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The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.

Manganese monophosphate MnP single crystal deserves attention due to its rich magnetic phase diagram, which is quite different depending on the direction of the applied magnetic field. Generally speaking, it has a Curie temperature around 291 K and several other magnetic arrangements at low temperatures ͑cone-, screw-, fan-, and ferromagnetic-type structures͒. This richness is due to the strong magnetocrystalline anisotropy. In this sense, the present paper makes a thorough description of the influence of this anisotropy on the magnetocaloric properties of this material. From a fundamental view we could point out, among those several magnetic arrangements, the most stable one. On the other hand, from an applied view, we could show that the magnetic entropy change around room temperature ranges from −4.7 to − 3.2 J / kg K, when the magnetic field ͑5 T͒ is applied along the easy and hard magnetization directions, respectively. In addition, we have shown that it is also possible to take advantage of the magnetic anisotropy for magnetocaloric applications, i.e., we have found a quite flat magnetic entropy change ͑with a huge relative cooling power͒, at a fixed value of magnetic field, only rotating the crystal by 90°.

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