An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey includes not only the Choquet ...integral (1954) 10, the Shilkret integral (1971) 66 and the Sugeno integral (1974) 71 and some of their properties, but also some more general and more recent concepts as universal integrals Klement et al. (2010) 27 and decomposition integrals Even (2014) 13, together with some of their properties, such as integral inequalities and convergence theorems.
Semicopula based integrals Jin, LeSheng; Kolesárová, Anna; Mesiar, Radko
Fuzzy sets and systems,
06/2021, Letnik:
412
Journal Article
Recenzirano
We define the notion of weak universal integral based on a semicopula, and introduce and discuss two particular classes of weak universal integrals based on semicopulas, which generalize the ...well-known Sugeno and Shilkret integrals. In special cases, when the considered semicopulas are bounded from above by the Łukasiewicz t-norm, all introduced integrals reduce to the corresponding smallest semicopula based universal integrals. Remarkably, when the product semicopula is considered, the proposed integrals generalizing the Shilkret integral belong to the class of aggregation functions, which is not the case of the minimum semicopula when the introduced integrals generalize the Sugeno integral.
In this paper we obtain the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for fractional integrals which generalize the two familiar fractional integrals namely, the Riemann–Liouville ...and the Hadamard fractional integrals into a single form. We prove that, in most cases, we obtain the Riemann–Liouville and the Hadamard equivalence just by taking limits when a parameter ρ→1 and ρ→0+, respectively.
This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based ...on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.
This paper introduces the family of CF-integrals, which are pre-aggregations functions that generalizes the Choquet integral considering a bivariate function F that is left 0-absorbent. We show that ...CF-integrals are 1→-pre-aggregation functions, studying in which conditions they are idempotent and/or averaging functions. This characterization is an important issue of our approach, since we apply these functions in the Fuzzy Reasoning Method (FRM) of a fuzzy rule-based classification system and, in the literature, it is possible to observe that non-averaging aggregation functions usually provide better results. We carry out a study with several subfamilies of CF-integrals having averaging or non-averaging characteristics. As expected, the proposed non-averaging CF-integrals obtain more accurate results than the averaging ones, thus, offering new possibilities for aggregating accurately the information in the FRM. Furthermore, it allows us to enhance the results of classical FRMs like the winning rule and the additive combination.
To deal with multicriteria decision making (MCDM) problems with interaction criteria, the Choquet integral (CI) is one of effective tools. This paper first proposes the reverse Choquet integral ...(RCI), which defines the importance of the ordered elements in an opposite principle to the CI. To show the principle of the RCI, we offer its concrete expression in view of the Möbius representation by which one can clearly see the difference and the relationship between the CI and the RCI. Then, we propose the “bi-direction Choquet integral” (BDCI), which is a convex combination of the CI and the RCI. To get the interactions of ordered coalitions comprehensively, this paper further proposes the generalized Shapley bi-direction Choquet integral (GSBDCI). Furthermore, the hybrid generalized Shapley bi-direction Choquet integral (HGSBDCI) is proposed, which defines the importance of ordered positions and the criteria with interactions simultaneously. With respect to these types of CIs, their exponent forms are also discussed. Finally, we use an application case to show the utilization of the proposed new CIs for MCDM. The proposed new Choquet integrals provide us a very useful way to deal with MCDM problems.
The current reprint presents studies of uncertain variational problems. It aims to develop the research in this field by formulating and demonstrating some characterization results of well-posedness ...and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations.
A new numerical method based on Haar wavelet is proposed for two-dimensional nonlinear Fredholm, Volterra and Volterra–Fredholm integral equations of first and second kind. The proposed method is an ...extension of the Haar wavelet method Aziz and Siraj-ul-Islam (2013), Siraj-ul-Islam et al. (2013) and Siraj-ul-Islam et al. (2014) from one-dimensional nonlinear integral equations (Fredholm and Volterra) to two-dimensional nonlinear integral equations (Fredholm, Volterra and Volterra–Fredholm). The main characteristic of the method is that, unlike several other methods, it does not involve numerical integration which results in an improved accuracy of the method. In order to show the effectiveness of the method, it is applied to several benchmark problems. The numerical results are compared with other methods existing in the recent literature.
The current reprint presents studies of uncertain variational problems. It aims to develop the research in this field by formulating and demonstrating some characterization results of well-posedness ...and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations.
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