Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

A study on the cardinality of some families of discrete operators through alternating sign matrices

[eng] Determining the number of discrete operators has been a topic of interest for the scientific community since the introduction of these operators. This paper represents a further stage within this topic in the field of operators defined on a finite chain. Mainly, two families of discrete operators are studied: discrete conjunctions that are smooth (a property that is usually considered the equivalent to continuity in discrete settings) and commutative discrete conjunctions with n, the greatest element of the chain, as neutral element, so that only associativity is missing to become discrete t-norms. To determine the cardinality of the first family, we study its explicit representation by alternating sign matrices, obtaining that the cardinality is preserved between both structures and allowing us to relate intrinsic properties of the family of discrete operators with intrinsic properties of such class of matrices. For commutative discrete conjunctions with n as neutral element, we have considered the concepts of n-Gog y n-Magog triangles, allowing us to transform properties of these operators into properties of these triangular arrays; in particular, their cardinality. In this way, an upper bound for the cardinality of discrete t-norms is achieved

An analysis of the asymptotic behavior of the cardinality of some classes of logical connectives and aggregation functions defined on finite chains

[eng] The enumeration of certain classes of logical connectives and aggregation functions defined on a finite chain has garnered significant attention in recent years, particularly due to its usefulness in various applied domains such as image processing and decision-making. However, in some instances, the sheer magnitude of this enumeration makes its exact value less critical. Instead, the focus often shifts to understanding its asymptotic growth order. This perspective is valuable for anticipating and planning computational costs, and more importantly, for assessing the restrictiveness of properties imposed on a class of logical operators. Consequently, this paper delves into the asymptotic behavior of several expressions already proposed in the literature, mainly in the enumeration of discrete negations, discrete implications and discrete aggregation functions and some subclasses. Additionally, in this paper a measure is proposed to quantify the degree of restrictiveness associated with an additional property within a class of logical operators

A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

[eng] The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions

On a total order on the set of Z-numbers based on discretefuzzy numbers

[eng] Z -numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as ameasure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z -numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers isinvestigated for the first time. Specifically, the total order is designed for discrete Z -numbers where the second component has membership values belonging to a finite and prefixed setof values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z -numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then atotal order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z -numbers is presented and a comparison with other ranking methodsis performed from which the strengths of our method are stressed

The intricacies of dinoflagellate pellicle cysts: The example of Alexandrium minutum cysts from a bloom-recurrent area (Bay of Baiona,NW Spain)

The terms “temporary”, “pellicle,” and “ecdysal” cysts have been employed arbitrarily in the literature of the dinoflagellate life cycle to describe a non-motile and single-layered-wall stage with no mandatory dormancy period, of asexual or sexual origin. These three terms have been used more or less synonymously, but more specific definitions, taking into account morphological and physiological aspects and their roles in dinoflagellate population dynamics, are still needed. To clarify the current terminology, we examine and discuss the usages and foundations of those terms. The background for this discussion is provided by a comparison of the morphology and germination times of three different types of Alexandrium minutum cysts collected during a seasonal bloom in the Bay of Baiona (NW Spain). The double-walled cysts were similar to the resting cysts reported for this species, but other, thin-walled and thecate cysts were also observed. These latter cysts needed between 1 and 17 days to germinate and were therefore considered as short-term cysts, in contrast to the 1.5-month dormancy period of resting (hypnozygotic) cysts. Our results showed that the temporal distribution of these short-term cysts during the bloom period followed a pattern very similar to that of vegetative cells. However, resting cysts were only detected at the end of the bloom. In the context of our present knowledge regarding the dormancy and quiescence of dinoflagellate cysts, “temporary” is a very misleading and uncertain term and must be rejected. The term “ecdysal” has been used in reference to thin-walled cysts when ecdysis has been proved; however, ecdysis is not unique to this type of cysts as thick-walled zygotic cysts can be formed thorough ecdysis of a thecate planozygote. In conclusion, based on our current understanding of cysts, the term “pellicle” more appropriately describes single-layered-wall stages.Versión del editor2,277

Bloom dynamics and life cycle strategies of two toxic dinoflagellates in a coastal upwelling system (NW Iberian Peninsula)

A study of Gymnodinium catenatum and Alexandrium minutum blooms on the Galician coast was conducted from 2005 to 2007 in order to increase knowledge of the mechanisms governing recurrent blooms of these species. Considerable differences in their bloom dynamics were observed. G. catenatum blooms occurred in autumn and winter, following the pattern previously reported in the literature: they began offshore and were advected to the Galician rias when a relaxation of the coastal upwelling occurred. On the other hand, A. minutum blooms developed inside embayments in spring and summer during the upwelling season and were associated with water stability and stratification. Both the vegetative population and the cyst distribution of A. minutum were related to less saline water from freshwater river outputs, which supports a saline-gradient relationship postulated herein for this species. Dinoflagellates may produce both long-term double-walled cysts (resting) and short-term pellicle cysts. Resting cyst deposition and distribution in sediments showed that seeding occurred during the blooms of both species. However, the relationship between the cyst distribution in the sediments in Baiona Bay and the intensity and occurrence of G. catenatum blooms, suggests that the latter are not directly related to resting cyst germination. Moreover, the results presented in the present study point to other difference between the two species, such as the detection of pellicle cysts only for A. minutum. Finally we discuss how the life cycle strategies of these two species may help to explain the different mechanisms of bloom formation reported herein.Versión del editor2,277

Acyloxylation of Cyclic Enones: Synthesis of Densely Oxygenated Guaianolides

The α′-acyloxylation of cyclic enones with linear carboxylic acids is described. The reaction is promoted by KMnO4 in the presence of a carboxylic acid and its corresponding carboxylic anhydride. The optimization of the reaction has been carried out using the statistical methodology known as design of experiments. The optimized reaction conditions have been evaluated in terms of substrate scope and compatibility with different functional groups. The methodology has been applied to the synthesis of densely oxygenated guaianes and guaianolides

Allylic Oxidation of Alkenes Catalyzed by a Copper−Aluminum Mixed Oxide

A strategy for the allylic oxidation of cyclic alkenes with a copper−aluminum mixed oxide as catalyst is presented. The reaction involves the treatment of an alkene with a carboxylic acid employing tert-butyl hydroperoxide as the oxidant. In all cases, the corresponding allylic esters are obtained. When L-proline is employed, the allylic alcohol or ketone is obtained. The oxidation of cyclohexene and valencene has been optimized by design of experiments (DoE) statistical methodology