Abstract This paper deals with the fractional calculus of zeta functions. In particular, study is focused on Hurwitz ? \zeta function. All results are based complex generalization Grünwald-Letnikov derivative. We state and prove functional equation together an integral representation by Bernoulli numbers. Moreover, we ...

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied in [SZ04], is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based ...

We generalize the Clausius (in)equality to overdamped mesoscopic and macroscopic diffusions in the presence of nonconservative forces. In contrast to previous frameworks, we use a decomposition scheme for heat which is based on an exact variant of the Minimum Entropy Production Principle as obtained from dynamical fluctuation theory. This new extended heat theorem holds true for arbitrary drivi...

A new method based on principal component analysis (PCA) and artificial neural networks (ANN) is proposed for fault diagnosis of gearboxes. Firstly the six different base wavelets are considered, in which three are from real valued and other three from complex valued. Two wavelet selection criteria Maximum Energy to Shannon Entropy ratio and Maximum Relative Wavelet Energy are used and compared...

Complex systems are often inherently non-ergodic and non-Markovian and Shannon entropy loses its applicability. Accelerating, path-dependent and aging random walks offer an intuitive picture for non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived from three of the Shannon–Khinchin axioms and by violating the fourth, the so-called comp...

In this paper, at first we derive a family of maximum Tsallis entropy distributions under optional side conditions on the mean income and the Gini index. Furthermore, corresponding with these distributions a family of Lorenz curves compatible with the optional side conditions is generated. Meanwhile, we show that our results reduce to Shannon entropy as $beta$ tends to one. Finally, by using ac...

Shannon entropy of a probability measure P , defined as − ∫ X dP dμ ln dP dμ dμ on a measure space (X,M, μ), is not a natural extension from the discrete case. However, maximum entropy (ME) prescriptions of Shannon entropy functional in the measure-theoretic case are consistent with those for the discrete case. Also it is well known that Kullback-Leibler relative entropy can be extended natural...

The general idea of information entropy provided by C.E. Shannon “hangs over everything we do” and can be applied to a great variety of problems once the connection between a distribution and the quantities of interest is found. The Shannon information entropy essentially quantify the information of a quantity with its specific distribution, for which the information entropy based methods have ...