Marsala, Christophe Bouchon-Meunier, Bernadette

Monotonicity is a major property of entropy measures. In this paper, we focus on entropy measures for intuitionistic fuzzy sets and interval-valued fuzzy sets. We consider their polar representation which provides an easy interpretation of monotonicity for several entropy measures introduced in the literature. Moreover, this polar representation en...

Vega, Frank

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is know...

Paulin, Christina

The French atomic energy commission (CEA, France) deploys specific numerical methods to carry out its hydrodynamics simulations under especially severe constraints : advection and deformation over long distances, isentropic evolutions and strong shocks, complex equations of state and material mixtures, coupling with other various and complex physic...

Randriamihamison, Nathanaël Vialaneix, Nathalie Neuvial, Pierre

Hierarchical Agglomerative Classification (HAC) with Ward’s linkage has been widely used since its introduction in Ward (1963). The present article reviews the different extensions of the method to various input data and the constrained framework, while providing applicability conditions. In addition, various versions of the graphical representatio...

Bouchon-Meunier, Bernadette Marsala, Christophe

Entropies and measures of information are extensively used in several domains and applications in Artificial Intelligence. Among the original quantities from Information theory and Probability theory, a lot of extensions have been introduced to take into account fuzzy sets, intuitionistic fuzzy sets and other representation models of uncertainty an...

Bustince, Humberto Mesiar, Radko Fernandez, Javier Galar, Mikel Paternain, Daniel Altalhi, Abdulrahman Dimuro, Graçaliz P. Bedregal, Benjamín Takáč, Zdenko
Published in
Information Processing and Management of Uncertainty in Knowledge-Based Systems

In this paper, in order to generalize the Choquet integral, we replace the difference between inputs in its definition by a restricted dissimilarity function and refer to the obtained function as d -Choquet integral. For some particular restricted dissimilarity function the corresponding d -Choquet integral with respect to a fuzzy measure is just t...

Arbel, Julyan Marchal, Olivier Nipoti, Bernardo
Published in
Journal of Inequalities and Applications

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution.

He, Xin Cui, Yunan Hudzik, Henryk
Published in
Journal of Inequalities and Applications

In this paper, the monotonicity is investigated with respect to Orlicz sequence space lΦ,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{\varPhi , p}$\end{document} ...

Khachatryan, Kh. A. Petrosyan, H. S.
Published in
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)

AbstractThe paper is devoted to the study of a class of integral equations with a symmetric kernel and with convex nonlinearity on the positive semiaxis. Existence and uniqueness theorems for a nonnegative and bounded solution are proved. The qualitative properties of the constructed solution are investigated. At the end of the paper, some particul...

Wehenkel, Antoine Louppe, Gilles

Monotonic neural networks have recently been proposed as a way to define invertible transformations. These transformations can be combined into powerful autoregressive flows that have been shown to be universal approximators of continuous probability distributions. Architectures that ensure monotonicity typically enforce constraints on weights and ...