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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Arrows of Times, Non-integer Operators, Self-Similar Structures, Zeta Functions and Riemann Hypothesis: a Synthetic Categorical Approach

Journal of Applied Nonlinear Dynamics 6(2) (2017) 283--301 | DOI:10.5890/JAND.2017.06.012

Alain Le Méhauté$^{1}$,$^{2}$; Philippe Riot$^{1}$,$^{2}$

$^{1}$ Physics Department, Kazan Federal University, Kremlenskaia 18, 42008 Kazan, Russia

$^{2}$ Institut Franco Qu´ebecois, 41 rue Jacob 75006 Paris, France

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Abstract

The authors have previously reported the existence of a morphism between the Riemann zeta function and the “Cole and Cole” canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar structures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics controlled by non-integer derivative operators. Furthermore it was already shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase attached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue without limited its epistemological consequences.

Acknowledgments

D. Tayursky (Vice Rector for Education at KFU ), E. Wimmer (CEO Materials design S.A & Inc.) and S. Raynal (Institut Franco-Quebecois Paris) for financial support; D. Tayurski ( Kazan Federal University KFU), F. Ben Adda (N.Y. University), A. Oustaloup and J Sabatier (Bordeaux University), A. El Kaabouchi, L. Nivanen (ISMANS) Fu Yao Ren (Fudan University) for scientific exchanges and in memoriam Per Bro (Tromsø Norway 1924 - San Fe USA 2004) and J.P. Badiali (Paris: 1941-2016).

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