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This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of a grey matrix and grey linear programming plays an important role in establishing a grey computational system.

Starting from the fact that missing information often appears in complex systems, and therefore that true values of elements are uncertain when the authors construct a matrix, as well as calculate its inverse. However, the authors can get their ranges, which are called the number-covered sets, by using grey computational rules. How to get the matrix-covered set of inverse grey matrix became a typical approach. In this paper, grey linear programming was explained in detail, for the point of grey meaning and the methodology to calculate the inverse grey matrix can successfully solve grey linear programming.

The results show that the ranges of grey value of inverse grey matrix and grey linear programming can be obtained by using the computational rules.

Because the matrix and the linear programming have been widely used in many fields such as system controlling, economic analysis and social management, and the missing information is a general phenomenon for complex systems, grey matrix and grey linear programming may have great potential application in real world. The methodology realizes the feasibility to control the complex system under uncertain situations.

The paper successfully obtained the ranges of uncertain inverse matrix and linear programming by using grey system theory, when the elements of matrix and the coefficients of linear programming are intervals and the results enrich the contents of grey mathematics.

This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth.

Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties.

N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied.

Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

Loss of control and air crashes are frequently caused by aircraft faults. Therefore, a practical control strategy can prevent aircrafts from losing control without control laws reconstruction. The purpose of this paper is to propose a sideslip trim fault-tolerant control strategy for wing damage and aileron stuck.

The six degree of freedom model of the damaged aircraft is constructed by using the non-center-of-mass approach on the basis of aerodynamic database, which is calculated in XFlow. This paper adopts the sideslip command for trim, combining with the adaptive nonlinear dynamic inversion control to achieve fault-tolerant control.

This strategy can effectively improve the control margin of the remaining control surface and guarantee maneuverability of the aircraft after serious faults.

The original and wing-damaged aircraft models are reconstructed in CATIA, and the aerodynamic data is calculated in XFlow. Sideslip angle is adopted to compensate additional roll moment caused by wing damage or aileron stuck. Adaptive nonlinear dynamic inversion control, combined with sideslip trim, is applied to achieve fault-tolerant control.

When recommending products to consumers, it is important to be able to accurately predict how consumers will rate them. However, existing collaborative filtering models are difficult to achieve a balance between rating prediction accuracy and complexity. Therefore, the purpose of this paper is to propose an accurate and effective model to predict users’ ratings of products for the accurate recommendation of products to users.

First, we introduce an attention mechanism that dynamically assigns weights to user preferences, highlighting key interaction information and enhancing the model’s understanding of user behavior. Second, a fold embedding strategy is employed to segment user interaction data, increasing the information density of each subset while reducing the complexity of the attention mechanism. Finally, a masking strategy is integrated to mitigate overfitting by concealing portions of user-item interactions, thereby improving the model’s generalization ability.

The experimental results demonstrate that the proposed model significantly minimizes prediction error across five real-world datasets. On average, the evaluation metrics root mean square error (RMSE) and mean absolute error (MAE) are reduced by 9.11 and 13.3%, respectively. Additionally, the Friedman test results confirm that these improvements are statistically significant. Consequently, the proposed model more accurately captures the intrinsic correlation between users and products, leading to a substantial reduction in prediction error.

We propose a novel collaborative filtering model to learn the user-item interaction matrix effectively. Additionally, we introduce a fold embedding strategy to reduce the computational resource consumption of the attention mechanism. Finally, we implement a masking strategy to encourage the model to focus on key features and patterns, thereby mitigating overfitting.

As a kind of NP-hard combinatorial optimization problem, pipe routing design (PRD) is applied widely in modern industries. In the offshore oil and gas industry, a semi-submersible production platform is an important equipment for oil exploitation and production. PRD is one of the most key parts of the design of semi-submersible platform. This study aims to present an improved ant colony algorithm (IACO) to address PRD for the oil and gas treatment system when designing a semi-submersible production platform.

First, to simplify PRD problem, a novel mathematical model is built according to real constraints and rules. Then, IACO, which combines modified heuristic function, mutation mechanism and dynamical parameter mechanism, is introduced.

Based on a set of specific instances, experiments are carried out, and the experimental results show that the performance of IACO is better than that of two variants of ACO, especially in terms of the convergence speed and swarm diversity. Finally, IACO is used to solve PRD for the oil and gas treatment system of semi-submersible production platform. The simulation results, which include nine pipe paths, demonstrate the practicality and high-efficiency of IACO.

