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\exp \pi  broadband  chain model  defence  food chain  function  group defence  group  model  population  structure pure  zeta function  {\rm 
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Preview: IMA Journal of Applied Mathematics - Advance Access

IMA Journal of Applied Mathematics Advance Access

Published: Mon, 20 Nov 2017 00:00:00 GMT

Last Build Date: Tue, 21 Nov 2017 00:48:20 GMT


Riemann’s zeta function and the broadband structure of pure harmonics


Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a) = \sum n^{-s} \exp (2\pi \,{\rm i}na)$ for $\Re s >1$, and extended to the complex plane via analytic continuation. Let $s_n = \sigma_n + {\rm i}t_n, \, t_n >0 $, denote the sequence of non-trivial zeros of the Riemann zeta function in the upper halfplane ordered according to non-decreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Cesàro means of the sequence $F_{s_n} (a)$ converge to the first harmonic $\exp (2\pi \,{\rm i} a)$ in the sense of periodic distributions. This reveals a natural broadband structure of the pure tone. The proof involves Fujii’s refinement of the classical Landau theorem related to the uniform distribution modulo one of the non-trivial zeros of $\zeta(s)$. We also discuss the applicable aspects of this phenomenon, focusing on broadband encodings of signals.

Investigation of an explosive food chain model with interference and inhibitory effects


In the current manuscript, we have investigated the temporal as well as spatio-temporal dynamics of a three species modified Leslie–Gower food chain model with Holling type IV and Crowley–Martin function responses. We have shown that explosion in the top predator population can be prevented if group defence is sufficiently strong at the lowest trophic levels. This demonstrates that group defence can act as a damping mechanism, and prevent population explosion of apex predators. We also show that the spatially explicit model can exhibit diffusion-driven instability, that depends strongly on the intensity of the group defence, in the prey population. Standard bifurcation analysis and the period doubling route to chaos are also investigated.