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We prove a result, similar to the ones known as Ishihara's First and Second Trick, for sequences of functions.

Bei der Planung von Satellitenmissionen ist die Auswahl geeigneter Bodenstationen zur Datenübertragung ein wichtiger Erfolgs- und Kostenfaktor. Das Ziel dieser Besonderen Lernleistung war die Entwicklung einer nutzerfreundlichen Webanwendung zur Planung von wissenschaftlichen Satellitenmissionen. Wesentlicher Teil der Besonderen Lernleistung war die Ermittlung der optimalen Abfolge von Kontakten zwischen Satellit und Bodenstationen zum Erreichen der maximal übertragbaren Datenmenge. Im Rahmen dieser Arbeit wurden die geografischen Daten und Antennenparameter vorhandener Bodenstationen in einer Datenbank zusammengetragen und für die Webapplikation aufbereitet. Die Bahnparameter des Satelliten zur Berechnung des Satellitenorbits werden zusammen mit den Antennenparametern des Satelliten vom Anwender vorgegeben. Für die Umsetzung der Webanwendung wurden JavaScript, HTML, CSS und das in der Programmiersprache Python geschriebene Webframework Django verwendet.

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and infinitesimals. The construction of this new set is done na\"ively in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its na\"ivety character, the set $\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years., Comment: 48 pages, 4 figures

We investigate the notion of a complete enough metric space that, while classically vacuous, in a constructive setting allows for the generalisation of many theorems to a much wider class of spaces. In doing so, this notion also brings the known body of constructive results significantly closer to that of classical mathematics. Most prominently, we generalise the Kreisel-Lacome-Shoenfield Theorem/Tseytin's Theorem on the continuity of functions in recursive mathematics., DOCUMENTA MATHEMATICA, Vol 24 (2019), p. 899-914

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and infinitesimals. The construction of this new set is done na��vely in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its na��vety character, the set $\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years., 48 pages, 4 figures

We prove some constructive results that on first and maybe even on second glance seem impossible.

We give a comprehensive treatment of the area known as constructive reverse mathematics, which aims to classify, over intuitionistic logic, various theorems in intuitionistic, constructive recursive, and classical mathematics. Our work provides an overview of many known results, but also many new results, and new proofs of known results. We cover omniscience principles, Markov’s principle, Brouwer’s fan theorem(s), recursive principles, Ishihara’s boundedness principle BD-N, as well as minor principles, and separation techniques. Diese Arbeit beschäftigt sich mit dem Gebiet der „konstruktiven reversen Mathematik“ („constructive reverse mathematics“), dessen Ziel es ist verschiedene Aussagen aus der intuitionistischen, der rekursiven und der klassischen Mathematik über intuitionistischer Logik zu klassifizieren. Es wird ein Überblick über bekannte Ergebnisse, aber auch viele neue Resultate und neue Beweise gegeben. Unter anderem werden „Allwissenheits“-Prinzipien, Markov’s Prinzip, die Brouwer’schen Fächersätze, Aussagen aus der rekursiven Mathematik und Ishiharas Beschränktheitsprinzip BD-N behandelt.

The Drinker Paradox is as follows. In every nonempty tavern, there is a person such that if that person is drinking, then everyone in the tavern is drinking. Formally, \[ \exists x \big(\varphi \rightarrow \forall y \varphi[x/y]\big) \ . \] Due to its counterintuitive nature it is called a paradox, even though it actually is a classical tautology. However, it is not minimally (or even intuitionistically) provable. The same can be said of its dual, which is (equivalent to) the well-known principle of \emph{independence of premise}, \[ \varphi \rightarrow \exists x \psi \ \vdash \ \exists x (\varphi \rightarrow \psi) \] where $x$ is not free in $\varphi$. In this paper we study the implications of adding these and other formula schemata to minimal logic. We show first that these principles are independent of the law of excluded middle and of each other, and second how these schemata relate to other well-known principles, such as Markov's Principle of unbounded search, providing proofs and semantic models where appropriate.

An introduction and overview of constructive reverse mathematics., Comment: version 1.3

We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit functions theorem. This leads not only to an a priori proof of continuity, but also to an alternative, fully constructive existence proof.

We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.