In this paper, we investigate qualitative integrals (generalizations of Sugeno integral) acting on recently introduced Dragonfly algebras. These algebras are designed for applications in data analysis (based on fuzzy relational compositions) when some data are unknown (e.g., missing). Unknown data are represented by the additional dummy value. Definitions of operations on Dragonfly algebras follow lower estimation strategy, i.e., results of operations can be interpreted as lower estimations of results obtained when unknown values are replaced by known ones. We define qualitative integrals on Dragonfly algebras and show their monotonicity. We prove results characterizing when these integrals return known (or unknown) values. Illustrative example and directions of further research are also presented.

SUMMARY: We present the LipidQuant 1.0 tool for automated data processing workflows in lipidomic quantitation based on lipid class separation coupled with high-resolution mass spectrometry. Lipid class separation workflows, such as hydrophilic interaction liquid chromatography or supercritical fluid chromatography, should be preferred in lipidomic quantitation due to the coionization of lipid class internal standards with analytes from the same class. The individual steps in the LipidQuant workflow are explained, including lipid identification, quantitation, isotopic correction and reporting results. We show the application of LipidQuant data processing to a small cohort of human serum samples. AVAILABILITY AND IMPLEMENTATION: The LipidQuant 1.0 is freely available at Zenodo https://doi.org/10.5281/zenodo.5151201 and https://holcapek.upce.cz/lipidquant.php. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Bioinformatics (2020) https://doi.org/10.1093/bioinformatics/btab644 In the originally published version of this manuscript, author Ondřej Peterka’s first name was repeated twice. The publisher apologizes for the error.

This paper continues and generalizes the line of research on ordinal sum of t-norms and t-conorms on bounded lattices. We introduce a new ordinal sum construction on bounded posets based on interior and closure operators. Our proposed method provides a simple tool to introduce new classes of t-norms and t-conorms. Several necessary and sufficient conditions are presented for ensuring whether our generalized ordinal sum on a bounded posets of arbitrary t-norms is, in fact, a t-norm. We show that in this general setting the existence of our ordinal sum for t-norms requires that the respective interior operators are t-norm preserving.

The aim of this paper is to introduce two types of integral transforms that naturally generalize the lower and upper fuzzy transforms for the residuated lattice valued functions. For the construction of such integral transforms we consider a multiplication based fuzzy (qualitative) integral and an integral kernel in the form of a special binary fuzzy relation. We analyze the basic properties of the proposed integral transforms.

Qualitative or equivalently fuzzy integrals are used as qualitative aggregation functions or as L-fuzzy quantifiers. In both cases they are generalisations of Sugeno integrals. The definitions of these fuzzy integrals are quite similar and coincide in particular cases, but surprisingly there is no deeper analysis of their relationship. The paper attempts to fill this gap and provides unified definitions of fuzzy quantifiers on the basis of which various links between these fuzzy integrals are studied. In order to make these links more visible and to emphasise their logical structure, we present them using the graded square and modern square of opposition.

The article stems from distinct studies on arithmetics of fuzzy numbers, orderings of fuzzy numbers, and metrics on fuzzy numbers. Trying to capture the existing knowledge in the mentioned areas and putting them together, we motivate the construction of metric-like spaces on fuzzy numbers by desirable connection to their arithmetics. The desirable "metric" should be mapping pairs of fuzzy numbers again to fuzzy numbers and thus, reflecting the vagueness of operation on fuzzy numbers. This leads to developing all such areas under the joint umbrella and connecting such basic notions as orderings of fuzzy numbers to arithmetics and elementary "metrics" such as the absolute value of the difference of two fuzzy numbers. This article focuses mainly on the orderings and investigation of the preservation of their most natural properties. However, links to further studies going towards applications are also foreshadowed and referred to.

The notion of the metric space that allows to measure a distance between objects of the given space, has a crucial importance for distinct parts of mathematics, for instance, for the approximation theory, interpolation methods, data analysis, optimization etc. In fuzzy mathematics, the same areas of applications have an analogous importance and thus, not surprisingly measuring the distance between objects possesses a desirable importance. In many cases, e.g., in fuzzy clustering, the use of the standard metric spaces is absolutely sufficient. However, if we deal with vague quantities represented by fuzzy numbers, though the application of a standard metric to fuzzy numbers is mathematically correct, it may lead to counterintuitive and undesirable results. Our investigation constructs the "metric-like" spaces enabling to measure the distance between two fuzzy numbers in a way that is not disconnected from the used arithmetic of fuzzy numbers. Following the analogy from the classical math where the most natural distance between two numbers is the absolute value of their difference, in the case of fuzzy numbers and under the assumption that the distance is connected to the arithmetic, the most natural distance of two fuzzy numbers is the absolute values of their difference too. But then, naturally, the distance should map fuzzy numbers again to fuzzy numbers, not to crisp numbers. This article is a contribution to this area that guides readers from the fundamental notions to the final construction supported by some theoretical results.

