 Grzegorczyk hierarchy vs Iterated differential equations? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Grzegorczyk hierarchy vs Iterated differential equations? (/showthread.php?tid=945) Grzegorczyk hierarchy vs Iterated differential equations? - MphLee - 01/03/2015 Inspired by JmsNxn's thread (http://math.eretrandre.org/tetrationforum/showthread.php?tid=818&pid=7139#pid7139) about the continuum sum I repost this obsevation about the link between the fractional calculus and the Hyperoperations. I guess that there can be interesting links... and probably is not the wrong way to approach the problem. I just found some results about something similar. M. Campagnolo, C. Moore -Upper and Lower Bounds on Continuous-Time Computation In this text I found a relation betwen a hierarchy of real valued function and the Grzegorczyk hierarchy. The interesting relations are betwen a hierarchy called and the hierarchy : Quote:1-Any function in is computable in 2-If then is the extension to the reals of some then 3-the converse holds: if is a function on the naturals of rank it has an extension in ------------------- The interesting thing is that the various levels of are defined via iterated solution of a special kind of functional equation...and that maybe can be linked with your knowledge in this field... Definition- is defined as follow Quote:I-the constants ,, and , the projection functions, are in II- is closed composition and linear integration in a recursive way we define Quote:III- contains the functions in IV- in we can find all the solutions to the equation (2) in this text ( http://languagelog.ldc.upenn.edu/myl/DK/CampagnoloMoore.pdf ) applied to the functions in V- is closed under composition and linear integration and if if