A global approach to nonlinear Brascamp-Lieb inequalities and related topics

In this thesis, we investigate global nonlinear Brascamp–Lieb inequalities and some related problems in multilinear harmonic analysis. The body of this thesis is split into three parts, the first is concerning the near-monotonicity properties of nonlinear Brascamp–Lieb functionals under heat-flow. We establish a global nonlinear analogy to the heat-flow monotonicity property enjoyed by linear Brascamp–Lieb inequalities, which we use to prove a slight improvement of the local nonlinear Brascamp–Lieb inequality due to Bennett, Bez, Buschenhenke, Cowling, and Flock, as well as a global stability property of the finiteness of nonlinear Brascamp–Lieb inequalities. In the second part we prove a diffeomorphism-invariant weighted nonlinear Brascamp–Lieb inequality for maps thatadmit a certain structure that generalises the class of polynomial maps. Like polynomials, they have a well-defined notion of degree, and the best constant in this inequality depends explicitly on only the degree of these maps, as well as the underlying dimensions and exponents. Lastly, we refine an induction-on-scales method due to Bennett, Carbery, and Tao to prove a global multilinear $L^2$ estimate on oscillatory integral operators in general dimensions

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

A Stalnakerian Analysis of Metafictive Statements

Because Stalnaker’s common ground framework is focussed on cooperative information exchange, it is challenging to model fictional discourse. To this end, I develop an extension of Stalnaker’s analysis of assertion that adds a temporary workspace to the common ground. I argue that my framework models metafictive discourse better than competing approaches that are based on adding unofficial common grounds

The Prague School and Theories of Structure

Diese Reihe untersucht Gemeinsamkeiten und Unterschiede von Natur- und Geisteswissenschaftlichen. Das Konzept des »Einflusses« bzw. des »gegenseitigen Einflusses« soll zugunsten eines dynamischeren Konzepts des »Interfacing« (Verbindung/Vernetzung) hinterfragt werden. Ein grundlegender Ausgangspunkt ist die Erkenntnis, dass die beiden Wissenssphären, die geistes- und die naturwissenschaftliche, häufig zur gleichen Zeit neue Untersuchungsmodelle entwickeln und damit auf komplexe wissenschaftliche und kulturelle Phänomene reagieren. Das Konzept des »Interfacing« impliziert eine integrierte Sicht neuer Wissensgebiete in neuen Kontexten. Nicht länger an der traditionellen Vorstellung von »Ursache und Wirkung« gebunden, impliziert der Isomorphismus Gleichzeitigkeit statt Konsequentialität. Nicht immer beeinflusst die eine Sphäre die andere; Isomorphismus impliziert gemeinsame Entdeckungen, durch die beide Bereichen zur gleichen Zeit neue investigative Modelle und Darstellungssysteme entwickeln. Dialog und gegenseitiges Verständnis zwischen den beiden sogenannten »zwei Kulturen« werden so stimuliert. Wichtige Forschungsbereiche sind Interfacing-Modelle und Paradigmen in den Natur- und Geisteswissenschaften, kulturell bedingte Darstellungen von Naturwissenschaft und Technologie, wissenschaftliche Entdeckungen und narrative Diskurse, Lebenserinnerungen von Wissenschaftlerinnen und Wissenschaftlern, das Überschreiten von Grenzen zwischen Natur- und Geisteswissenschaft durch Lernen sowie die Bereicherung der Geisteswissenschaften durch angewandte Naturwissenschaften, einschließlich der Informationstechnologien. Die Reihe umfasst sowohl Monographien als auch Essaysammlungen in englischer, deutscher, französischer und italienischer Sprache. Das Nebeneinander verschiedener Sprachen zeugt von der Intention von Herausgeberschaft und wissenschaftlichem Beirats, ein integriertes Wissen aus europäischer Perspektive herauszubilden

Constructions and justifications of a generalization of Viviani's theorem.

Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.This qualitative study actively engaged a group of eight pre-service mathematics teachers (PMTs) in an evolutionary process of generalizing and justifying. It was conducted in a developmental context and underpinned by a strong constructivist framework. Through using a set of task based activities embedded in a dynamic geometric context, this study firstly investigated how the PMTs experienced the reconstruction of Viviani’s theorem via the processes of experimentation, conjecturing, generalizing and justifying. Secondly, it was investigated how they generalized Viviani’s result for equilateral triangles, further across to a sequence of higher order equilateral (convex) polygons such as the rhombus, pentagon, and eventually to ‘any’ convex equi-sided polygon, with appropriate forms of justifications. This study also inquired how PMTs experienced counter-examples from a conceptual change perspective, and how they modified their conjecture generalizations and/or justifications, as a result of such experiences, particularly in instances where such modifications took place. Apart from constructivsm and conceptual change, the design of the activities and the analysis of students’ justifications was underpinned by the distinction of the so-called ‘explanatory’ and ‘discovery’ functions of proof. Analysis of data was grounded in an analytical–inductive method governed by an interpretive paradigm. Results of the study showed that in order for students to reconstruct Viviani’s generalization for equilateral triangles, the following was required for all students: *experimental exploration in a dynamic geometry context; *experiencing cognitive conflict to their initial conjecture; *further experimental exploration and a reformulation of their initial conjecture to finally achieve cognitive equilibrium. Although most students still required the aforementioned experiences again as they extended the Viviani generalization for equilateral triangles to equilateral convex polygons of 4 sides (rhombi) and five sides (pentagons), the need for experimental exploration gradually subsided. All PMTs expressed a need for an explanation as to why their equilateral triangle conjecture generalization was always true, and were only able to construct a logical explanation through scaffolded guidance with the means of a worksheet. The majority of the PMTs (i.e. six out of eight) extended the Viviani generalization to the rhombus on empirical grounds using Sketchpad while two did so on analogical grounds but superficially. However, as the PMTs progressed to the equilateral pentagon (convex) problem, the majority generalized on either inductive grounds or analogical grounds without the use of Sketchpad. Finally all of them generalized to any convex equi-sided polygon on logical grounds. In so doing it seems that all the PMTs finally cut off their ontological bonds with their earlier forms or processes of making generalizations. This conceptual growth pattern was also exhibited in the ways the PMTs justified each of their further generalizations, as they were progressively able to see the general proof through particular proofs, and hence justify their deductive generalization of Viviani’s theorem. This study has also shown that the phenomenon of looking back (folding back) at their prior explanations assisted the PMTs to extend their logical explanations to the general equi-sided polygon. This development of a logical explanation (proof) for the general case after looking back and carefully analysing the statements and reasons that make up the proof argument for the prior particular cases (i.e. specific equilateral convex polygons), namely pentagon, rhombus and equilateral triangle, emulates the ‘discovery’ function of proof. This suggests that the ‘explanatory’ function of proof compliments and feeds into the ‘discovery’ function of proof. Experimental exploration in a dynamic geometry context provided students with a heuristic counterexample to their initial conjectures that caused internal cognitive conflict and surprise to the extent that their cognitive equilibrium became disturbed. This paved the way for conceptual change to occur through the modification of their postulated conjecture generalizations. Furthermore, this study has shown that there exists a close link between generalization and justification. In particular, justifications in the form of logical explanations seemed to have helped the students to understand and make sense as to why their generalizations were always true, but through considering their justifications for their earlier generalizations (equilateral triangle, rhombus and pentagon) students were able to make their generalization to any convex equi-sided polygon on deductive grounds. Thus, with ‘deductive’ generalization as shown by the students, especially in the final stage, justification was woven into the generalization itself. In conclusion, from a practitioner perspective, this study has provided a descriptive analysis of a ‘guided approach’ to both the further constructions and justifications of generalizations via an evolutionary process, which mathematics teachers could use as models for their own attempts in their mathematics classrooms