Conjecture on superfunctions

Sledge by and \(~y=\sin^n(\pi/2)-\sin^n(x)~\) with \(~n\!=\!100~\)

Conjecture on superfunctions (Конжекция о суперфункциях) is hypothesis, concept about application of superfunctions in physics and other sciences.

Two conjectures are formulated below.

Conjecture 1

WIth methods described in articles , for any physically-meaningful transfer function \(T\), I can construct the physically-meaningful superfunction \(F\), satisfying the transfer equation \(F(z\!+\!1)=T(F(z))\).

Conjecture 2

Let there be some physical dependence denoted with function \(\Phi\).
Let there is a set of experimental data as a table of values of the argument and corresponding values of the function, and also estimates of the error of the measurement.
Let the function \(\Phi\) is assumed to be smooth (real holomorphic), but no physical model to describe its shape is available.
Let there is at least one approximation of function \(\Phi\) in terms of special functions, that are built-in at the programming languages available for year 2003,
that fits the experimental data with satisfactory precision (id est, gives so many correct decimal digits, as the data have),
and Let this fit uses \(M\) adjusting parameters, and let \(~M\!>\!10~\).

Then, with use of superfunctions and the non-integer iterates, I can reduce the number of fitting parameters, keeping the same error of fitting,
or/and reduce the norm of deviation of the fit from experimental data, keeping the same number \(M\) of the fitting parameters.

3. Almost elementary function

Let \(f\) be entire periodic function with period \(P\)

Let \( c \) be positive real constant.

Let \( F(z)=f(z)+cz~\)   for all \(z\in \mathbb C \)

Then, the set of values of function \(F\) is set of complex numbers, perhaps, except a single number.

4. Existence of real superfunction

Let \(T\) be real-entire function growing along along the real axis.

Then, there exist positive number(varepsilon\) such that there exist and real-holomorphic superfunction \(F\) as solution of the transfer equation \(F(z+1)=F(T(z)) \)
such that the range of holomorphism of \( F(z)\) covers the half-strip
\( \Re(z) > 0 \)
\( |\Im(z)| < \varepsilon \)

Example

Use of superfunctions and the resulting non-integer iterates greatly extends the arsenal of functions available for fitting of any physical dependences. One may expect, that use of superfunctions allow to improve the approximation (and extrapolation) of various data. In more specific form (to satisfy the TORI axioms), this expectation is formulated as The figure at right illustrates fitting of the shape of the sledge runner with the 100th iteration of sin with single parameter; superfunction and abelfunction of sin can be used to evaluate the iteration.

This article presents the first attempt to formulate the expectation about this applicability as refutable conjectures, as a concept, that satisfies the TORI axioms.

Copyleft 2013 by Dmitrii Kouznetsov.

Keywords

Abel function, Exotic iteration Iteration, Philosophy, Regular iteration, Religion, Superfunction, TORI, TORI axioms,