In this paper, we introduce thick r-sensitivity, multi-r-sensitivity and block thick r-sensitivity for $r\ge 2$. We first give a characterization of a minimal system which is block thickly r-sensitive. Then we obtain a sufficient condition of a minimal system which is thickly r-sensitive. The maximal pattern entropy of a multi-r-sensitive topological dynamical system is also discussed.

As the first part in the present paper, we study a class of backward stochastic differential equation (BSDE, for short) driven by Teugels martingales associated with some Lévy processes having moment of all orders and an independent Brownian motion. We obtain an existence and uniqueness result for this type of BSDEs when the final time is allowed to be random. As the second part, we prove, under a monotonicity condition, an existence and uniqueness result for fully coupled forward–backward stochastic differential equation (FBSDE, for short) driven by Teugels martingales in stopping time duration. As an illustration of our theoretical results, we deal with a portfolio selection in Lévy-type market.

In this article, we consider the statistical inferences of the unknown parameters of a generalized inverted exponential distribution based on the Type II progressively hybrid censored sample. By applying the expectation–maximization (EM) algorithm, the maximum likelihood estimators are developed for estimating the unknown parameters. The observed Fisher information matrix is obtained using the missing information principle, and it can be used for constructing asymptotic confidence intervals. By applying the bootstrapping technique, the confidence intervals for the parameters are also derived. Bayesian estimates of the unknown parameters are obtained using the Lindley’s approximation. Monte Carlo simulations are implemented and observations are given. Finally, a real data set representing the spread factor of micro-drops is analyzed to illustrative purposes.

Let $K_0={\mathbb {Q}}\left( \sqrt{\delta }\right) $ be a quadratic field. For those $K_0$ with odd class number, much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension $K={\mathbb {Q}}\left( \sqrt{\delta },\sqrt{d}\right) $ over ${\mathbb {Q}}$. When $\delta =2$ or p with $p\equiv 1\bmod 4$ a prime and K is real, it was described in Yue (Ramanujan J 21:17–25, 2010) and Bae and Yue (Ramanujan J 24:161–181, 2011). In this paper, we describe the Hilbert genus field of K explicitly when $K_0$ is real and K is imaginary. In fact, we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.

Given that the strength of statistics lies in modelling, we are motivated to do a comparative statistical study between two types of ragas, one being aesthetically known to be restful and the other restless in nature. We first try to distinguish these two types through statistical modeling. To substantiate our findings, two more statistical features are considered in the paper to separate the two categories of ragas, namely the rate of change of pitch and inter-onset interval. The experimental results are encouraging.

Fuzzy entropy is an important concept to measure the fuzzy information. Measure of fuzziness of a fuzzy set is the measure of its fuzziness. In the present communication, we have defined an exponential fuzzy entropy of order-$(\alpha , \beta )$. Besides establishing the validity of the proposed measure, we have also discussed some of its properties. At last, we have given the application of the proposed measure in multiple attribute decision-making problems. In this section, we have considered two cases for the weights of attributes: One is the case when weights are completely unknown to us, and the other is the case when weights are partially known to us.