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Abstract This paper investigates the use of fuzzy set theory in approximating solutions to differential equations under uncertainty. By defining lower and upper bounds, a robust framework is develope…

Abstract This paper investigates the use of fuzzy set theory in approximating solutions to differential equations under uncertainty. By defining lower and upper bounds, a robust framework is developed for modeling transitions between function extrema, extending classical methods to fuzzy initial value problems. The study demonstrates universal approximation properties and proposes numerical techniques that ensure stability and convergence. Applications highlight the practical utility of this approach in engineering and computational mathematics, solidifying fuzzy set theory as a powerful tool for addressing uncertainty in mathematical modeling.  

Abstract This paper introduces an enhanced framework for performing division operations on interactive fuzzy numbers characterized by complete correlation. Unlike traditional methods reliant on the i…

Abstract This paper introduces an enhanced framework for performing division operations on interactive fuzzy numbers characterized by complete correlation. Unlike traditional methods reliant on the independence assumption, we build on the sup-J extension framework to support correlated input fuzzy values. The proposed method establishes precise conditions under which the result aligns with, diverges from, or subsumes conventional divisions such as Zadeh’s and the generalized Hukuhara division. Additionally, we investigate invertibility conditions for the proposed division with respect to multiplication. These refinements offer valuable theoretical insights and have implications for models involving uncertainty, including difference equations.  

We observe an unknown function of d variables f(t), t ∈ [0, 1]^d, in the Gaussian white noise model of intensity ε > 0. We assume that the function f is regular and that it is a sum of…

We observe an unknown function of d variables f(t), t ∈ [0, 1]^d, in the Gaussian white noise model of intensity ε > 0. We assume that the function f is regular and that it is a sum of k-variate functions, where k varies from 1 to s (1 ≤ s ≤ d). These functions are unknown to us and only a few of them are nonzero. In this article, we address the problem of identifying the nonzero components of f in the case when d = d_ε → ∞ as ε → 0 and s is either fixed or s = s_ε → ∞, s = o(d) as ε → ∞. This may be viewed as a variable selection problem. We derive the conditions when exact variable selection in the model at hand is possible and provide a selection procedure that achieves this type of selection. The procedure is adaptive to a degree of model sparsity described by the sparsity parameter β ∈ (0, 1). We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.

Abstract   Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given univer…

Abstract   Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate  quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism  of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will  prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra).   

Abstract Background: Dyslipidemia and distorted fatty acid (FA) metabolism are frequent biochemical abnormalities associated with anorexia nervosa (AN). Gut microbiota is supposed to play an important…

Abstract Background: Dyslipidemia and distorted fatty acid (FA) metabolism are frequent biochemical abnormalities associated with anorexia nervosa (AN). Gut microbiota is supposed to play an important role in the etiopathogenesis of AN. Apart from the digestive function of bile acids (BAs), these compounds have multiple metabolic functions due to the activation of specific receptors. Objective/aims: The aims of the study were to investigate biochemical measures, including plasma lipids (lipoproteins, respectively), fatty acid (FA) patterns, and the profile of plasma Bas, in AN patients and healthy controls (CON). Methods: Plasma phospholipid FA and BAs profiles were analyzed in 39 women with a restrictive type of AN (AN-R; median age 17 years) and in 35 CON women (median age 20 years). Results: Compared to CON, AN had an increased concentration of HDL-C, increased content of palmitic acid, and decreased proportion of linoleic acid. Moreover, AN had a drop in the level of the sum of PUFAn-6 and increased delta 9 desaturase activity for stearic acid. In AN, we found decreased levels of plasma tauroursodeoxycholic acid (TUDCA). In AN, concentrations of 22:5n-6, 16:0, 20:3n-6 and fat mass index were predic-tors of HDL-C levels (R2 = 0.43). Conclusions: Patients with AN-R had an increased concentration of HDL-C, decreased levels of total PUFA n-6, and increased activity of D9D for stearic acid. Furthermore, AN exerted decreased levels of TUDCA. Therefore, a decreased level of TUDCA could potentially serve as a marker of AN.

