Recently, the concept of generalized entropy has been proposed in the literature of information theory. In the present paper, we introduce and study the notion of generalized entropy in the interval (t₁, t₂) as an uncertainty measure. It is shown that the suggested information measure uniquely determines the distribution function. Also, its properties has been studied. Some results have been obtained and some distributions such as uniform, exponential, Pareto, power series and finite range have been characterized by doubly truncated (interval) generalized entropy. Further, we describe a few orders based on this entropy and show its properties.

Minimal entropy martingale measure (MEMM) and geometric Levy process has been introduced as a pricing model for the incomplete financial market. This model has many good properties and is applicable to very wide classes of underlying asset price processes. MEMM is the nearest equivalent martingale measure to the original probability in the sense of Kullback-Leibler distance and is closely related to the large deviation theory .Those good properties has been explained. MEMM is also justified for option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion and Markov-modulated exponential Levy model.

Last two decades of research history on water, are focused on the Jacques Benveniste famous adventure on the “Ultra-High Dilutions” that ended with the definition of water memory. This approach was unreasonably destroyed by a referee articles appeared on Nature. Later the interest on water physics, biology and chemistry has been developed and improved a lot all over the world. Peculiarly, basing the observations that the effects of “Ultra-High Dilutions” were abolished by magnetic fields J. Benveniste suggested that signaling might involve EM waves potentially transmittable to cells and water by electromagnetic means. Here is developed a system approach to the water memory by using the information theory and peculiarly the Bekenstein Law concerning the quantity of information contained in a sphere. Actually, the information theory, by itself, does negate completely the wording of  Nature referee. Observing almost two decades, it appears that  physics, biology and chemistry researchers devote a lot of effort to find relevant ways and methods according to which water maintain information memory and to transfer data among and inside molecules. Those researches are based upon physical, biologic and chemical processes but non the pure information aspect is neglected. These are typically, Nobel Price Luc Montagnier work, V. Voeikov work, Jerry Pollack  and  many others. However, for the coded information inside the water molecule it is applicable all the information theory that is much more than the mechanical Lock and Key criteria. So just limiting this work to the simple calculation of the information quantity included in a water molecule enough to negate the article appeared on Nature. Life principles processes could be grasped by introducing the water molecule information theory of the memory based on the Giuliano Preparata Coherence Dominium theory too.

In classical mechanics, we have individual particle and invariant density in the phase space. In quantum mechanics, any particle is sensitive in a different way from all other particles, for its position and also to the measure process. Thus, we substitute the classical probability in the phase space with the conditional probability in the network of communicating particles. Any probability and entropy are functions of the phase position conditioned by the position of the other particles. Therefore, for different measures we have different conditional entropies. The space of the entropies is a curved and possible torque multidimensional space where the derivative is the covariant derivative on a manifold of the entropic space. At the zero quantum field, the covariant derivative commutes and Fisher matrix is part of the kinetic terms in the Lagrangian where the derivative is the covariant derivative. With Lagrange minimum condition and the entropic space it is possible to show a connection between entropy space and Bohm potential in quantum mechanics. Entropy multidimensional space includes dependence and entanglement as geometric structure of the entropy. Now we can create a non-zero quantum field approach when the covariant derivative does not commute so we have curvature and torsion. The non-zero quantum field can be the Casimir field of forces. Therefore, Casimir force as gravity in the space-time is modelled by curvature and torsion of the entropic space. Useful connection between dependence and covariant derivatives are obtained by copula (dependence measure) and quantum mechanics.

In this paper, we discuss different estimators of the records Weibull distributionparameters and also we apply the Kullback-Leibler divergence of survival function methodto estimate record Weibull parameters. Finally, these methods have been compared usingMonte Carlo simulation.

The Fisher information (FI) measure is an important concept in statistical estimation theory and information theory. However, it has received relatively little consideration in image processing. In this paper, a novel algorithm is developed based on the nonparametric FI measure. The proposed method determines the optimal threshold based on the FI measure by maximizing the measure of the separability of the resultant classes over all of the gray levels. This method is compared with several classic thresholding methods on a variety of images, including some nondestructive testing (NDT) images and text document images. The experimental results show the effectiveness of the new method.