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The aim of this research is to provide guidance for drawing Euler diagrams. People draw Euler diagrams to visualise information about sets using closed curves, illustrat- ing how they intersect and whether they are disjoint or in a subset relationship. As a widespread and popular visualisation technique, considerable research efort has been devoted to the problem of automatically drawing Euler diagrams. To draw efective Euler diagrams, however, insight is needed into the layout choices being made and their impact on user comprehension. Through surveying the literature we established five guides spanning three categories of layout choice: descriptional, topological and graph- ical. The literature accords robust layout guides for both the descriptional and topo- logical choices. However, the literature yields limited guidance regarding the numerous graphical choices that have to be made such as the shape and colour of curves. This lack of guidance is perhaps surprising given it is well documented by information visu- alisation theorists that people are visually sensitive to graphical choices. Importantly, such theory tells us that graphical choices can significantly impact the comprehension of diagrams.

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In this work we present a symbolic calculus for ΨDOs on a smooth manifold M based on a suitable notion of the global phase function. In previous literature on the topic, either local coordinates or connections have been used to define the phase functions, symbols and oscillatory integrals defining ΨDOs. We obtain global formulae for the compositions and adjoints of ΨDOs. Applications are given to elliptic ΨDOs, boundedness on Sobolev spaces and functional calculus for elliptic ΨDOs.

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No description available.