For a given incomplete context, object-induced approximate concepts have been defined, and this type of approximate concept can induce a type of decision rule. Based on the duality principle, another set of approximate concepts can be defined from the perspective of attributes, i.e., attribute-induced approximate concepts. Although object induced approximate concepts and attribute induced approximate concepts are symmetrical by duality principle, their induced decision rules exhibit different properties and the connections between attribute induced decision rules and object induced decision rules in incomplete formal contexts are not clear. To this end, a type of attribute-induced approximate concept and a method of extracting attribute-induced decision rules are presented. More importantly, it is revealed that given a type of decision rules, there must be corresponding decision rules of the other type, and both of them can provide some useful information, but they are not equivalent to each other. In other words, each type of decision rule can provide some unique and irreplaceable information.

Granule description is a fundamental problem in granular computing. However, how to describe indefinable granules is still an open, interesting, and important problem. The main objective of this paper is to give a preliminary solution to this problem. Before proceeding, the framework of approximate description is introduced. That is, any indefinable granule is characterized by an ordered pair of formulas, which form an interval set, where the first formula is the $\beta$-prior approximate optimal description and the second formula is the $\alpha$-prior approximate optimal description. More concretely, given an indefinable granule, by exploring the description of its lower approximate granule, its $\beta$-prior approximate optimal description is obtained. Likewise, by consulting the description of its upper approximate granule, its $\alpha$-prior approximate optimal description can also be derived. Following this idea, the descriptions of indefinable granules are investigated. Firstly, ∧-approximate descriptions of indefinable granules are investigated based on the classical concept lattice, and $(\wedge ,\vee )$-approximate descriptions of indefinable granules are given via object pictorial diagrams. And then, it is revealed from some examples that the classical concept lattice is no longer effective and negative attributes must be taken into consideration. Therefore, a three-way concept lattice is adopted instead of the classical concept lattice to study $(\wedge ,\neg )$-approximate descriptions and $(\wedge ,\vee ,\neg )$-approximate descriptions of indefinable granules. Finally, some discussions are presented to show the differences and similarities between our study and existing ones.

Planar underactuated manipulators are widely used in aerospace and deep-sea exploration. Due to their passive joints without actuation, it is difficult to perform trajectory planning and direct control. In this paper, an angular velocity trajectory planning method is proposed, aiming to calculate the velocity trajectory by using a superposition of basis functions and transforming the planning problem into an optimization problem by using a metaheuristic algorithm. According to the underactuated constraint relationship, the planning trajectory of the passive joint is obtained, which ensures that the passive joint can be indirectly controlled to reach the target angle while tracking the trajectory of the active joint. For the tracking control problem, a sliding mode controller with an exponential convergence rate is used to track the trajectory. Lastly, three sets of simulations are run to demonstrate the efficacy of the suggested approach.

We introduce the notion of asymptotically weighted statistical equivalence of order α in probability for sequences of positive linear operators and study it within a Korovkin‐type approximation framework. The emphasis is not only on the convergence of a single operator sequence to a target function, but on the asymptotic comparison of two approximation processes in a weighted probabilistic setting. After establishing the basic structural properties of the proposed equivalence relation and clarifying its connection with weighted statistical convergence in probability and with the classical statistical setting, we prove a Korovkin‐type comparison theorem for normalized reference schemes. More precisely, we show that if a normalized sequence of positive linear operators has asymptotically small second moments and if two operator sequences are asymptotically weighted statistically equivalent in probability on the Test functions 1, x , and x ² , then this equivalence extends to all functions in C [0, 1]. We also derive quantitative estimates for the deviation between the two approximation processes by means of the modulus of continuity and for functions in the Lipschitz class. The theory is illustrated by examples based on Bernstein operators, including a comparison with genuinely distinct positive linear schemes, showing that the present framework applies beyond the classical uniform approximation setting.

In this work, we introduce new classes of lacunary sequence spaces based on Nörlund-type means, a sequence of modulus functions and the order parameter $0<\alpha \le 1$. The main novelty is the incorporation of these components into a weighted Nörlund-lacunary framework. We establish inclusion and equivalence relations with classical Nörlund-type and lacunary sequence spaces under explicit conditions on the weight and lacunary sequences. We further characterize weighted lacunary statistical convergence of order $\alpha$ and investigate its relationships with the corresponding convergence spaces. Finally, we prove the completeness and topological properties of the resulting paranormed sequence spaces. These results extend existing frameworks for lacunary and Nörlund-type sequence spaces.