Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z)$. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(\Omega)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partial\Omega)$) then $\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(\Omega)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.Comment: 33 pages, 8 figures. arXiv admin note: text overlap with arXiv:1507.0433

Exploration through drawings in the conceptual stages of product design

This paper argues that sequences of exploratory drawings - constructed by designer's movements and decisions - trace systematic and logical paths from ideas to designs. This argument has three parts. First, sequences of exploratory sketches produced by product designers, against the same task specification, are analyzed in terms of the cognitive categories of reinterpretation, emergence and abstraction. Second, a computational model is outlined for the process of exploration through drawing and third the model is applied to elucidate the logic in the sequences of exploratory sketches examined earlier

Making meaning: developing an understanding of form in distance design education

Design education throughout the world provides students with a variety of experiences that help them develop an understanding of form and shape. The conventional model of such education requires students to participate in studio and workshop-based projects to develop skills through the creation of models and prototypes. However, with the increase in distance education worldwide we need to explore new ways for students to create and manipulate form remotely. This paper presents new work at the Open University, UK which set out to engage design students in form-making from a distance. Participants were given access to technical and design support that took rough sketches of chair designs and converted these into tangible scale models which were mailed back to the students. Several cycles of this activity generated data on how such supported modelling activity stimulated students' creative ability, design knowledge and motivation. This paper proposes new priorities for distance design education