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The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C , including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2 p -th power with p ≥ 1 of the numerical radius of the off-diagonal operator matrix 0 A B * 0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B = A * , where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory. [ABSTRACT FROM AUTHOR]

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the λ -pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (1) and T 3 (2) for the functions in this family. Further, we investigate several special cases and consequences of our results. [ABSTRACT FROM AUTHOR]

This paper investigates the spectral properties of block circulant matrices with high-order symmetric (or Hermitian) blocks. We analyze cases with dependent or sparse independent entries within these blocks. Additionally, we analyze the distribution of singular values for the product of independent circulant matrices with non-Hermitian blocks. [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]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Nowadays, deep representations have gained significant attention due to their outstanding performance in a wide range of tasks. However, the interpretability of deep representations in specific applications poses a significant challenge. For instances where the generated quantity matrices exhibit symmetry, this paper introduces a variant of deep matrix factorization (deep MF) called deep L p smooth symmetric non-negative matrix factorization (DSSNMF), which aims to improve the extraction of clustering structures inherent in complex hierarchical and graphical representations in high-dimensional datasets by improving the sparsity of the factor matrices. We successfully applied DSSNMF to synthetic datasets as well as datasets related to post-traumatic stress disorder (PTSD) to extract several hierarchical communities. Specifically, we identified non-disjoint communities within the partial correlation networks of PTSD psychiatric symptoms, resulting in highly meaningful clinical interpretations. Numerical experiments demonstrate the promising applications of DSSNMF in fields like network analysis and medicine. [ABSTRACT FROM AUTHOR]

This paper focuses on reducing the computational cost of a GNC Algorithm for deblurring images when dealing with full symmetric Toeplitz block matrices composed of Toeplitz blocks. Such a case is widespread in real cases when the PSF has a vast range. The analysis in this paper centers around the class of gamma matrices, which can perform vector multiplications quickly. The paper presents a theoretical and experimental analysis of how γ -matrices can accurately approximate symmetric Toeplitz matrices. The proposed approach involves adding a minimization step for a new approximation of the energy function to the GNC technique. Specifically, we replace the Toeplitz matrices found in the blocks of the blur operator with γ -matrices in this approximation. The experimental results demonstrate that the new GNC algorithm proposed in this paper reduces computation time by over 20 % compared with its previous version. The image reconstruction quality, however, remains unchanged. [ABSTRACT FROM AUTHOR]

The present paper deals with the external identification of a reciprocal, special passive, 2 n -port network under measurement uncertainties. In the present context, the multiport model is represented by an admittance matrix and the condition that the network is 'reciprocal special passive' refers to the assumption that the real part of the admittance matrix is symmetric and positive-definite. The key point is to reformulate the identification problem as a matrix optimization program over the matrix manifold S + (2 n) × S (2 n) . The optimization problem requires a least-squares criterion function designed to cope with over-determinacy due to the incoherent data pairs whose cardinality exceeds the problem's number of degrees of freedom. The present paper also proposes a numerical solution to such an optimization problem based on the Riemannian-gradient steepest descent method. The numerical results show that the proposed method is effective as long as reasonable measurement error levels and problem sizes are being dealt with. [ABSTRACT FROM AUTHOR]

( R , S )-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the ( R , S )-(skew) symmetric solutions to the quaternion matrix equation A X B = C . We use a real representation A τ to obtain the necessary and sufficient conditions for A X B = C to have ( R , S )-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares ( R , S )-(skew) symmetric solution to the above matrix equation. [ABSTRACT FROM AUTHOR]

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

In order to realize the stable clamping of large cabin parts, this paper studied the clamping force optimization of the clamping mechanism for large-mass and large-size cabin parts. Firstly, three kinds of contact models are introduced. Then, a clamping matrix is constructed for a particular clamping configuration. The nonlinear friction cone constraint at the contact point is transformed into a linear affine constraint in a smooth Riemannian manifold using the special structure of the positive definite symmetric matrix. Finally, a large cabin part used in the aerospace field is used as an example for calculating and simulating the clamping force optimization. Different optimization algorithms are used to calculate the initial value which is put into the gradient flow optimization method of the clamping force for optimization calculation. Meanwhile, the normal clamping force value of a 2 t object is measured using relevant experimental equipment. The simulation results and experimental results show that the gradient flow optimization method of the clamping force can quickly complete the clamping force optimization of large-mass and large-size cabin parts. The actual measured value of the normal clamping force is close to the simulated convergence value. The distribution of the normal force of the clamping mechanisms and the convergence value of the clamping force for each clamping mechanism can provide some references for determining the output clamping force of the clamping mechanism and confirming a reasonable distribution of the clamping mechanism. It also confirms the feasibility and effectiveness of this method applied to the clamping force optimization for a large axial part. [ABSTRACT FROM AUTHOR]

