05/25/2021, 07:32 PM
Hi, I like this summary! I was already familiar with the outline of you frac-calc approach from 2015 but now this seems really tidy and easier to follow. Im happy to see those two commutative diagrams! xD
About the logical structure, I just skimmed it... I need to study it line by line. But I find some passages unclear or interesting so I hope you can help me (and some typos).
1 Introdution
p.2; "As a formal sequence, we can call a hyper-operator chain," that object is indeed more general than an Hyperoperation sequence. Regardless of the initial conditions "chain" seems a nice name. There at U of T have you received some comments on that "chain" equation?
2 Fractional derivative
p.3; after the exp-fixpoint eq.: "to complex values, and [deride]"
p.4; \( S_\theta \) is missing the point 0 right? The area enclosed by the two rays \( t(\cos(\theta)+\sin(\theta)i \) and \( t(\cos(\theta)-\sin(\theta)i \) where t>0 and the point 0 removed?
Are arcs assumed to be injective, no winding and do not cross over themselves? Sure, to be precise you consider an arc ad just the subset to integrate over, the image of the parametrization, so that different parametrizations can map [0,+\infty) to the same arc but running on it on different velocities.
p.4; after you define the set of endofuntion boldface \( {\mathbb S}_\theta \) you say: "Then there exists a correspondence between functions F(z)". A correspondence between those bounded F and what?
In other words in that line you are referring to the correspndence between boldface \( {\mathbb S}_\theta \) and boldface \( {\mathbb E}_\theta \) you define at page 6 ?
p.5; let's double check my understanding.
Theorem 2.1 (Euler) takes \( f\in {\mathbb S}_\theta \) and maps it to \( {{\mathfrak E}_w[f]}\in {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C}) \), i.e.
where \( \Gamma(z){{\mathfrak E}_w[f]}(-z)={\sum_{n=0}^\infty}f^{(n)}(w)\frac{(-\gamma(1))^n}{n!(n+z)}+\int_{\gamma[1,\infty)}f(w-y)y^{z-1}dy \) and \( {{\mathfrak E}[f]}:{{\mathbb C}_{\Re (z)<1}}\to{\mathbb C} \).
Theorem 2.2 (Ramanujan) takes \( H\in {\mathbb E}_\theta \)(?) and maps it to \( {{\mathfrak R}[H]}\in {\mathbb C}^{{\mathbb C}} \), i.e.
where \( {{\mathfrak R}[H]}(w)={\sum_{n=0}^\infty}H(n)\frac{w^n}{n!} \) and \( {{\mathfrak R}[H]}(w):{\mathbb C}\to{\mathbb C} \)
and claims
(a)\( ({\mathfrak E}_0[{\mathfrak R}[H]])(z)=H(z) \) and
(b)\( ({\mathfrak E}_w[{\mathfrak R}[H]])(z)={\mathfrak R}[H\circ S^z](w) \) (S is the successor so S^z(n)=z+n)
Now we know where the trasforms take value but not exactly where they land (can we compose them?):
Thm 2.1 \( {{\mathfrak E}_w:{\mathbb S}_\theta\to {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C}) \)
Thm 2.1 \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb C}^{{\mathbb C}} \) and
(a)\( {\mathfrak E}_0\circ {{\mathfrak R}=id \)
Observation: (b) implies trivially (a) as the tail of the series vanish setting w=0 and we keep the leading coefficient H(z). I can't fully parse (b) yet. I can say that (a) is possible iff we can feed Ramanujan into Euler, i.e. \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta \): in fact you prove this later
p.7; By theorem 2.3 \( {{\mathfrak E}_0:{\mathbb S}_\theta\to{\mathbb E}_\theta \). When you say "and conversely" do you mean that also \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta \) holds right?
Why don't you have to prove theorem 2.3 BEFORE you can even claim in theorem 2.2 that you can apply \( {{\mathfrak E}_0} \) to\( f(w)={{\mathfrak R}[H]}(w) \)?
In fact if thm 2.3 is true then we can apply thm 2.2 eq. (a), i.e. \( {\mathfrak E}_0\circ {{\mathfrak R}=id_{{\mathbb E}_\theta} \). This equation alone implies that:
- \( {\mathfrak E}_0 \) is surjective (every function in boldface E is de differintegral at w=0 of some boldface S function);
- \( {\mathfrak R} \) is injective (if two boldface E functions define the same auxilliary function then they are the same function).
But thm 2.4 also add that those two functions are also inverse hence \( {{\mathbb E}_\theta}\simeq {{\mathbb S}_\theta} \) are in bijection
Question 1: for which \( \theta \) those spaces are in bijection?
Question 2: Do this bijection preserve some stucture? Idk... are those functions paces closed under piecewise sum, scalar multiplication, piecewise multiplication, do have a metric or topological structure (a system of open sets), a norm?
Question 3: take \( \theta<\kappa \) we have \( S_\theta\subseteq S_\kappa \). What is the relationship between \( {\mathbb S}_\theta \) and \( {\mathbb S}_\kappa \) o between \( {\mathbb E}_\theta \) and \( {\mathbb E}_\kappa \)?
