On the Chebyshev type inequality for seminormed fuzzy integral

The Chebyshev type inequality for seminormed fuzzy integral is discussed. The main results of this paper generalize some previous results obtained by the authors. We also investigate the properties of semiconormed fuzzy integral, and a related inequality for this type of integral is obtained.     

Inequalities and Convergence Concepts of Fuzzy and Rough Variables

It is well-known that Markov inequality, Chebyshev inequality, Hölder's inequality, and Minkowski inequality are important and useful results in probability theory. This paper presents the analogous inequalities in fuzzy set theory and rough set theory. In addition, sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper presents four types of convergence concept of fuzzy/rough sequence: convergence almost surely, convergence in credibility/trust, convergence in mean, and convergence in distribution. Some mathematical properties of those new convergence concepts are also given.     

Chebyshev’s inequality for Choquet-like integral

The Chebyshev inequality for the Choquet-like integral is investigated. As an application, a Markov’s inequality for this type of integral is proven. Some previous results obtained by others are generalized.     

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.     

A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov’s theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel–Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.     

The Iyengar Type Inequalities with Exact Estimations and the Chebyshev Central Algorithms of Integrals

In this paper, both low order and high order extensions of the Iyengar type inequality are obtained. Such extensions are the best possible in the same sense as that of the Iyengar inequality. hzrthermore, the Chebyshev central algorithms of integrals for some function classes and some related problems are also considered and investigated.     

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.     

Turán Quadrature Formulas and Christoffel Type Functions for the Chebyshev Polynomials of the Second Kind

An explicit representation for the Cotes numbers of Turán quadrature formulas based on the zeros of the Chebyshev polynomials of the second kind and its asymptotic behavior are given. The asymptotic formula for the corresponding Christoffel type functions is also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于第二类Chebyshev权函数的Landau’s型不等式

首先建立了第二类Chebyshev多项式Un(x)的Landau’s型不等式.利用Un(x)的正交性,建立了代数多项式pn(x)的加权Landau’s型不等式,并且指出其不等式的系数在某种意义上是最好可能的.     

Multivariate Markov polynomial inequalities and Chebyshev nodes

This article considers the extension of V.A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.     

A generalized real-valued measure of the inequality associated with a fuzzy random variable

Fuzzy random variables have been introduced by Puri and Ralescu as an extension of random sets. In this paper, we first introduce a real-valued generalized measure of the “relative variation” (or inequality) associated with a fuzzy random variable. This measure is inspired in Csiszár's f-divergence, and extends to fuzzy random variables many well-known inequality indices. To guarantee certain relevant properties of this measure, we have to distinguish two main families of measures which will be characterized. Then, the fundamental properties are derived, and an outstanding measure in each family is separately examined on the basis of an additive decomposition property and an additive decomposability one. Finally, two examples illustrate the application of the study in this paper.     

Can the bounds in the multivariate Chebyshev inequality be attained?

Chebyshev’s inequality was recently extended to the multivariate case. In this paper we prove that the bounds in the multivariate Chebyshev’s inequality for random vectors can be attained in the limit. Hence, these bounds are the best possible bounds for this kind of regions.     

Chebyshev type inequalities for pseudo-integrals

Chebyshev type inequalities for two classes of pseudo-integrals are shown. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function g. Another one concerns the pseudo-integrals based on a semiring ([a,b],max,⊙), where ⊙ is generated. Moreover, a strengthened version of Chebyshev’s inequality for pseudo-integrals is proved.     

Fuzzy partitions and relations; an axiomatic basis for clustering

In this paper some connections between fuzzy partitions and similarity relations are explored. A new definition of transitivity for fuzzy relations yields a relation-theoretic characterization of the class of all psuedo-metrics on a fixed (finite) data set into the closed unit interval. This notion of transitivity also links the triangle inequality to convex decompositions of fuzzy similarity relations in a manner which may generate new techniques for fuzzy clustering. Finally, we show that every fuzzy c-partition of a finite data set induces a psuedo-metric of the type described above on the data.     

Turán type inequalities for generalized complete elliptic integrals

In this paper our aim is to establish some Turán type inequalities for Gaussian hypergeometric functions and for generalized complete elliptic integrals. These results complete the earlier result of P. Turán proved for Legendre polynomials. Moreover we show that there is a close connection between a Turán type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind. At the end of this paper we prove a recent conjecture of T. Sugawa and M. Vuorinen related to estimates of the hyperbolic distance of the twice punctured plane. Dedicated to my son Koppány.     

Molnár-dependence in a pre-Hilbert module over an H*-algebra

Molnár-dependence is related to the strong Cauchy-Schwarz inequality in a pre-Hilbert module over an H*-algebra analogously as the linear dependence is related to the Cauchy–Schwarz inequality in a pre-Hilbert space. Necessary and sufficient conditions for two elements of a pre-Hilbert module to be Molnár-dependent are established in this article, what is enabled by proving a stronger inequality than the strong Cauchy–Schwarz one. Furthermore, it is shown that Molnár-dependence is a transitive relation.     

Reliable solution of an elasto–plastic torsion problem

A variational inequality with uncertain coefficients, which are given in certain bounded intervals, is considered. The inequality corresponds to a torsion problem of an elasto–plastic orthotropic bar, when employing the Haar–Kármán principle. Two maximization problems with respect to the admissible coefficients are formulated. The solvability of continuous and approximate problems is proven and a convergence analysis is presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Analog of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> with a subadditive measure

For a subadditive fuzzy measure (not assumed finite), a Minkowski type triangle inequality, with Choquet integrals in place of Lebesgue integrals, is shown to hold. It is immediate that the set of functions for which a certain positive power of the absolute values have finite Choquet integrals is closed under addition, leading to a linear space analogous to the Lebesgue space L ^p , with a metric related to the integral of that power. Under the additional condition that the subadditive fuzzy measure is inner continuous (Sugeno), the space is shown to be complete. Consequences of the Minkowski type inequality are illustrated in two specific instances.      

Hermite–Hadamard inequality for fuzzy integrals

In this paper we prove a Hermite–Hadamard type inequality for fuzzy integrals. Some examples are given to illustrate the results.