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Complex systems in the real world often can be modeled as network structures, and community discovery algorithms for complex networks enable researchers to understand the internal structure and implicit information of networks. Existing community discovery algorithms are usually designed for single-layer networks or single-interaction relationships and do not consider the attribute information of nodes. However, many real-world networks consist of multiple types of nodes and edges, and there may be rich semantic information on nodes and edges. The methods for single-layer networks cannot effectively tackle multi-layer information, multi-relationship information, and attribute information. This paper proposes a community discovery algorithm based on multi-relationship embedding. The proposed algorithm first models the nodes in the network to obtain the embedding matrix for each node relationship type and generates the node embedding matrix for each specific relationship type in the network by node encoder. The node embedding matrix is provided as input for aggregating the node embedding matrix of each specific relationship type using a Graph Convolutional Network (GCN) to obtain the final node embedding matrix. This strategy allows capturing of rich structural and attributes information in multi-relational networks. Experiments were conducted on different datasets with baselines, and the results show that the proposed algorithm obtains significant performance improvement in community discovery, node clustering, and similarity search tasks, and compared to the baseline with the best performance, the proposed algorithm achieves an average improvement of 3.1% on Macro-F1 and 4.7% on Micro-F1, which proves the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this study, we introduce a system called Matrix Researcher2vec (MResearcher2vec) which generates researcher embedding vectors from their papers and research projects in researchmap and KAKENHI databases. The system includes data on 276,841 researchers, 6,161,592 papers, and research projects. Utilizing natural language processing techniques, the MResearcher2vec model extracts researcher vectors from the papers and research project summaries of KAKENHI grant recipients. The similarity between reseachers is then computed to visualize inter-researcher relationships. The machine learning results have been integrated into a web service, providing a novel approach for academic relationship mining. It can be applied in the matching of research contents and researchers in evaluation of industry-government-academia collaboration and joint research. It contributes in four aspects: (1) exchanges between researchers, (2) creation of opportunities for researchers and companies to connect, (3) further promotion of interdisciplinary research, and (4) reduction of lost opportunities for research institutions to acquire talents. [ABSTRACT FROM AUTHOR]

The paper addresses the issue of classification machine learning algorithm performance based on a novel probabilistic confusion matrix concept. The paper develops a theoretical framework which associates the proposed confusion matrix and the resulting performance metrics with the regular confusion matrix. The theoretical results are verified based on a wide variety of real-world classification problems and state-of-the-art machine learning algorithms. Based on the properties of the probabilistic confusion matrix, the paper then highlights the benefits of using the proposed concept both during the training phase and the application phase of a classification machine learning algorithm. [ABSTRACT FROM AUTHOR]

The class of generalized S D D 1 (G S D D 1 ) matrices is a new subclass of H -matrices. In this paper, we focus on the subdirect sum of G S D D 1 matrices, and some sufficient conditions to ensure that the subdirect sum of G S D D 1 matrices with strictly diagonally dominant (S D D) matrices is in the class of G S D D 1 matrices are given. Moreover, corresponding examples are given to illustrate our results. The class of generalized matrices is a new subclass of -matrices. In this paper, we focus on the subdirect sum of matrices, and some sufficient conditions to ensure that the subdirect sum of matrices with strictly diagonally dominant matrices is in the class of matrices are given. Moreover, corresponding examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE - T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in a Grünbaum paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership. [ABSTRACT FROM AUTHOR]

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

Real-time process data are the foundation for the successful implementation of intelligent manufacturing in the chemical industry. However, in the actual production process, process data may randomly be missing due to various reasons, thus affecting the practical application of intelligent manufacturing technology. Therefore, this paper proposes the application of appropriate matrix completion algorithms to impute the missing values of real-time process data. Considering the characteristics of online missing value imputation problems, this paper proposes an improved method for a matrix completion algorithm that is suitable for real-time missing data imputation. By utilizing real device data, this paper studies the impact of algorithm parameters on the effect of missing value imputing and compares it with several classical missing value imputing methods. The results show that the introduced method achieves higher accuracy in data imputation compared to the baseline method. Furthermore, the proposed enhancement significantly improves the speed performance of algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation A X B   =   C , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations A X   =   C and X B   =   C . Finally, we provide a numerical example to demonstrate the findings of this paper. [ABSTRACT FROM AUTHOR]

