The minimalist notion of a phase has often been investigated with a view to the interfaces. ‘Phases’ provides a strictly syntax-internal perspective. If phases are fundamental, they should provide the grounds for a unifying treatment of different syntactic phenomena. Concentrating on displacement, the book argues that this expectation is borne out: there is an empirical clustering of properties, whereby the phrases that undergo pied-piping are also the phrases that host intermediate traces of cyclic movement. The same phrases also host partial and secondary movement. Finally, the immediate complements within these phrases never strand the embedding heads. The phrases that show this behaviour are the phases (CP, vP, DP, and PP). To account for the cluster of properties, phases are claimed to have two special properties: their complement is inaccessible to operations outside, the Phase Impenetrability Condition; their heads may be endowed with unvalued features that are neither connected to the categorical status of the phase nor interpreted on it. It is shown how the cluster of empirical properties flows naturally from these two assumptions, supporting the idea that phases are indeed a fundamental construct in syntax.
The minimalist notion of a phase has often been investigated with a view to the interfaces. ‘Phases’ provides a strictly syntax-internal perspective. If phases are fundamental, they should provide the grounds for a unifying treatment of different syntactic phenomena. Concentrating on displacement, the book argues that this expectation is borne out: there is an empirical clustering of properties, whereby the phrases that undergo pied-piping are also the phrases that host intermediate traces of cyclic movement. The same phrases also host partial and secondary movement. Finally, the immediate complements within these phrases never strand the embedding heads. The phrases that show this behaviour are the phases (CP, vP, DP, and PP). To account for the cluster of properties, phases are claimed to have two special properties: their complement is inaccessible to operations outside, the Phase Impenetrability Condition; their heads may be endowed with unvalued features that are neither connected to the categorical status of the phase nor interpreted on it. It is shown how the cluster of empirical properties flows naturally from these two assumptions, supporting the idea that phases are indeed a fundamental construct in syntax.
These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization. We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.
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