Asap I'll go on the other sections.
Regards
About the logical structure, I just skimmed it... I need to study it line by line. But I find some passages unclear or interesting so I hope you can help me (and some typos).
1 Introdution
p.2; "As a formal sequence, we can call a hyper-operator chain," that object is indeed more general than an Hyperoperation sequence. Regardless of the initial conditions "chain" seems a nice name. There at U of T have you received some comments on that "chain" equation?
2 Fractional derivative
p.3; after the exp-fixpoint eq.: "to complex values, and [deride]"
p.4; \( S_\theta \) is missing the point 0 right? The area enclosed by the two rays \( t(\cos(\theta)+\sin(\theta)i \) and \( t(\cos(\theta)-\sin(\theta)i \) where t>0 and the point 0 removed?
Are arcs assumed to be injective, no winding and do not cross over themselves? Sure, to be precise you consider an arc ad just the subset to integrate over, the image of the parametrization, so that different parametrizations can map [0,+\infty) to the same arc but running on it on different velocities.
p.4; after you define the set of endofuntion boldface \( {\mathbb S}_\theta \) you say: "Then there exists a correspondence between functions F(z)". A correspondence between those bounded F and what?
In other words in that line you are referring to the correspndence between boldface \( {\mathbb S}_\theta \) and boldface \( {\mathbb E}_\theta \) you define at page 6 ?
p.5; let's double check my understanding.
Theorem 2.1 (Euler) takes \( f\in {\mathbb S}_\theta \) and maps it to \( {{\mathfrak E}_w[f]}\in {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C}) \), i.e.
where \( \Gamma(z){{\mathfrak E}_w[f]}(-z)={\sum_{n=0}^\infty}f^{(n)}(w)\frac{(-\gamma(1))^n}{n!(n+z)}+\int_{\gamma[1,\infty)}f(w-y)y^{z-1}dy \) and \( {{\mathfrak E}[f]}:{{\mathbb C}_{\Re (z)<1}}\to{\mathbb C} \).
Theorem 2.2 (Ramanujan) takes \( H\in {\mathbb E}_\theta \)(?) and maps it to \( {{\mathfrak R}[H]}\in {\mathbb C}^{{\mathbb C}} \), i.e.
where \( {{\mathfrak R}[H]}(w)={\sum_{n=0}^\infty}H(n)\frac{w^n}{n!} \) and \( {{\mathfrak R}[H]}(w):{\mathbb C}\to{\mathbb C} \)
and claims
(a)\( ({\mathfrak E}_0[{\mathfrak R}[H]])(z)=H(z) \) and
(b)\( ({\mathfrak E}_w[{\mathfrak R}[H]])(z)={\mathfrak R}[H\circ S^z](w) \) (S is the successor so S^z(n)=z+n)
Now we know where the trasforms take value but not exactly where they land (can we compose them?):
Thm 2.1 \( {{\mathfrak E}_w:{\mathbb S}_\theta\to {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C}) \)
Thm 2.1 \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb C}^{{\mathbb C}} \) and
(a)\( {\mathfrak E}_0\circ {{\mathfrak R}=id \)
Observation: (b) implies trivially (a) as the tail of the series vanish setting w=0 and we keep the leading coefficient H(z). I can't fully parse (b) yet. I can say that (a) is possible iff we can feed Ramanujan into Euler, i.e. \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta \): in fact you prove this later
p.7; By theorem 2.3 \( {{\mathfrak E}_0:{\mathbb S}_\theta\to{\mathbb E}_\theta \). When you say "and conversely" do you mean that also \( {{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta \) holds right?
Why don't you have to prove theorem 2.3 BEFORE you can even claim in theorem 2.2 that you can apply \( {{\mathfrak E}_0} \) to\( f(w)={{\mathfrak R}[H]}(w) \)?
In fact if thm 2.3 is true then we can apply thm 2.2 eq. (a), i.e. \( {\mathfrak E}_0\circ {{\mathfrak R}=id_{{\mathbb E}_\theta} \). This equation alone implies that:
- \( {\mathfrak E}_0 \) is surjective (every function in boldface E is de differintegral at w=0 of some boldface S function);
- \( {\mathfrak R} \) is injective (if two boldface E functions define the same auxilliary function then they are the same function).
But thm 2.4 also add that those two functions are also inverse hence \( {{\mathbb E}_\theta}\simeq {{\mathbb S}_\theta} \) are in bijection
Question 1: for which \( \theta \) those spaces are in bijection?
Question 2: Do this bijection preserve some stucture? Idk... are those functions paces closed under piecewise sum, scalar multiplication, piecewise multiplication, do have a metric or topological structure (a system of open sets), a norm?
Question 3: take \( \theta<\kappa \) we have \( S_\theta\subseteq S_\kappa \). What is the relationship between \( {\mathbb S}_\theta \) and \( {\mathbb S}_\kappa \) o between \( {\mathbb E}_\theta \) and \( {\mathbb E}_\kappa \)?
Asap I'll go on the other sections.
Regards
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