This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over Z 2 . This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over . [ABSTRACT FROM AUTHOR]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation A X B = C has been extensively studied. However, there is currently limited information on matrix equation A X B = C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation A X B = C , and present the expression for the general solution when it is solvable. As an application, we derive the ϕ -Hermitian solutions for dual quaternion matrix equation A X A ϕ = C , where the ϕ -Hermitian extends the concepts of Hermiticity and η -Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. [ABSTRACT FROM AUTHOR]

Accurate evaluation of the critical nodes in the system is essential work for a multiplatform avionics system (MPAS) for resource allocation and other works. However, current evaluation methods are either limited to the aircraft level or the function module level. There is a lack of research on the evaluation using the information of these two levels. In view of this situation, this paper researches the two-level iterative method of evaluating the importance of aircraft function modules. The influence matrix was constructed by using the node access probability calculated by the PageRank algorithm and the function module weight calculated based on centrality. In addition, the importance of aircraft nodes was used to carry out two-level iteration, and finally, the importance of aircraft function modules was obtained. The experimental results show that this method can comprehensively utilize the information on aircraft cooperative network and function module cooperative network, solve the key problems of two-level iterative evaluation, and meet the requirement of evaluating critical nodes in a system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation Σni=1 AiXiBi=Ewith sub-matrix constraints. With the help of LC-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, GH-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and suficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration. [ABSTRACT FROM AUTHOR]

During the course of actual oilfield development, judicious selection and design of well placement are paramount due to cost constraints and operating conditions. This paper introduces the Matrix Directional Continuous Elements Summation Algorithm (MDCESA), which is utilized to identify that segment with the largest summation for a given length in a 2D or a 3D matrix. An additional function that accounts for the distance between segments was added when searching for multiple segments to avoid intersections or overlaps between segments. The well placement optimization was transformed into a segment summation on a 3D matrix. Our findings reveal significant advancements in well placement optimization. Employing the MDCESA method, six producers were identified and their production performance was compared against two previously selected producers using a reservoir numerical simulator. The results demonstrated that the wells selected through MDCESA exhibited a substantial improvement in production efficiency. Specifically, there was an 11.6% increase in average cumulative oil production over a 15-year period compared to the wells selected by traditional methods. This research not only presents a significant leap in well placement optimization but also sets a foundation for further innovations in reservoir management and development strategies in offshore oilfields. [ABSTRACT FROM AUTHOR]

In this paper we give an explicit description of linear maps preserving the Cullis determinant of rectangular matrices of the size (n+1)×n. Unlike the result about the ordinary determinant, it appears that linear preservers of Cullis determinant can be singular. We provide the corresponding examples and characterize the case when these maps are non-singular. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for solving a two-person zero-sum fuzzy matrix game with goals, payoffs, and decision variables represented as triangular fuzzy rough numbers. We created a pair of fully fuzzy rough linear programming problems for players. Triangular fuzzy rough numbers can be used to formulate two fuzzy linear programming problems for the first player in the form of upper approximation intervals and lower approximation intervals. Two problems for the second player can be created in the same way. These problems have been split into five sub-crisp problems for the player first and five sub-crisp problems for the player second. The solution to the game can be obtained by solving these ten fuzzy linear programming problems. To demonstrate the method, a numerical example is provided. Using Wolfram Cloud, optimal strategies and game values are calculated for various parameters. Sensitivity analysis is carried out by altering the values of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present some generalizations and further refinements for the numerical radii of sectorial matrices and Schatten p-norms inequalities of accretive-dissipative matrices, which generalized some results of Kittaneh et al. Moreover, we also give some n-tuple power inequality for sectorial matrices by Yang [22]. [ABSTRACT FROM AUTHOR]

This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge–Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step, and thus, the trapezoidal rule is always employed in the first sub-step. This paper demonstrates for the first time that the proposed s-sub-step implicit method with s ≤ 6 can reach sth-order accuracy when achieving dissipation control and unconditional stability simultaneously. Hence, this paper develops, analyzes, and compares four cost-optimal high-order implicit algorithms within the present s-sub-step method using three, four, five, and six sub-steps. Each high-order implicit algorithm shares identical effective stiffness matrices to achieve optimal spectral properties. Unlike the published algorithms, the proposed high-order methods do not suffer from the order reduction for solving forced vibrations. Moreover, the novel methods overcome the defect that the authors' previous algorithms require an additional solution to obtain accurate accelerations. Linear and nonlinear examples are solved to confirm the numerical performance and superiority of four novel high-order algorithms. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

