See On Denotational Inconsistency.
At 22 minutes 13 seconds, the mapping of the plane onto a torus could be done using the projective circle described at 32 minutes 13 seconds. The vague idea I have here is to repeatedly wind the graph around the torus faster and faster as it approaches the horizontal and vertical asymptotes. The rate at which the line wraps being given by the rate at which the projected point moves as it approaches the point [0, 0] along the circumference of the circle.
This might open the way to a definition of the n'th roots of unity as a kind of generalization of Casteljau-Bézier iterated convex combinations to cyclotomic fields, with the successive approximations being determined using a kind of Vernier principle, where the corrections are functions of the beat frequencies that arise from the difference between the actual value of the function and that of the approximant. See this video, released a day after I first posted this:
Note that these differences can be detected between harmonics, so there is a possibility that error correction could be carried out simultaneously at different levels. This is basically the idea behind the Fourier Transform, which this video explains in very graphical terms:
This article by Terence Tao, which is a bit beyond my technical ability to comprehend, looks like it has some interesting observations about Pontryagin Duality which gives a connection between Fourier methods and cyclotomic fields: The Fourier Transform. I would expect this projective model of the growth rates of composite functions would suffice for Jean-Yves Girard's theorem of comparison of hierachies discussed in Proof-theory and Logical Complexity II.
See this video on the problem we are trying to solve:
I am imagining all this going on in the framework outlined here: The Power of Thought.
At 22 minutes 13 seconds, the mapping of the plane onto a torus could be done using the projective circle described at 32 minutes 13 seconds. The vague idea I have here is to repeatedly wind the graph around the torus faster and faster as it approaches the horizontal and vertical asymptotes. The rate at which the line wraps being given by the rate at which the projected point moves as it approaches the point [0, 0] along the circumference of the circle.
This might open the way to a definition of the n'th roots of unity as a kind of generalization of Casteljau-Bézier iterated convex combinations to cyclotomic fields, with the successive approximations being determined using a kind of Vernier principle, where the corrections are functions of the beat frequencies that arise from the difference between the actual value of the function and that of the approximant. See this video, released a day after I first posted this:
Note that these differences can be detected between harmonics, so there is a possibility that error correction could be carried out simultaneously at different levels. This is basically the idea behind the Fourier Transform, which this video explains in very graphical terms:
This article by Terence Tao, which is a bit beyond my technical ability to comprehend, looks like it has some interesting observations about Pontryagin Duality which gives a connection between Fourier methods and cyclotomic fields: The Fourier Transform. I would expect this projective model of the growth rates of composite functions would suffice for Jean-Yves Girard's theorem of comparison of hierachies discussed in Proof-theory and Logical Complexity II.
See this video on the problem we are trying to solve:
I am imagining all this going on in the framework outlined here: The Power of Thought.
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