The present paper is devoted to the study of Renyi entropy in algebraic structures. We define Renyi entropy of order q and  its Conditional version for a partition of an algebraic structure, and we study their properties. In particular, we show that the  Renyi entropy of a partition is monotonically decreasing.

In this paper, we characterize symmetric distributions based on Renyi entropy of order statistics in subsamples. A test of symmetry is proposed based on the estimated Renyi entropy. Critical values of the test are computed by Monte Carlo simulation. Also we compute the power of the test under different alternatives and show that it behaves better that the test of Habibi and Arghami (1386).

In this paper, we obtain the Renyi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Renyi entropy of an irreducible Markov chain. Finally, we show that the bound for the Renyi entropy rate is the Shannon entropy rate.

The purpose of this study is to define the concepts of Tsallis entropy and Conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for this information measure and prove some properties about the entropy of independent fuzzy partitions. Some results of the relationship between the Tsallis entropy and Conditional Tsallis entropy of fuzzy partitions are obtained and, by using Conditional Tsallis entropy of fuzzy partitions, we show that the subadditivity property for Tsallis entropy of fuzzy partitions is not established in the case that the parameter of this entropy is smaller than one. In general, the Tsallis entropy of fuzzy partitions has similar properties to the shannon entropy, where the parameter of this entropy is larger than one, and therefore can be used in addition to the Shannon entropy, to measure the amount of information to be extracted from a fuzzy experiment.

Maximum of the Renyi entropy and the Tsallis entropy are generalization of the maximum entropy for a larger class of Shannon entropy. In this paper we introduce the maximum Renyi entropy and some of the attributes of distributions which have maximum Renyi entropy investigated. The form of distributions with maximum Renyi entropy is power so we state some properties of these distributions and we have a new form of the Renyi entropy. After pointing the topics of minimum Renyi divergence, some other points in this relation have been discussed. An another form of Renyi divergence have also obtained. Therefore we discussed some of the economic applications of the maximum entropy. Meanwhile, the review of the Csiszar information measure, the general form of distributions with minimum Renyi divergence have obtained.

This contribution deals with the mathematical modeling of R-norm entropy and R-norm divergence in quantum logics. We extend some results concerning the R-norm entropy and Conditional R-norm entropy given in (Inf. Control 45, 1980), to the quantum logics. Firstly, the concepts of R-norm entropy and Conditional R-norm entropy in quantum logics are introduced. We prove the concavity property for the notion of R-norm entropy in quantum logics and we show that this entropy measure does not have the property of sub-additivity in a true sense. It is proven that the monotonicity property for the suggested type of Conditional version of R-norm entropy, holds. Furthermore, we introduce the concept of R-norm divergence of states in quantum logics and we derive basic properties of this quantity. In particular, a relationship between the R-norm divergence and the R-norm entropy of partitions is provided.

In most practical hydraulic engineering problems, accurate flow measurements are required. Understanding flow quantities is an important point in water resources management. Therefore, providing the most appropriate velocity distribution estimation relation that is consistent with the measurement data has always been of interest to researchers. With the development of entropy theory, these methods have been used in a wide range of engineering sciences, including hydraulics and fluid mechanics. In the present study, using the Renyi entropy method, the effective parameter "m" on the Renyi entropy parameter "G" was investigated and the velocity distribution in a circular pipe in the conditions that 36. 2, 50, and 70% of the circular pipe fills. It was estimated at two points by measuring the velocity at depths (0. 1D-0. 9D), (0. 2D-0. 8D), and (0. 3D-0. 7D) relative to the water level. In order to determine the accuracy of estimating the velocity distribution using the Renyi method, the correlation coefficient and the root mean square error was used and also to determine the accuracy of entropy parameters, normalized root mean square error was used. The results showed that the Renyi entropy method has high accuracy with observational data. Also, velocity measurement at depth (0. 9D-0. 1D) from the water surface for 36. 2, 50, and 70% of the circular pipe, with the normalized root mean square error to 0. 2325, 2. 36, and 0. 51 respectively, has higher accuracy.

In this paper, we first show that Renyi distance between any member of a parametric family and its perturbations is proportional to its Fisher information. We, then, prove some relations between the Renyi distance of two distributions and the Fisher information of their exponentially twisted family of densities. Finally, we show that the partial ordering of families induced by Renyi distance is the same as that induced by Fisher information.