In this paper the construction of C k basis functions on meshes with hanging nodes for hp -finite element methods is proposed. The construction is limited to axis-parallel, rectangular meshes, which can, however, be of any dimension. The basis functions are composed by using hierarchical tensor product shape functions which are defined on patches of so-called constraining nodes and combine univariate Hermite interpolation polynomials and Gegenbauer polynomials. The paper discusses how to integrate the constructed C k basis functions into the assembling process of stiffness matrices and load vectors by using connectivity matrices. Numerical examples illustrate the applicability of the proposed concept. [ABSTRACT FROM AUTHOR]

In this paper, we address the extremal inverse eigenvalue problem for pentadiagonal matrices. We provide sufficient conditions for their existence and realizability through new constructions that consider spectral data of its leading principal submatrices. Finally, we present some examples generated from the algorithmic procedures derived from our results. [ABSTRACT FROM AUTHOR]

In this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the impulsive synchronization of a class of nonlinear multiple neural networks (MNNs) with multi-delays was considered under a dynamic event-based mechanism. To achieve a more comprehensive synchronization outcome and mitigate the conservativeness of impulsive control due to predetermined time sequences, we integrated a dynamic event-triggered strategy. This approach formed a novel control framework for generalized MNNs, where impulsive inputs were applied only under specific conditions governed by event-triggering rules. Towards the above objectives, the impulsive jumping system, resulting from dynamic component, and matrix measure method were invoked to directly increase the computational simplicity and extensibility of the study. As the outcome, the synchronization criteria for the MNNs could be achieved, and the exponential convergence rate is resolved by considering both the generalized comparison principle regarding impulsive systems and the variable parameter formula. Moreover, Zeno-freeness of the achieved triggering regulation is ensured. Finally, two numerical examples confirmed the validity of the designed approach. [ABSTRACT FROM AUTHOR]

The correlation filter object tracking algorithm has gained extensive attention from scholars in the field of tracking because of its excellent tracking performance and efficiency. However, the mathematical modeling relationships of correlation filter tracking frameworks are unclear. Therefore, many forms of correlation filters are susceptible to confusion and misuse. To solve these problems, we attempted to review various forms of the correlation filter and discussed their intrinsic connections. First, we reviewed the basic definitions of the circulant matrix, convolution, and correlation operations. Then, the relationship among the three operations was discussed. Considering this, four mathematical modeling forms of correlation filter object tracking from the literature were listed, and the equivalence of the four modeling forms was theoretically proven. Then, the fast solution of the correlation filter was discussed from the perspective of the diagonalization property of the circulant matrix and the convolution theorem. In addition, we delved into the difference between the one-dimensional and two-dimensional correlation filter responses as well as the reasons for their generation. Numerical experiments were conducted to verify the proposed perspectives. The results showed that the filters calculated based on the diagonalization property and the convolution property of the cyclic matrix were completely equivalent. The experimental code of this paper is available at https://github.com/110500617/Correlation-filter/tree/main. [ABSTRACT FROM AUTHOR]

Plagued by heterogeneous clutter, it is a serious challenge for airborne radars to detect low-altitude, weak targets. To overcome this problem, a novel matrix information geometry detector for airborne multi-channel radar is proposed in this paper. The proposed detector applies the given steering vector and array structure information to the matrix information geometry detection method so that it can be used for space–time adaptive detection. While improving the detection performance, the matrix information geometry detector's original anti-clutter advantage is enhanced as well. The simulation experiment results indicate that the proposed detector has advantages in several of the properties related to space–time adaptive detection, while its computational complexity does not increase significantly. Moreover, experiment results based on the measured data verify the superior performance of the proposed method. Sea-detecting data-sharing-program data, mountaintop data, and phased-array radar data are employed to verify the performance advantage of the proposed method in heterogeneous clutter and the ability for weak target detection. [ABSTRACT FROM AUTHOR]