In this contribution, we introduce gradual numbers and intervals of gradual numbers to be able to prove the strong law of large numbers for random variables that are valued by such elements. The results show a novel point of view in the investigation of strong law of large numbers for random variables that are valued by non-exactly specified numbers.

This paper introduces a novel construction of test spaces in the Ritz-Galerkin method. These spaces are established on the basis of linear spaces of functions that are linear combinations of polynomials and fuzzy sets of a generalized uniform fuzzy partition.

This article follows the arithmetics of extensional fuzzy numbers that are based on the transitive closure of the used similarity relations and that form particular MI-prefields. The direction of the contribution goes towards defining the ordering of extensional fuzzy numbers in order to construct further notions from the mathematical analysis in the environment of extensional fuzzy numbers, especially the metric spaces.

Fuzzy relational compositions based on fuzzy quantifiers naturally do not preserve all the properties that are preserved for “standard” fuzzy relational compositions and, in many cases, the property is preserved only in a weaker form. For example, the associativity, that is preserved in the standard case derived from the universal and the existential quantifiers, generally does not hold for the case of compositions based on fuzzy quantifiers. However, is it the case that only the standard quantifiers lead to the preservation of such properties? Without any restriction on the shape of the fuzzy relations, the answer is positive.

In this paper, we propose a numerical method based on the F-transform for solving a certain type of partial differential equations (PDEs) with Dirichlet boundary conditions and initial conditions. We show how the PDEs after the application of the Crank-Nicolson scheme for time discretization can be approximated by a system of linear equations with the direct F-transform components as their variables. The numerical solution of PDEs is obtained by solving the system of linear equations and the application of the inverse F-transform. The proposed method is then adjusted to the two-dimensional boundary value problem illustrated on two examples including the Black-Scholes equation well-known in financial modeling.

In this paper, we reformulate the multivariate fuzzy transform (Fm-transform) of higher degree proposed originally for an approximation of continuous functions to the discrete case. We introduce the Fm-transform components as multivariate polynomials using the matrix algebra, the inverse Fm-transform is established to obtain a continuous function. We provide an analysis of basic properties of Fm-transform.

In this paper, we reformulate the fuzzy transform of higher degree (Fm-transform) proposed originally for an approximation of continuous functions to the discrete case. We introduce two types of Fm-transform which components are defined using polynomials in the first case and using specific values of these polynomials in the second case. We provide an analysis of basic properties of Fm-transform.

In this paper, we will focus on the application of fuzzy transform (F-transform) in the analysis of time series. We assume that the time series is decomposed into two constituents: the trend-cycle and random noise. We will demonstrate that using the F-transform we can reduce the variability of random noise which consequence is an extraction of the trend-cycle.

Up to our best knowledge, distinct so far existing arithmetics of fuzzy numbers, usually stemming from the Zadeh's extensional principle, do not preserve some of the important properties of the standard arithmetics of classical (real) numbers. Obviously, although we cannot expect that a generalization of standard arithmetic will preserve precisely all its properties however, at least the most important ones should be preserved. We present a novel framework of arithmetics of extensional fuzzy numbers that preserves more or less all the important (algebraic) properties of the arithmetic of real numbers and thus, seems to be an important seed for further investigations on this topic. The suggested approach arithmetics of extensional fuzzy numbers is demonstrated on many examples and besides the algebraic properties, it is also shown that it carries some desirable practical properties.

In the first part of this contribution, we proposed extensional fuzzy numbers and a working arithmetic for them that may be abstracted to so-called many identities algebras (MI-algebras, for short). In this second part, we show that the proposed MI-algebras give a framework not only for the arithmetic of extensional fuzzy numbers, but also for other arithmetics of fuzzy numbers and even more general sets of real vectors used in mathematical morphology. This entitles us to develop a theory of MI-algebras to study general properties of structures for which the standard algebras are not appropriate. Some of the basic concepts and properties are presented here.

In the classical set theory, there is the well known correspondence between the equipollence of finite sets and the equality of natural numbers expressing their cardinalities. In the fuzzy set theory, an equivalent correspondence between the equipollence of finite fuzzy sets and the equality of generalized cardinals has been proposed in [1]. This paper is devoted to the situation when we consider the mentioned relations to be graded, that means, the crisp equipollence of fuzzy sets is replaced now by the graded equipollence of fuzzy sets and the equality of fuzzy sets by their similarity.

In this contribution, we study a very important semantic property of generalized quantifiers called the monotonicity for fuzzy quantifiers of type 〈1,1〉 defined using fuzzy measures and Sugeno type of fuzzy integrals. We show that fuzzy integrals can ensure under some natural conditions the monotonicity of fuzzy quantifiers. Finally, we propose the concept of concave fuzzy quantifiers and prove that each concave fuzzy quantifier is expressible as the conjunction of a non-increasing and a non-decreasing fuzzy quantifier.

In this paper we present a new definition of fuzzy quantifiers of type 〈1, 1〉 determined by fuzzy measures and fuzzy integrals. We show basic semantic properties, namely, permutation and isomorphism invariance, extension and conservativity, of these fuzzy quantifiers.