Both the Kullback-Leibler and the Tsallis divergence have a strong limitation: if the value zero appears in probability distributions (p 1 , · · · , p n ) and (q 1 , · · · , q n ), it must appear in the same positions for the sake of significance. In order to avoid that limitation in the framework of Shannon statistics, Ferreri introduced in 1980 hypoentropy: "such conditions rarely occur in practice". The aim of the present paper is to extend Ferreri's hypoentropy to the Tsallis statistics. We introduce the Tsallis hypoentropy and the Tsallis hypodivergence and describe their mathematical behavior. Fundamental properties, like nonnegativity, monotonicity, the chain rule and subadditivity, are established.

The order/disorder characteristics of a compartment fire are researched based on experimental data. From our analysis performed by new, pioneering methods, we claim that the parametric Jensen-Shannon complexity can be successfully used to detect unusual data, and that one can use it also as a means to perform relevant analysis of fire experiments. Thoroughly comparing the performance of different algorithms (known as permutation entropy and two-length permutation entropy) to extract the probability distribution is an essential step. We discuss some of the theoretical assumptions behind each step and stress that the role of the parameter is to fine-tune the results of the Jensen-Shannon statistical complexity. Note that the Jensen-Shannon statistical complexity is symmetric, while its parametric version displays a symmetric duality due to the a priori probabilities used.Symmetry 2020, 12, 22 2 of 12 which were fixed at key locations (see Figure 1) and connected to a data acquisition logger. Flames were observed to impinge on the ceiling and exit through the opening, and we also noted the ignition of crumpled newspaper and stages of fire development that are known as indicators of flashover.Symmetry 2020, 12, x FOR PEER REVIEW 2 of 12

Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality are given.2 ,

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