One key aspect of soft set theory is abstraction. The main idea in soft set theory revolves around making decisions with the guidance of experts. The combination of soft sets and rough sets led to the creation of soft-rough sets. This study expands the concept of soft-rough matrices to square soft-rough matrices (ss-r m). Various noteworthy properties of ss-r m are examined in this paper. [ABSTRACT FROM AUTHOR]

In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method. [ABSTRACT FROM AUTHOR]

Automatic control of robots with flexible links has been a pivotal subject in control engineering and robotics due to the challenges posed by vibrations during repetitive movements. These vibrations affect the system's performance and accuracy, potentially causing errors, wear, and failures. LQR control is a common technique for vibration control, but determining the optimal weight matrices [Q] and [R] is a complex and crucial task. This paper proposes a methodology based on genetic algorithms to define the [Q] and [R] matrices according to design requirements. MATLAB and Simulink, along with data provided by Quanser, will be used to model and evaluate the performance of the proposed approach. The process will include testing and iterative adjustments to optimize performance. The work aims to improve the control of robots with flexible links, offering a methodology that allows for the design of LQR control under the design requirements of controllers used in classical control through the use of genetic algorithms. [ABSTRACT FROM AUTHOR]

Transformer has emerged as one of the modern neural networks that has been applied in numerous applications. However, transformers' large and deep architecture makes them computationally and memory-intensive. In this paper, we propose the block g-circulant matrices to replace the dense weight matrices in the feedforward layers of the transformer and leverage the DCT-DST algorithm to multiply these matrices with the input vector. Our test using Portuguese-English datasets shows that the suggested method improves model memory efficiency compared to the dense transformer but at the cost of a slight drop in accuracy. We found that the model Dense-block 1-circulant DCT-DST of 128 dimensions achieved the highest model memory efficiency at 22.14%. We further show that the same model achieved a BLEU score of 26.47%. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) β -minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF β -covering approximation space (i.e., an IF environment) with a high cardinality, it would be tedious and complicated to use IF set representations to calculate them. Therefore, it is necessary to find a quick method to obtain them. In this paper, we present the notion of IF β -maximal description based on the definition of IF β -minimal description, along with the concepts of IF granular matrix and IF reduction. Moreover, we propose matrix calculation methods for IF covering-based rough sets, such as IF β -minimal descriptions, IF β -maximal descriptions, and IF reductions. Firstly, the notion of an IF granular matrix is presented, which is used to calculate IF β -minimal description. Secondly, inspired by IF β -minimal description, we give the notion of IF β -maximal description. Furthermore, the matrix representations of IF β -maximal descriptions are presented. Next, two types of reductions for IF β -covering approximation spaces via IF β -minimal and fuzzy β -minimal descriptions are presented, along with their matrix representations. Finally, the new calculation methods are compared with corresponding set representations by carrying out several experiments. [ABSTRACT FROM AUTHOR]

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2 × 2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2 × 2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2 × 2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas. [ABSTRACT FROM AUTHOR]

With the advancement of information technology, academics have steadily strengthened their research and creation of a complete assessment system for college sports flipped classrooms, so as to improve school education quality. Teachers can use classroom evaluations to quickly identify the flaws in their own teaching approaches and switch to a more effective teaching mode. The old paper assessment approach, on the other hand, is inefficient, which wastes time and money. This paper will look at how to use an AHP-fuzzy matrix to create a thorough assessment system for college athletics flipped classrooms. The main algorithm of the AHP-fuzzy matrix is proposed in this work, as well as the fuzzy comprehensive evaluation approach. The stability and operating efficiency of the system are constantly rising. Stability rose from 77% in the first experiment to 85% in the 10th experiment, an 8% increase. The operating efficiency has risen from 85% at the beginning to 87%. Although the rise is not large, it has been stable. This also shows that the system designed in this paper not only has strong stability but also has high operating efficiency. [ABSTRACT FROM AUTHOR]

