In fuzzy rough sets, the traditional approach primarily employs fuzzy rough approximation operators and fuzzy neighborhood operators to solve attribute reduction problems. However, measures constructed by these operators fail to capture the correlation between attributes, whereas the generalized Shapley value(SHV), as a non-additive measure, takes into account the correlation between attributes from a global perspective. Based on the global attribute correlation problems, SHV for fuzzy neighborhoods utilizing pseudo-overlap functions are proposed to deal with the problem of attribute reduction. Firstly, in the fuzzy β covering approximation spaces( β -FCASs), based on pseudo-overlap functions and their corresponding residual implications(( I _PO, PO )), ( I _PO, PO )-fuzzy β neighborhood operators ((I _PO,PO)−β−FNoperators) and four pairs of ( I _PO, PO ) fuzzy β neighborhood measures(( I _PO, PO )-fuzzy β−NMs) based on these operators are constructed, thereby extending the representational capacity of fuzzy covering-based rough sets. Secondly, four pairs of SHVs utilize on ( I _PO, PO )-fuzzy β−NMs are introduced to evaluate attribute significance from a global perspective. And a new method of attribute reduction of fuzzy β covering information decision tables( β -FCIDTs) based on SHV is proposed. Thirdly, four pairs of Choquet integrals(CHIs) based on SHV are constructed. On this basis, a method for addressing the attribute reduction of β -FCIDT is proposed by considering the correlation of global attributes, and the proposed method is used to deal with the classification problem specifically. Finally, the validity and practicality of the proposed methods are confirmed through the use of several public datasets.

Traditional fuzzy and rough set models usually have a hard time in providing sufficient ambiguity, hesitancy, and partiality to information that is common in actual hospitality decision-making. In order to overcome these problems, this paper suggests a new decision-support model that is built upon pessimistic multi-granulation rough sets combined with cubic intuitionistic fuzzy soft relation. The model proposed is a unification of interval-valued membership, intuitionistic non-membership and soft binary relations in a structure of pessimistic rough approximations. The framework characterizes multi-dimensional uncertainty of complex hospitality appraisals by modelling membership, non-membership, hesitation and boundary regions together in a variety of granulations. Soft binary relations defined in terms of foresets and aftersets are used to form lower and upper pessimistic multi-granulation approximations of internal cubic intuitionistic fuzzy sets. Approximation operators are formally defined in two pairs and the algebraic property of the operators is explored. In addition, measures of similarity among internal cubic intuitionistic fuzzy sets on soft relational structures are created to facilitate the robust alternative comparison. On the grounds of these theoretical bases, a systematic decision-making scheme that is composed of two structured algorithms is set up. The relevance of the suggested framework can be proved by a practical hotel selection case study with the usage of several factors and professional evaluations. Comparative and sensitivity analysis demonstrates that the model proposed gives consistent rankings and minimizes the effects of boundary ambiguity in different levels of uncertainty, which can be used as a trusted and risk-conscious decision support device in hospitality management.

The ongoing electrification of the transport sector, driven by the numerous advantages of electric vehicles (EVs), introduces new challenges related to charging logistics, particularly due to long charging durations and uncertain conditions, posing significant negative impacts on grid stability and user satisfaction. While existing literature on EV charging scheduling often assumes deterministic charging durations, real-world conditions introduce randomness due to uncontrollable factors such as battery state-of-charge (SoC), fluctuating grid demand, and ambient temperature. In this paper, we address the Electric Vehicle Charging Scheduling Problem (EVCSP) under uncertain charging durations. First, we introduce a novel, flexible multi-objective scheduling model operating on a continuous time horizon, considering stochastic charging durations and incorporating controlled preemptions during charging, where the non-preemptive mode is a particular case. Then, we prove that finding a feasible assignment of EVs to chargers is strongly NP-hard under this uncertainty, even assuming identical chargers. Our model accounts for realistic constraints, including heterogeneous charger power levels and vehicle-charger compatibility, aiming to minimize the conditional expected values of grid overload and total tardiness, while also minimizing the undelivered energy to users. Given the problem’s computational complexity, we adapt four evolutionary algorithms (EAs), namely, extensions of the Non-Dominated Sorting Genetic Algorithm (NSGA), namely NSGA-II and NSGA-III, alongside other state-of-the-art multi-objective metaheuristics, including the Multi-Objective Cuckoo Search (MOCS) algorithm, and the Multi-Objective Grey Wolf Optimizer (MOGWO) by defining problem-specific operators to explore the search space and efficiently approximate the optimal Pareto front. Assuming lognormally distributed charging durations, we conducted a comparative experimental analysis on real-world data to evaluate the four methods and revealed that MOCS algorithm outperforms the other competitors. • Introduce the first multi-objective EV charging scheduling model under stochastic charging durations. • Prove that EV-to-charger assignment under uncertain charging durations is strongly NP-hard. • Derive explicit formulas for conditional expected total tardiness under multiple probability distributions. • Adapt four evolutionary algorithms with problem-specific operators for Pareto front approximation. • Demonstrate Multi-Objective Cuckoo Search (MOCS) algorithm superiority through comprehensive experiments on real-world ACN dataset.

Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as D ∝ 1 / ρ at low densities ρ . Our numerical approach generalizes the Dissipation-Assisted Operator Evolution algorithm: in the spirit of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we effectively approximate nonlocal operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low-temperature transport.

This paper proposes a quasi-dynamic Volt–Var control strategy for radial distribution networks based on the optimal sizing of a distribution static synchronous compensator (D-STATCOM) using a genetic algorithm (GA). The objective is to enhance voltage regulation and reduce technical energy losses under variable loading conditions while preserving nonlinear AC power flow fidelity. The IEEE 33-bus test system was modeled in DIgSILENT PowerFactory (v2021), and the D-STATCOM installation bus was selected based on a rigorous literature-supported placement criterion derived from optimization-based studies. Three representative demand scenarios—minimum, average, and maximum loading—were defined to approximate quasi-dynamic operation over a daily cycle. The GA was implemented in MATLAB (R2023b) to solve a normalized nonlinear multi-objective optimization problem that simultaneously minimizes total active power losses and the aggregate voltage deviation index. The optimized reactive power capacities obtained were 0.49 Mvar, 1.1933 Mvar, and 2.30 Mvar for the minimum, average, and maximum demand scenarios, respectively. These configurations achieved active power loss reductions of 27.5%, 24.602%, and 23.44% under the corresponding loading levels while improving voltage regulation at the critical bus (bus 18) and maintaining system voltages within the admissible 0.95–1.05 p.u. range. Through quasi-dynamic interpolation of operating points, the daily performance assessment showed a 24.11% reduction in total energy losses and a 38.28% decrease in the average voltage deviation. A statistical robustness analysis confirmed stable convergence behavior across independent executions. The results demonstrate that the proposed framework provides a computationally efficient, planning-oriented approach for reactive power compensation in distribution systems subject to demand variability.

We consider properties of L-fuzzy relations on extended residuated lattice L and show a characterization theorem of $$*$$ -confluent L-fuzzy relation, which includes many cases of L-fuzzy relations such as, reflexive, symmetric, transitive, Euclidean and so on, by using only the notion of Galois connection of operators defined by L-fuzzy relation. Since our method is based on operator-base, it enables us to provide simpler and shorter proofs to the results obtained so far and to be able to get new results about properties of L-fuzzy relations of other algebras L.

Conventional fuzzy or rough set models struggle to capture the ambiguity, hesitancy, and partial information that are frequently present in healthcare decision-making. To address this challenge, we propose a novel rough cubic intuitionistic fuzzy soft relational framework that combines interval-valued membership, intuitionistic non-membership, and soft binary relations within a harmonious rough set paradigm. The model captures multi-dimensional uncertainty by representing membership, non-membership, hesitation, and boundary approximations simultaneously. To this end, we use soft binary relations, defined by foresets and aftersets, to approximate an internal cubic intuitionistic fuzzy set. Initially, two pairs of rough approximation operators for an internal cubic intuitionistic fuzzy set, concerning foresets and aftersets, and their different algebraic features are examined. Furthermore, a variety of similarity relations between internal cubic intuitionistic fuzzy sets related to soft binary relations are discussed. We develop a decision-making scheme utilizing two algorithms with methodical procedural steps to demonstrate the effectiveness of the proposed framework. A comprehensive real-world breast cancer risk identification and hospital selection case study demonstrates the model’s effectiveness, achieving higher diagnostic reliability than classical fuzzy, intuitionistic fuzzy, and rough set approaches. A detailed comparative analysis with specific prevailing techniques reveals that the presented strategy achieves higher decision accuracy and reduced boundary ambiguity across multiple evaluation metrics.

Abstract Synthetic aperture radar (SAR) imaging is crucial for radar cross section (RCS) measurement. In near-field RCS measurement with growing demand for rapid, convenient testing, it faces challenges of large data volume and high computational complexity. Compressed sensing is adopted to reduce sampling rates below the Nyquist limit, alleviate data burdens, and improve efficiency and imaging performance. This paper proposes a near-field sparse SAR imaging method based on multi-constraint optimization, integrating high-order total variation (HOTV) regularization, exponential regularization, hybrid sparsity constraints, and a time–frequency domain constraint. HOTV introduces a second-order gradient norm to preserve details, suppress noise, and enhance target features, outperforming first-order models. Exponential regularization adaptively adjusts gradient penalty strength, balancing detail preservation and noise suppression. The time–frequency domain constraint further improves imaging quality, while an approximate observation operator reduces computational cost significantly. The optimization problem is solved via alternating direction method of multiplier. Simulation and real-data experiments validate the algorithm’s superior image quality and effective noise suppression, making it suitable for near-field RCS measurement.

