Global dynamics of discrete mathematical models of tuberculosis.

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0(E∗) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0≤ 1 and unstable if R0>1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E∗)<1

Mem-fractive properties of mushrooms.

Memristors close the loop forI-Vcharacteristics of the traditional, passive, semi-conductor devices. A memristor is a physical realisation of the material implication and thus is a universal logical element. Memristors are getting particular interest in the field of bioelectronics. Electrical properties of living substrates are not binary and there is nearly a continuous transitions from being non-memristive to mem-fractive (exhibiting a combination of passive memory) to ideally memristive. In laboratory experiments we show that living oyster mushroomsPleurotus ostreatusexhibit mem-fractive properties. We offer a piece-wise polynomial approximation of theI-Vbehaviour of the oyster mushrooms. We also report spiking activity, oscillations in conduced current of the oyster mushrooms.

Some elements for a history of the dynamical systems theory.

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.