1
Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?
« Last post by varsigma on Yesterday at 23:19:22 »I think I can say a few things about the modular structure of this graph.
In that, a free additive module is just the idea of repeated addition, a sequence like "+ 1 + 1 + 1 . . .". This, in modular addition, say mod 2, will mean you have the sequence 0 + 1 -> 1 + 1 -> 0 + 1 -> . . .
This means you can define a loop, you can identify the two points, {0,1} and make the distance between them + 1, as the loop so this represents the action of addition on both points, as a single point "{0,1}". So it maps a loop, representing addition modulo 2, to a set of points. This works for {0,1,2, . . .,n}.
So then you have 0 + 1 + 1 + . . . n times, = 0 (mod n). Yay. That's because 0 is the kernel of a function that maps to a homomorphism as the image of the next , or is it the previous? I need to figure that out, function (the homomorphism between additive groups) in the chain.
A Turing machine that models this is one that reads a 0 and writes a 1, or reads a 1 and writes a 0 ( adds 1 mod 2). So there are two read/write loops on a single state, and no halt condition.
In that, a free additive module is just the idea of repeated addition, a sequence like "+ 1 + 1 + 1 . . .". This, in modular addition, say mod 2, will mean you have the sequence 0 + 1 -> 1 + 1 -> 0 + 1 -> . . .
This means you can define a loop, you can identify the two points, {0,1} and make the distance between them + 1, as the loop so this represents the action of addition on both points, as a single point "{0,1}". So it maps a loop, representing addition modulo 2, to a set of points. This works for {0,1,2, . . .,n}.
So then you have 0 + 1 + 1 + . . . n times, = 0 (mod n). Yay. That's because 0 is the kernel of a function that maps to a homomorphism as the image of the next , or is it the previous? I need to figure that out, function (the homomorphism between additive groups) in the chain.
A Turing machine that models this is one that reads a 0 and writes a 1, or reads a 1 and writes a 0 ( adds 1 mod 2). So there are two read/write loops on a single state, and no halt condition.