On the equivalence classes induced by L-fuzzy rough approximation operators

ABSTRACT Although parallel prefix adders (PPAs) optimize data path through prefix operators, Ling adders show exceptional computing performance. Compared to traditional adders, however, Ling adders require additional components for generating Ling carries, so they are more complex. This paper proposes an architecture for approximate Ling adders based on parallel prefix topologies, referred to as approximate parallel prefix Ling adders (AxPPLAs). It is realized by using approximate prefix operators (AxPOs) to simplify the exact function of Ling carries for some less significant bits. The more significant bits are accurately computed. Various parallel prefix topologies and approximate processing bits are implemented to balance hardware costs and computing accuracy. The proposed AxPPLAs are compared with state‐of‐the‐art designs in terms of hardware cost, accuracy, and other figures of merit. Experimental results show that the proposed 16‐bit AxPPLAs improve hardware efficiency by significantly reducing delay and energy by up to 38.51% and 19.46%, on average, respectively, while maintaining competitive accuracy.

The Hartree-Fock exchange potential is fundamental for capturing quantum mechanical exchange effects but faces critical challenges in large-scale applications due to its nonlocal and computationally intensive nature. This study introduces a generalized framework for constructing approximate Fock exchange operators in Hartree-Fock theory, addressing the computational bottlenecks caused by the nonlocal nature. By employing low-rank decomposition and incorporating adjustable variables, the proposed method ensures high accuracy for occupied orbitals while maintaining Hermiticity and structural consistency with the exact Fock exchange operator. This low-rank approximation constitutes the core contribution of the presented study. Meanwhile, a two-level nested self-consistent field iteration strategy is developed to decouple the exchange operator stabilization (outer loop) and electron density refinement (inner loop), significantly reducing overall computational costs. Numerical experiments on several molecules demonstrate that the approximate exchange operators achieve near-identical energies compared to that of the exact exchange operator and the NWChem references with substantial improvements in computational efficiency.

This paper develops a structural and functional framework for operator algebras acting on nonseparable Banach spaces (NSBS). While classical operator algebras—such as 𝐶∗ - and 𝑊∗ -algebras—are traditionally constructed on separable Hilbert spaces, many physical and mathematical contexts require non-separable or even transfinite structures: quantum field theories with infinitely many degrees of freedom, infinite tensor product systems, and algebras associated with non-measurable state spaces. We extend the classical operator-algebraic formalism to NSBS by introducing approximate operator algebras, defined through directed nets of weakly compact projections and local separable subspaces. This approach restores the analytic machinery of functional calculus, spectra, and representations, while preserving topological and dual properties within locally separable components. The paper establishes several new results concerning approximate ideals, bicommutants, spectral continuity, and weak operator topologies in NSBS. Furthermore, we analyse the correspondence between approximate representations of 𝐶∗ -algebras on NSBS and physical observables in quantum mechanics and field theory. From a physical perspective, the proposed framework provides a rigorous mathematical description of systems with non-countable degrees of freedom, extending von Neumann’s theory of operator algebras beyond separability. Applications include the representation of infinite spin systems, algebras of observables in non-separable Hilbert–Banach settings, and generalised state spaces in quantum statistical mechanics.

This paper introduces a new hybrid operator based on combining the Phillips concept with a sequence of lambda-Bernstein operators. This operator represents a qualitative improvement over classical Bernstein-Durrmeyer operators, which faced significant limitations in controlling the behavior of functions at critical points such as the zero point and suffered from a significantly slow rate of convergence. The developed operator overcomes these challenges, achieving a substantial improvement in the quality and accuracy of convergence. To demonstrate the effectiveness of this operator, the study proves a set of basic theoretical results. First, the paper proves the regular convergence theorem for the operator. This is followed by establishing the error estimation theorem using a continuum measure, which in turn confirms the achievement of first-order convergence. Finally, the study presents a precise Voronovskaya-type asymptotic formula that reveals the detailed behavior of the operator's approximation rate when studying functions regularly.