The main contribution of this study is the development of method for solving PRD of a semi-submersible production platform based on the novel mathematical model and the proposed IACO.

The paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.

The fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.

Different from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.

According to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

In community question answering (CQA), people who answer questions assume readers have mastered the content in the answers. Nevertheless, some readers cannot understand all content. Thus, there is a need for further explanation of the concepts that appear in the answers. Moreover, the large number of question and answer (Q&A) documents make manual retrieval difficult. This paper aims to alleviate these issues for CQA websites.

In the paper, an algorithm for recommending explanatory Q&A documents is proposed. Q&A documents are modeled with the biterm topic model (BTM) (Yan et al., 2013). Then, the growing neural gas (GNG) algorithm (Fritzke, 1995) is used to cluster Q&A documents. To train multiple classifiers, three features are extracted from the Q&A categories. Thereafter, an ensemble classification model is constructed to identify the explanatory relationships. Finally, the explanatory Q&A documents are recommended.

The GNG algorithm shows good clustering performance. The ensemble classification model performs better than other classifiers. The both effect and quality scores of explanatory Q&A recommendations are high. These scores indicate the practicality and good performance of the proposed recommendation algorithm.

The proposed algorithm alleviates information overload in CQA from the new perspective of recommending explanatory knowledge. It provides new insight into research on recommendations in CQA. Moreover, in practice, CQA websites can use it to help retrieve Q&A documents and facilitate understanding of their contents. However, the algorithm is for the general recommendation of Q&A documents which does not consider individual personalized characteristics. In future work, personalized recommendations will be evaluated.

A novel explanatory Q&A recommendation algorithm is proposed for CQA to alleviate the burden of manual retrieval and Q&A overload. The novel GNG clustering algorithm and ensemble classification model provide a more accurate way to identify explanatory Q&A documents. The method of ranking the explanatory Q&A documents improves the effectiveness and quality of the recommendation. The proposed algorithm improves the accuracy and efficiency of retrieving explanatory Q&A documents. It assists users in grasping answers easily.

The purpose of this paper is to find an omnidirectional robust gust response stabilization (GRS) scheme with anti-disturbance and state-limited features.

Disturbance observer and barrier Lyapunov techniques, which can, respectively, estimate the lumped disturbances of the dynamic system in real-time and ensure the middle states within some prescribed ranges according to some flight safety indexes.

In the existing literature, almost all of the GRS controllers are either only for the longitudinal dynamics or only for the latitudinal dynamics. Few studies have considered the gust response alleviation problem with omnidirectional wind disturbance and full aircraft model.

This paper proposes a fresh scheme to deal with a more holistic GRS problem; the disturbance observer based (DOB) barrier Lyapunov backstepping longitudinal controller has been put forward; DOB nonlinear dynamic inversion to handle the multi-input-multi-output lateral dynamics; and to closely connect the two loops of the latitudinal dynamics, a manipulating variable conversion method is proposed.

The purpose of this paper is to investigate the feasibility of solving turbulent flows based on smoothed finite element method (S-FEM). Then, the differences between S-FEM and finite element method (FEM) in dealing with turbulent flows are compared.

The stabilization scheme, the streamline-upwind/Petrov-Galerkin stabilization is coupled with stabilized pressure gradient projection in the fractional step framework. The Reynolds-averaged Navier-Stokes equations with standard k-epsilon model are selected to solve turbulent flows based on S-FEM and FEM. Standard wall functions are applied to predict boundary layer profiles.

This paper explores a completely new application of S-FEM on turbulent flows. The adopted stabilization scheme presents a good performance on stabilizing the flows, especially for very high Reynolds numbers flows. An advantage of S-FEM is found in applying wall functions comparing with FEM. The differences between S-FEM and FEM have been investigated.

The research in this work is limited to the two-dimensional incompressible turbulent flow.

The verification and validation of a new combination are conducted by several numerical examples. The new combination could be used to deal with more complicated turbulent flows.

The applications of the new combination to study basic and complex turbulent flow are also presented, which demonstrates its potential to solve more turbulent flows in nature and engineering.

This work carries out a great extension of S-FEM in simulations of fluid dynamics. The new combination is verified to be very effective in handling turbulent flows. The performances of S-FEM and FEM on turbulent flows were analyzed by several numerical examples. Superior results were found compared with existing results and experiments. Meanwhile, S-FEM has an advantage of accuracy in predicting boundary layer profile.

A nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].

Mathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.

Though it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.

This study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.