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Copulas are functions that link an n-dimensional distribution function with its one-dimensional margins. In this contribution we show how n-variate copulas with given values at two arbitrary points can be constructed. Thereby, we also answer a so far open question whether lower and upper bounds for n-variate copulas with given value at a single arbitrary point are achieved. We also introduce and discuss the concept of an $$\mathbf{F}$$ F -copula which is needed for proving our results.

Several extensions of the family of (bivariate) Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas) are considered. Some of them are well-known from the literature, others have recently been suggested (copulas based on quadratic constructions, based on some forms of convexity, and polynomial copulas). For each of these extensions we analyze which properties of EFGM copulas are preserved (or even improved) and which are (partly) lost. Such properties can be structural (order theoretical or topological) in nature, or algebraic (symmetry or being a polynomial) or analytic (absolute continuity). Other examples are forms of convexity, quadrant dependence, and symmetry with respect to copula transformations. The last group of properties considered here is related to some dependence parameters.

Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a,b,c), i.e., to some set of polynomials in two variables of degree 1:p(x,y) =ax+by+c. The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1, 1[ if polynomial copulas of arbitrary degree are considered.

Starting with [Goguen, J.A. 1967. Journal of Mathematical Analysis and Applications], several generalizations of the original definition of a fuzzy set have been proposed. In one popular case, one ...

We start with some binary (“outer”) copula, apply it to an arbitrary binary (“inner”) copula and its dual (the latter being transformed by some real function) and ask under which conditions the result is again a binary copula. Sufficient convexity conditions for the transformation function and for the “outer” copula (ultramodularity and Schur concavity) are given, thus generalizing the scenarios considered in Klement et al. (J Math Inequal 11(2):361–381, 2017) and Manstavicius and Bagdonas (Fuzzy Sets Syst 354:48–62, 2019). In general, these sufficient conditions are not necessary (and a counterexample is provided), but in some distinguished cases necessary and sufficient conditions can be given. Several well-known families of copulas can be obtained in this way. We also present a few extensions for special “outer”/“inner” copulas and/or transformation functions, as well as some counterexamples.

Picture fuzzy sets, recently introduced by B. C. Cuong and V. Kreinovich, are a special case of L-fuzzy sets. We discuss the set of truth values for these fuzzy sets as well as aggregation functions for these truth values, paying special attention to t-norms and t-conorms. The important role of representable t-norms and t-conorms is emphasized.

In this paper we study the smallest and the greatest M-Lipschitz continuous n-ary aggregation functions with a given diagonal section. We show that several properties that were studied for the smallest and the greatest 1-Lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while considering different Lipschitz constants. Just as in the binary case, we show that the smallest n-quasi-copula with a given diagonal section coincides with the smallest 1-Lipschitz n-ary aggregation function with that diagonal section. Additionally, we show that the smallest n-quasi-copula with a given diagonal section, called the Bertino n-quasi-copula, is supermodular for any n ⩾ 2 .

We study a generalized Frank functional equation in the broader framework of associative aggregation functions and show that, up to the two projections, we obtain exactly the same set of solutions as in the original paper by M. J. Frank (Aequationes Mathematicae 19, 1979).

Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being a fixed point of the transformation under consideration. Finally, an application to the construction of so-called imprecise copulas is given.

Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively.

This paper presents a novel approach for detecting cracked or broken reciprocating compressor valves under varying load conditions. The main idea is that the time frequency representation of vibration measurement data will show typical patterns depending on the fault state. The problem is to detect these patterns reliably. For the detection task, we make a detour via the two dimensional autocorrelation. The autocorrelation emphasizes the patterns and reduces noise effects. This makes it easier to define appropriate features. After feature extraction, classification is done using logistic regression and support vector machines. The method׳s performance is validated by analyzing real world measurement data. The results will show a very high detection accuracy while keeping the false alarm rates at a very low level for different compressor loads, thus achieving a load-independent method. The proposed approach is, to our best knowledge, the first automated method for reciprocating compressor valve fault detection that can handle varying load conditions.

Generalizing a first approach by L. A. ZADEH (J. Math. Anal. Appl. 23, 1968), expected values of fuzzy events are studied which are (up to standard boundary conditions) only required to be monotone. They can be seen as an extension of capacities, i.e., monotone set functions satisfying standard boundary conditions. Some of these expected values can be characterized axiomatically, others are based on some distinguished integrals (Choquet, Sugeno, Shilkret, universal, and decomposition integral).

In classical measure theory there are a number of convergence theorems, such as the Egorov, the Riesz and the Lusin theorem, among others. We consider monotone measures (i.e., monotone set functions vanishing in the empty set and defined on a measurable space) and discuss, how and to which extent classical convergence theorems can be carried over to this more general case.

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.

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy set. Finally we also discuss some questionable notations, in particular, those of &ldquo, intuitionistic&rdquo, and &ldquo, Pythagorean&rdquo, fuzzy sets.

Picture fuzzy sets can be seen as particular "neutrosophic sets", without forming a sublattice. On the other hand, they can be isomorphically related to 3-fuzzy sets, and then they form a bounded lattice. We discuss some possible algebraic looks on picture fuzzy sets and related aggregation functions, including several examples.

In the last two decades, multi-objective evolutionary algorithms (MOEAs) have become ever more used in scientific and industrial decision support and decision making contexts the require an a posteriori articulation of preference. The present work is focused on a comparative analysis of the performance of two master-slave parallelization (MSP) methods, the canonical generational scheme and the steady-state asynchronous scheme. Both can be used to improve the convergence speed of multi-objective evolutionary algorithms that must use computationally-intensive fitness evaluation functions. Both previous and present experiments show that a correct choice for one or the other parallelization method can lead to substantial improvements with regard to the overall duration of the optimization process. Our main aim is to provide practitioners of MOEAs with a simple but effective method of deciding which MSP option is better given the particularities of the concrete optimization process. This in turn, would give the decision maker more time for articulating preferences (i.e., more flexibility). Our analysis is performed based on 15 well-known MOOP benchmark problems and two simulation-based industrial optimization processes from the field of electrical drive design. For the first industrial MOOP, when comparing with a preliminary study, applying the steady-state asynchronous MSP enables us to achieve an overall speedup (in terms of total wall-clock computation time) of ? 25 % . For the second industrial MOOP, applying the steady-state MSP produces an improvement of ? 12 % . We focus our study on two of the best known and most widely used MOEAs: the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and the Strength Pareto Evolutionary Algorithm (SPEA2).