Abstract Aristotle's Square also known as Square of Opposition, is a mathematical diagram dating back to Greek philosophy and exhibiting the connection between four logical propositions in a…

Abstract Aristotle's Square also known as Square of Opposition, is a mathematical diagram dating back to Greek philosophy and exhibiting the connection between four logical propositions in a simple graphical form. Fuzzy Relational Concept Analysis (FRCA) is a technique for extracting special clusters called fuzzy concepts from a Fuzzy Relational Context Family (FRCF), which is a dataset organized as multiple fuzzy object-attribute and object-object relations. The primary FRCA tools to obtain information from data are special fuzzy quantifiers viewed as interpretations in a model of formulas of the formal theory of the intermediate generalized quantifiers. This work focuses on the issue of generating a collection of fuzzy concepts from a certain FRCF, by choosing one of four particular FRCA quantifiers: the positive universal quantifier 𝒮1, the negative universal quantifier 𝒮−1, the positive existential quantifier 𝒮∃, and the negative existential quantifier 𝒮−∃. Certainly, the selection of the quantifier is crucial in the FRCA procedure since it affects the final concept classification: diverse fuzzy concepts arise from varying quantifiers. As the initial objective, this article introduces the logical relations involving 𝒮1,𝒮−1, 𝒮∃, and 𝒮−∃, in order to arrange them in a graded version of the Aristotelian square. The second goal of this study is to examine the connections among fuzzy concepts produced by distinct quantifiers in {𝒮1,𝒮−1,𝒮∃,𝒮−∃}. Therefore, our findings provide a twofold contribution to the advancement of Aristotle's square. Indeed, they reveal a novel interpretation of the square of opposition within the framework of Fuzzy Relational Concept Analysis, emphasizing its potential as a valuable tool for the analysis of data.

Computing maximum likelihood estimates in linear structural equation models is generally a difficult problem. The critical equations are usually non-linear and have numerous solutions, even for purely…

Computing maximum likelihood estimates in linear structural equation models is generally a difficult problem. The critical equations are usually non-linear and have numerous solutions, even for purely observational data. The block-coordinate descent (BCD) algorithm proposed by Drton et al. (2019)[1] is an efficient way to solve the optimization problem by decomposing it into a series of sub-problems with closed-form solutions, and which works with observational data. In this work, we describe the general problem of a BCD-type scheme for computing maximum likelihood estimates in linear structural equation models without hidden variables, integrating multiple observational and interventional environments. With interventional data, the degrees of both the original likelihood equations and the block-coordinate update equations could increase greatly. We study special setups in which the block optimization subproblems have a degree of at most 2 and provide closed-form solutions in these cases. Additionally, we discuss the potential applications of the model and algorithm to health and well-being data.

This paper examines the approximation capabilities of lattice integral transforms and their compositions in reconstructing lattice-valued functions. By introducing an integral kernel 𝑄 on the function…

This paper examines the approximation capabilities of lattice integral transforms and their compositions in reconstructing lattice-valued functions. By introducing an integral kernel 𝑄 on the function domain, we define the concept of a 𝑄-inverse integral kernel, which generalizes the traditional inverse kernel defined as a transposed integral kernel. Leveraging these 𝑄-inverses, we establish upper and lower bounds for a transformed version of the original function induced by the integral kernel 𝑄. The quality of approximation is analyzed using a lattice-based modulus of continuity, specifically designed for functions valued in complete residuated lattices. Additionally, under specific conditions, we demonstrate that the approximation quality for extensional functions with respect to the kernel 𝑄 can be estimated through the integral of the square of 𝑄, and in certain cases, these extensional functions can be perfectly reconstructed. The theoretical findings, illustrated through examples, provide a strong foundation for further theoretical advancement and practical applications.

A new concept, termed virtual neural networks, is introduced, where the count of trainable parameters is kept constant, and scalability is attained purely through computational resources. This concept…

A new concept, termed virtual neural networks, is introduced, where the count of trainable parameters is kept constant, and scalability is attained purely through computational resources. This concept is an abstract framework that can be realized using any standard convolutional neural network. It merges siamese neural networks with a deep ensemble technique by generating numerous virtual models that share weights derived from a small set of physical models. The ensemble comprises up to hundreds of trained models simultaneously. All virtual networks take the same input, and their interconnected structure induces an internal distortion that boosts the entire ensemble robustness. The accuracy of the ensemble improves as the number of virtual networks increases, without changing the capacity. Virtual neural networks outperform larger capacity models, typical deep ensembles, and contemporary approaches like SWA and Masksembles. Additionally, the highest performing individual model from the ensemble surpasses other models trained individually, even those with a greater number of parameters.