In 1989, Bakhtin proved a version of the Banach contraction principle in the context of b-metric spaces. More precisely, let (X, d; s) be a complete b-metric space with parameter s ⩾ 1 and let T a contractive map on X, that is a self-map T of X satisfying d(Tx, Ty) ⩽ λd(x, y), ∀x, y ∈ X, (Bλ) with some λ ∈ [0, 1). Bakhtin proved that if λ ∈ [0, 1 s), then T has a unique fixed point. In the two last decades many research papers were produced by many authors in the setting of b-metric spaces. In 2021, the author published a note on the Banach contraction principle in b-metric spaces in Divilgaciones Mat'ematicas dealing with some complements to Bakhtin results. The aim of this paper is to provide other complements. In particular, we discuss the remaining case where λ ∈ [1 s, 1). We give an evaluation of the order of convergence for the iterative Picard process and a posteriori error estimate for this process and we estimate the diameter of the T-orbits. We investigate well-posedness of the fixed problem of a map T satisfying Bλ when X is Torbitally complete and we establish two shadowing properties for these maps without requiring completeness of X. [ABSTRACT FROM AUTHOR]

This paper aims to consider the extended Perron complements for the collection of Mmatrices. We first exhibit the connection between the extended Perron complements of M-matrices and nonnegative matrices. Moreover, we present some common inequalities involving extended Perron complements, Schur complements, and principal submatrices of irreducibleM-matrices by utilizing the properties of M-matrices. We also discuss the monotonicity of the extended Perron complements and minimum eigenvalue. For the collection of M-matrices, we demonstrate that all (extended) Perron complements are M-matrices. Especially, we deduce that M-matrices and their Perron complements share the same minimum eigenvalue. Finally, a simple example is presented to illustrate our findings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is first to introduce the notation of matrix intuitionistic fuzzy normed spaces, and then by virtue of this notation to study the Hyers-Ulam stability results concerning the mixed type additive-quadratic functional equation 2k[ f (x + ky) + f (kx + y)] = k(1 - s + k + ks + 2k2) f (x + y) + k(1 - s - 3k + ks + 2k2) f (x - y) + 2k f (kx) + 2k(s + k - ks - 2k2) f (x) + 2(1 - k - s) f (ky) + 2ks f (y) in the setting of matrix intuitionistic fuzzy normed spaces by applying two different methods, where s is a parameter, k > 1 and s, 1 - 2k. Moreover, the interdisciplinary relation between the theory of matrix intuitionistic fuzzy normed spaces and the theory of functional equations are also presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [2]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs R(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and R(G2) and S(H1) and R(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs. [ABSTRACT FROM AUTHOR]

This paper redefines picture fuzzy soft matrices (pfs-matrices) because of some of their inconsistencies resulting from Cuong's definition of picture fuzzy sets. Then, it introduces several distance measures of pfs-matrices. Afterward, this paper proposes a new kNN-based classifier, namely the Picture Fuzzy Soft k-Nearest Neighbor (PFS-kNN) classifier. The proposed classifier utilizes the Minkowski's metric of pfs-matrices to find the k-nearest neighbor. Thereafter, it performs an experimental study utilizing four UCI medical datasets and compares to the suggested approach using the state-of-the-art kNN-based classifiers. To evaluate the performance of the classification, it conducts ten iterations of five-fold cross-validation on all the classifiers. The findings indicate that PFS-kNN surpasses the state-of-the-art kNN-based algorithms in 72 out of 128 performance results based on accuracy, precision, recall, and F1-score. More specifically, the proposed method achieves higher accuracy and F1-score results compared to the other classifiers. Simulation results show that pfs-matrices and PFS-kNN are capable of modeling uncertainty and real-world problems. Finally, the applications of pfs-matrices to supervised learning are discussed for further research. [ABSTRACT FROM AUTHOR]

In this paper, we present some further refinements of the Tan-Xie inequality for sector matrices and its applications due to Nasiri and Furuichi [J. Math. Inequal., 15 (2021), 1425-1434]. [ABSTRACT FROM AUTHOR]

In this paper, we present two lower bounds for the smallest singular value of nonsingular matrices. Moreover, we illustrate with numerical examples that these bounds are better than the existing bounds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identity of these numbers. [ABSTRACT FROM AUTHOR]

In this paper, the authors compare the features and the overall performance of the two high-level numerical computing and modeling software environments: the freeware Scilab and commercially available industry-standard MATLAB. The motivation for the work emanated from the educational use of these tools at the college and university level, but with a perspective to their professional and scientific use as well. Their performance is tested by measuring the execution times of several combined-task benchmarks implemented as test functions, built upon nine common numerical tasks that are often found in programs for solving standard engineering problems. They include basic algebra and matrix calculations, signal generation, signal analysis, and storing and retrieving data to and from the hard disk drive. Although MATLAB outperforms Scilab in all the benchmarks except the disk file manipulations, in the presumed vectorization versions of the benchmarks, it is not for much. The overall performance of the freeware rival is very satisfactory, making it a good choice not only for educational use but also for scientific and professional purposes, especially when funding is critical. [ABSTRACT FROM AUTHOR]

