijections between these spaces and their inverses are computable. We also prove that Jordan decomposition is computable on each of these spaces

The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function

In this article we treat a notion of continuity for a multi-valued function $F$ and we compute the descriptive set-theoretic complexity of the set of all $x$ for which $F$ is continuous at $x$. We give conditions under which the latter set is either a $G_\delta$ set or the countable union of $G_\delta$ sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of $F$ is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function $F$ whose graph is a Borel set and the set of points of continuity of $F$ is not a Borel set. Finally we give some analogous results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in [{\em Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra}, {\sl submitted}].Comment: 22 page

Representing Probability Measures using Probabilistic Processes

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names ” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.

Representations of measurable sets in computable measure theory

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly countably infinite, over some alphabet {\Sigma}. As a basic computability structure we consider a computable measure on a computable $\sigma$-algebra. We introduce and compare w.r.t. reducibility several natural representations of measurable sets. They are admissible and generally form four different equivalence classes. We then compare our representations with those introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our representations is the most useful one for studying computability on measurable functions

Effective Choice and Boundedness Principles in Computable Analysis

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem and others. We also explore how existing classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into this picture. We compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example

An ancient compound rediscovered: perspectives of aphid control in organic hop growing by the use of quassia products

In the first three decades of the 20th century, quassia extract was widely used in hop growing as a chemical agent to control Phorodon humuli and other insect pests. In the first years of the 21st century this compound was rediscovered by German organic hop growers. In nine efficacy trials conducted in five field seasons, quassia products proved to be effective control agents for P. humuli in organically grown aroma cultivars. As the best method of application a systemic variant was developed by painting a suspension of quassia extract to the bines. This method proved not only to be very effective but was also best from an environmental point of view as sprayed quassia extracts had side effects on non-target organisms. As an optimal systemic application rate 24 g/ha of the active ingredient quassine was determined