omg... I don't feel ready for the integrals yet but... something is moving in the back of my head. I need to reread all of your posts at least 10 times more.
The bigger obstacle is that I'll need to study what contours, Jordan Curves and singularities really are... I need to study many key parts of your papers. But I'll get there.
Now I'd like to answer to all the points. Tonight or tomorrow I'll do it.
Now I have time only to say this: the setting is already as general as can be in this sense: I never assumed somewhere that the monoid T is a group, let alone being commutative.
All of this holds for abstract monoids T as long as we have a theory that makes us able to define the partial derivative operator (differentiating functions taking values in the monoid): I used additive notation only as a philosophical aid. I'll get back on this. In the proofs the only facts I use of T are the existence of the identity (that's how we can get the infinitesimal generator factorization) and associativity (that's how we derive theorem 1).(*)
The eventuality that the time is not commutative deserves a longer comment. In fact I have something on this. There is (trivial) canonical way to extend every N-action (integer iteration) to an E-action where E is a new monoid. E is the monoid of what I like to call "the functions with rank=1 relative to the initial function".
The interesting part btw is that If I'll be able to finally understand how the integrals work I could come up with a monoid of paths (arcs? A category of paths? The fundamental groupoid of the time-object?) over a top-space and maybe we could rebuild all of this theory of the composition integral in term of homotopy theory... big maybe...
(*)of course if for classical partial derivative to exists T needs to be assumed commutative that's another story.
Edit.
I went on wiki looking for some basic path-contour-residue theorem business. I won't annoy you with "first -course-on-complex-analysis" questions. But it is interesting how it seems that we can define a category of paths, where objects are points of the plane and arrows are oriented paths (curves): it looks promising because monotonicity of the parametrization boils down to functoriality (imho) and considering equivalence classes modulo parametrization boils down to modding out by homotopy (I guess). It is unbelievable to me how the value of the integral on a closed curve does not depends on the parametrization chosen, how it only depends on the set of singularities that it encloses, and that it doesn't changes when you permute them... I'm not gonna lie..to me this is black magic.
Edit 2. That's amazing... it is beginning to make some sense. When the domain is simply connected the functions that loops around a point contract themselves to identity (just like paths homotopic to the point) but if the domain is punctured (by the singularity) they can't contract just as the circle (a punctured disk) is not topological equivalent to the disk (that is really a point) because it has an hole! It's so marvelous... the residue classes (in your theory) are class of conjugate functions. There are many doubts but the bigger now is: how you define \( {\rm Rsd}(f,\zeta;z) \)?
Edit 3. I went back to your old post in the bullet notation thread. That method really comes from Euler method as you say... It is curious that there is no known link between the infinitesimal generator f and the solutions to y'=f(y).
In your anulus example you path \rho realizing that conjugation of the two paths gamma is the interval [-\delta,\delta]. What happens if you measure the superfunctions of two elements of the residue class by some invariants attached to the paths (like the length)? Maybe finding the shortest path/curve?
The bigger obstacle is that I'll need to study what contours, Jordan Curves and singularities really are... I need to study many key parts of your papers. But I'll get there.
Now I'd like to answer to all the points. Tonight or tomorrow I'll do it.
Now I have time only to say this: the setting is already as general as can be in this sense: I never assumed somewhere that the monoid T is a group, let alone being commutative.
All of this holds for abstract monoids T as long as we have a theory that makes us able to define the partial derivative operator (differentiating functions taking values in the monoid): I used additive notation only as a philosophical aid. I'll get back on this. In the proofs the only facts I use of T are the existence of the identity (that's how we can get the infinitesimal generator factorization) and associativity (that's how we derive theorem 1).(*)
The eventuality that the time is not commutative deserves a longer comment. In fact I have something on this. There is (trivial) canonical way to extend every N-action (integer iteration) to an E-action where E is a new monoid. E is the monoid of what I like to call "the functions with rank=1 relative to the initial function".
The interesting part btw is that If I'll be able to finally understand how the integrals work I could come up with a monoid of paths (arcs? A category of paths? The fundamental groupoid of the time-object?) over a top-space and maybe we could rebuild all of this theory of the composition integral in term of homotopy theory... big maybe...
(*)of course if for classical partial derivative to exists T needs to be assumed commutative that's another story.
Edit.
I went on wiki looking for some basic path-contour-residue theorem business. I won't annoy you with "first -course-on-complex-analysis" questions. But it is interesting how it seems that we can define a category of paths, where objects are points of the plane and arrows are oriented paths (curves): it looks promising because monotonicity of the parametrization boils down to functoriality (imho) and considering equivalence classes modulo parametrization boils down to modding out by homotopy (I guess). It is unbelievable to me how the value of the integral on a closed curve does not depends on the parametrization chosen, how it only depends on the set of singularities that it encloses, and that it doesn't changes when you permute them... I'm not gonna lie..to me this is black magic.
Edit 2. That's amazing... it is beginning to make some sense. When the domain is simply connected the functions that loops around a point contract themselves to identity (just like paths homotopic to the point) but if the domain is punctured (by the singularity) they can't contract just as the circle (a punctured disk) is not topological equivalent to the disk (that is really a point) because it has an hole! It's so marvelous... the residue classes (in your theory) are class of conjugate functions. There are many doubts but the bigger now is: how you define \( {\rm Rsd}(f,\zeta;z) \)?
Edit 3. I went back to your old post in the bullet notation thread. That method really comes from Euler method as you say... It is curious that there is no known link between the infinitesimal generator f and the solutions to y'=f(y).
In your anulus example you path \rho realizing that conjugation of the two paths gamma is the interval [-\delta,\delta]. What happens if you measure the superfunctions of two elements of the residue class by some invariants attached to the paths (like the length)? Maybe finding the shortest path/curve?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)