This paper mainly studies the least-squares solutions of matrix equation A X B = C under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given. [ABSTRACT FROM AUTHOR]

Aiming at addressing the security and efficiency challenges during image transmission, an efficient image cryptosystem utilizing difference matrix and genetic algorithm is proposed in this paper. A difference matrix is a typical combinatorial structure that exhibits properties of discretization and approximate uniformity. It can serve as a pseudo-random sequence, offering various scrambling techniques while occupying a small storage space. The genetic algorithm generates multiple ciphertext images with strong randomness through local crossover and mutation operations, then obtains high-quality ciphertext images through multiple iterations using the optimal preservation strategy. The whole encryption process is divided into three stages: first, the difference matrix is generated; second, it is utilized for initial encryption to ensure that the resulting ciphertext image has relatively good initial randomness; finally, multiple rounds of local genetic operations are used to optimize the output. The proposed cryptosystem is demonstrated to be effective and robust through simulation experiments and statistical analyses, highlighting its superiority over other existing algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization, and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients—including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries—are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature. Funding: X. Li was partially supported by National Key R&D Program of China [Grants 2020YFA0711900, 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107, 62141407], the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], the "Chenguang Program" by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission [Grant 19CG02], the Shanghai Science and Technology Program [21JC1400600]. Y. Luo and A. R. Zhang were partially supported by the National Science Foundation [CAREER-2203741]. [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

Emotion recognition research has attracted great interest in various research fields, and electroencephalography (EEG) is considered a promising tool for extracting emotion-related information. However, traditional EEG-based emotion recognition methods ignore the spatial correlation between electrodes. To address this problem, this paper proposes an EEG-based emotion recognition method combining differential entropy feature matrix (DEFM) and 2D-CNN-LSTM. In this work, first, the one-dimensional EEG vector sequence is converted into a two-dimensional grid matrix sequence, which corresponds to the distribution of brain regions of the EEG electrode positions, and can better characterize the spatial correlation between the EEG signals of multiple adjacent electrodes. Then, the EEG signal is divided into equal time windows, and the differential entropy (DE) of each electrode in this time window is calculated, it is combined with a two-dimensional grid matrix and differential entropy to obtain a new data representation that can capture the spatiotemporal correlation of the EEG signal, which is called DEFM. Secondly, we use 2D-CNN-LSTM to accurately identify the emotional categories contained in the EEG signals and finally classify them through the fully connected layer. Experiments are conducted on the widely used DEAP dataset. Experimental results show that the method achieves an average classification accuracy of 91.92% and 92.31% for valence and arousal, respectively. The method performs outstandingly in emotion recognition. This method effectively combines the temporal and spatial correlation of EEG signals, improves the accuracy and robustness of EEG emotion recognition, and has broad application prospects in the field of emotion classification and recognition based on EEG signals. [ABSTRACT FROM AUTHOR]

The Conway group Co1 is one of the 26 sporadic simple groups. It is the largest of the three Conway groups with order 4157776806543360000 = 221.39.54.7².11.13.23 and has 22 conjugacy classes of maximal subgroups. In this paper, we discuss a group of the form G = N : G, where N = 211 and G = M24. This group G = N : G = 211 : M24 is a split extension of an elementary abelian group N = 211 by a Mathieu group G = M24. Using the computed Fischer matrices for each class representative g of G and ordinary character tables of the inertia factor groups of G, we obtain the full character table of G. The complete fusion of G into its mother group Co1 is also determined using the permutation character of Co1. [ABSTRACT FROM AUTHOR]

In this paper, we introduced novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We use our new characterizations of majorization to derive an improved entropy inequality. [ABSTRACT FROM AUTHOR]

Minor perturbations to a linear operator may drastically change its spectrum, and hence the difficulty of deciding whether or not a numerically computed quantity is zero causes problems in spectral theory. The purpose of this expository paper is to illustrate such instability phenomena by some examples with Toeplitz-like operators and matrices. [ABSTRACT FROM AUTHOR]