Context. The problem of approximating the values of continuous functions of two variables based on known information about them on stripes, the boundaries of which are parallel to the coordinate axes, is considered. The object of the study is the process of approximating the values of functions based on incomplete information about them, which is given on the system of stripes.Objective. The goal of the work is the review of information operators of Lagrangian interstripation and features of the construction of information approximation operators for some cases of the mutual arrangement of stripes in some region, which allow to significantly simplify the calculation of approximate values of the function in unknown subregions of the region.Method. Methods for approximating the values of continuous functions of two variables with incomplete information about them on some limited area are proposed. Information about the function is known only on a system of stripes limited by straight lines parallel to the coordinate axes. A method for approximating the values of continuous functions of two variables, information about which is known on two stripes, as a result of union of which only some rectangular subregion remains unknown in the region, is proposed. A method for approximating the values of continuous functions of two variables, information about which is known on three stripes, as a result of union of which only some rectangular subregion remains unknown in the region, is proposed. A method for approximating the values of continuous functions of two variables, information about which is known on four stripes, as a result of union of which only some rectangular subregion remains unknown in the region, is proposed. A method for approximating the values of continuous functions of two variables, the information about which is known on two stripes, as a result of union of which four rectangular subregions remain unknown in the region, is proposed. For all the considered cases, approximation operators are given that allow calculating the approximate form of the function in the unknown subregions in the analytical form.Results. The information operators of Lagrangian interstripation are implemented programmatically and investigated in problems of approximating the values of functions of two variables from known information about them on the systems of stripes.Conclusions. The experiments confirmed the accuracy of approximation of the values of continuous functions of two variablesof the proposed information interstripation operators for different systems of stripes. Approximation operators are given for special cases of the location of stripes in the region, the difference of which from the information interstripation operators of the general form lies in the significant simplification of the approximation operators without losing the accuracy of the approximation with a smaller number of arithmetic operations, which can be a decisive factor in some cases. Prospects for further research lie in the application of the proposed information operators in the problems of digital image processing, seismic mineral exploration data and remote sensing data etc.

This paper develops a unified analytical framework for non-separable Banach spaces (NSBS) grounded in the concepts of approximate Schauder bases, weakly compact approximation operators, and local–global interpolation structures. The traditional limitations associated with the absence of separability and sequential compactness are overcome by replacing sequences with directed nets and by localising all analytical arguments within separable hulls. Within this setting, the notions of approximate interpolation couples and approximate real and complex interpolation spaces are introduced and analysed in detail. We establish that boundedness, compactness, duality, and stability properties extend naturally from the classical separable case, providing new generalisations of the Lions–Peetre and Riesz–Thorin theorems. Spectral theory is reformulated for bounded linear operators in NSBS, proving spectral stability under approximate compactness and continuity of spectra under analytic perturbations. Applications to operator theory and mathematical physics are discussed, including spectral–interpolation correspondences, weak compactness criteria, and models for quantum and statistical systems where non-separable structures arise intrinsically. These results yield a coherent generalisation of fundamental principles of functional analysis and open new avenues for research in operator algebras, evolution equations, and non-separable

Exploration of rough approximation operators with supervised justifiable granularity principle

One axiom characterization for (L, M)-fuzzy rough approximation operators

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as Hn(g;u)=∑ƿ=0∞hƿ(n+u;ƛ)∫01Kn(ƿ, ʈ)g(ʈ)dʈ, where hƿ(nu;ƛ) is derived from degenerate Appell polynomials (nu denotes the product of n and u) and Kn(ƿ, ʈ) is a Beta-type kernel. We establish the linearity and positivity of these operators and derive crucial moment estimates. Approximation properties are examined via Korovkin-type theorems, and the asymptotic behavior is investigated through a Voronovskaja-type theorem. The results extend and unify earlier work on Appell-based approximation operators and offer new tools for approximating functions in weighted spaces. Numerical examples and error estimates are provided to illustrate the efficacy of the proposed operators. In addition, the inherent symmetry in the structure of the proposed operators-arising from the symmetric nature of the Beta kernel and the generating functions of degenerate Appell polynomials is discussed. Such symmetry plays a key role in ensuring balanced approximation and convergence characteristics.

Optimizations of approximation operators in covering rough set theory

Fractional calculus has emerged as an important research area for the analysis and solution of complex engineering problems. However, because exact implementation of fractional-order (FO) operators is not possible, various integer-order approximations are used for implementation. Agreement of the time and frequency responses obtained with these approximation methods with the analytical responses of the FO models is critical for application accuracy and performance. This study aims to reduce the difference between analytical and approximation-based frequency responses through optimization for better implementation performance. After the success of the proposed method was proved for an FO operator, it was applied to an FOPID controller and an FO filter. Notable improvements were observed in both frequency and time response. To test the practicality of the method, the proposed method and other approximation methods were tested on an FPGA using the Vitis Model Composer Hub system generator block within the MATLAB Simulink environment. Also, the proposed method was experimentally implemented for an FO operator on a Nexys 4 DDR Artix-7 FPGA. It was observed that FPGA implementation and simulation results were in good agreement with each other.