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   Event cameras, which feature pixels that independently respond to changes in brightness, are becoming increasingly popular in high- speed applications due to their lower latency, reduced bandwidth requirements, and enhanced dynamic range compared to traditional frame- based cameras. Numerous imaging and vision techniques have leveraged event cameras for high- speed scene understanding by capturing high- framerate, high- dynamic range videos, primarily utilizing the temporal advantages inherent to event cameras. Additionally, imaging and vision techniques have utilized the light field—a complementary dimension to temporal information—for enhanced scene understanding.In this work, we propose "Event Fields", a new approach that utilizes innovative optical designs for event cameras to capture light fields at high speed. We develop the underlying mathematical framework for Event Fields and introduce two foundational frameworks to capture them practically: spatial multiplexing to capture temporal derivatives and temporal multiplexing to capture angular derivatives. To realize these, we design two complementary optical setups— one using a kaleidoscope for spatial multiplexing and another using a galvanometer for temporal multiplexing. We evaluate the performance of both designs using a custom-built simulator and real hardware prototypes, showcasing their distinct benefits. Our event fields unlock the full advantages of typical light fields—like post- capture refocusing and depth estimation—now supercharged for high- speed and high- dynamic range scenes. This novel light- sensing paradigm opens doors to new applications in photography, robotics, and AR/VR, and presents fresh challenges in rendering and machine learning. 

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5. Ray class groups and ray class fields for orders of number fields

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   We contribute to the theory of orders of number fields. We define a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. We show existence of ray class fields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. We give exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes. 

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