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See manuscript PDF file for complete Abstract:We consider the ordinary differential equations defined by a trigonometric polynomial field, we prove that any solution $x$ admits a "rotation vector" $\rho\in \mathbb{R}^n$. More precisely, the function $t\mapsto x(t)-\rho t$ is bounded on time and it is a "weak almost periodic" function of "slope" $\rho$.

We prove that ζ(s)/s ≠ ζ(1−s)/1−s for every ℜ(s) ∈ (0, 1/2) and ℑ(s) > 0 where ζ is the Riemann Zeta function. In the end of the paper, we give a discussion about the Riemann hypothesis.

In this work, we describe a simple strategy for the preparation of a low-cost electrode material based on polypyrrole (PPy) film grown on an insulating cellulosic paper substrate (Pap) via in-situ oxidative polymerization technique and functionalized by silver nanoparticles (AgNPs) uniformly dispersed on its surface. The properties of the obtained AgNPs-PPy composites were characterized using FTIR-ATR spectroscopy (FTIR-ATR), X-ray photoelectron spectroscopy (XPS), Scanning Electron Microscope (SEM) coupled with energy dispersive X-ray spectrometry (EDX), X-ray diffraction spectroscopy (DRX) and electrochemical impedance spectroscopy (SIE).