Signal Detection, Target Tracking and Differential Geometry

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I intend to do work with a lot of Quantum Field Theory, and maybe try out general relativity (though not to specialize in it, just to see why its not comptable with qm). Please elaborate with a less hand-waving description. According to my lecturer John Steele, it is "computationally the easiest, notationally the neatest, aesthetically the best, makes the definitions more natural and the proofs of the two major theorems easier. The stone which we have read was the stone of the altar at Delos.

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