Generalized Cauchy means

Given two means M and N, the operator MM,NMM,N assigning to a given mean μ the mean MM,N(μ)(x,y)=M(μ(x,N(x,y)),μ(N(x,y),y)) was defined in Berrone and Moro (Aequationes Math 60:1–14, 2000) in connection with Cauchy means: the Cauchy mean generated by the pair f, g of continuous and strictly monotonic functions is the unique solution μ to the fixed point equation MA(f),A(g)(μ)=μ, where A(f) and A(g) are the quasiarithmetic means respectively generated by f and g. In this article, the operator MM,NMM,N is studied under less restrictive conditions and a general fixed point theorem is derived from an explicit formula for the iterates MnM,NMM,Nn . The concept of class of generalized Cauchy means associated to a given family of mixing pairs of means is introduced and some distinguished families of pairs are presented. The question of equality in these classes of means remains a challenging open problem.Fil: Berrone, Lucio Renato. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Mangiferin: A Promising Anticancer Bioactive

Of late, several biologically active antioxidants from natural products have been investigated by the researchers in order to combat the root cause of carcinogenesis, i.e., oxidative stress. Mangiferin, a therapeutically active C-glucosylated xanthone, is extracted from pulp, peel, seed, bark and leaf of Mangifera indica. These polyphenols of mangiferin exhibit antioxidant properties and tend to decrease the oxygen-free radicals, thereby reducing the DNA damage. Indeed, its capability to modulate several key inflammatory pathways undoubtedly helps in stalling the progression of carcinogenesis. The current review article emphasizes an updated account on the patents published on the chemopreventive action of Mangiferin, apoptosis induction made on various cancer cells, along with proposed antioxidative activities and patent mapping of other important therapeutic properties. Considering it as promising polyphenol, this paper would also summarize the diverse molecular targets of Mangiferin

New generalized fuzzy metrics and fixed point theorem in fuzzy metric space

In this paper, in fuzzy metric spaces (in the sense of Kramosil and Michalek (Kibernetika 11:336-344, 1957)) we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric. First, inspired by the ideas of Grabiec (Fuzzy Sets Syst. 125:385-389, 1989), we define a new G-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by M Grabiec). Next, inspired by the ideas of Gregori and Sapena (Fuzzy Sets Syst. 125:245-252, 2002), we define a new GV-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by V Gregori and A Sapena). Moreover, we provide the condition guaranteeing the existence of a fixed point for these single-valued contractions. Next, we show that the generalized pseudodistance J:X×X→[0,∞) (introduced by Włodarczyk and Plebaniak (Appl. Math. Lett. 24:325-328, 2011)) may generate some generalized fuzzy metric NJ on X. The paper includes also the comparison of our results with those existing in the literature

Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the $(M,N)$-convexity property of functions and also for H\"older--Minkowski type inequalities

Means and covariance functions for geostatistical compositional data: an axiomatic approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on the geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences on the spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability

A composite functional equation from algebraic aspect

In this paper we discuss the composite functional equation f(x+2f(y))=f(x)+y+f(y) on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers

Translation Representations and Scattering By Two Intervals

Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states