Decay Estimates for q (Theorem M1)

This chapter examines the proof for Theorem M1, deriving decay estimates for the quantity q for k ≤ k small + 20 derivatives. To this end, it uses the wave equation satisfied by q. The spacetime M is decomposed as M = (int)M u (ext)M and that u is an outgoing optical function on (ext)M while u is an ingoing optical function. The chapter relies on the global frame defined in section 3.5, and r and m denote the corresponding scalar functions associated to it. It also proves two theorems on improved weighted estimates, as well as flux decay estimates for q.

Space, Vision, and Faith

Linear perspective served as a useful tool in the pursuit of verisimilitude in early modern art. But the technique’s enduring appeal among visual artists from this period did not stem from its ability to transcribe the world’s appearance alone. Rather, as this Reflection highlights, it was the perspectival image’s capacity to build a direct, one-to-one rapport with a viewer that most excited art practitioners. Painters like Piero della Francesca and Giovanni Bellini took full advantage of linear perspective’s potential for simulating vision. The spaces they constructed in altarpieces and other devotional imagery often showed supernatural realms that clearly defy human optical function. This essay presents several examples of early modern works that evoke spiritual vision through linear perspective constructions.