Complex dynamical networks with time-varying connections have characteristics that allow a better representation of real-world complex systems, especially interest in their not static behavior and topology. Their applications reach areas such as communication systems, electrical systems, medicine, robotic, and more. Both continuous and discrete-time complex dynamical networks and the pinning control technique have been studied. However, even with interest in the research on complex networks combining characteristics of discrete-time, time-varying connections, pinning control, and impulsive control, there are few studies reported in the literature. There are some previous studies dealing with impulsively pin-controlling a discrete-time complex network. Nevertheless, they neglect to deal with time-varying connections; they deal with these systems by experimentally using continuous-time methods or linearizing the node dynamics. In this manner, this paper presents a control scheme that not only deals with pin control on discrete-time complex networks but also includes time-varying connections. This paper proposes an impulsive pin control to a zero state using passivity degrees considering a discrete-time complex network with undirected, linear, and diffusive couplings. Additionally, a corresponding mathematical analysis, which allows the representation of the dynamics as a set of symmetric matrices, is presented. With this, certain kinds of time-varying connections can be integrated into the analysis. Moreover, a particular criterion for selecting nodes to pin is also presented. The behavior of the controller for the non-varying and time-varying coupling cases is shown via numeric simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real variables. However, there is a very efficient representation of the CM called the standard form (SF) that reduces the degrees of freedom to four real variables, while preserving all the relevant information on the state. The SF can be easily evaluated using a set of symplectic invariants. The paper starts from the SF, introducing an architecture that implements with primitive components the given two-mode Gaussian state having the CM with the SF. The architecture consists of a beam splitter, followed by the parallel set of two single–mode real squeezers, followed by another beam splitter. The advantage of this architecture is that it gives a precise non-redundant physical meaning of the generation of the Gaussian state. Essentially, all the relevant information is contained in this simple architecture. [ABSTRACT FROM AUTHOR]

In this paper, a generalized nonlinear Schrödinger (gNLS) equation with time-varying coefficients is analytically studied using its Lax representation and the associated Riemann-Hilbert (RH) problem equipped with a symmetric scattering matrix in the Hermitian sense. First, Lax representation and the associated RH problem of the considered gNLS equation are established so that solution of the gNLS equation can be transformed into the associated RH problem. Secondly, using the solvability of unique solution of the established RH problem, time evolution laws of the scattering data reconstructing potential of the gNLS equation are determined. Finally, based on the determined time evolution laws of scattering data, the long-time asymptotic solution and N-soliton solution of the gNLS equation are obtained. In addition, some local spatial structures of the obtained one-soliton solution and two-soliton solution are shown in the figures. This paper shows that the RH method can be extended to nonlinear evolution models with variable coefficients, and the curve propagation of the obtained N-soliton solution in inhomogeneous media is controlled by the selection of variable–coefficient functions contained in the models. [ABSTRACT FROM AUTHOR]

This paper analyses the structures of covariance functions of digital electroencephalography measurement vectors and digital vectors of two coronavirus images. For this research, we used the measurement results of 30-channel electroencephalography (E1–E30) and digital vectors of images of two SARS-CoV-2 variants (cor2 and cor4), where the magnitudes of intensity of the electroencephalography parameters and the parameters of the digital images of coronaviruses were encoded. The estimators of cross-covariance functions of the digital electroencephalography measurements' vectors and the digital vectors of the coronavirus images and the estimators of auto-covariance functions of separate vectors were derived by applying random functions constructed according to the vectors' parameter measurement data files. The estimators of covariance functions were derived by changing the values of the quantised interval k on the time and image pixel scales. The symmetric matrices of correlation coefficients were calculated to estimate the level of dependencies between the electroencephalography measurement results' vectors and the digital vectors of the coronavirus images. The graphical images of the normalised cross-covariance functions for the electroencephalography measurement results' vectors and the digital vectors of the coronavirus images within the period of all measurements are asymmetric. For all calculations, a computer program was developed by applying a package of Matlab procedures. A probabilistic interdependence between the results of the electroencephalography measurements and the parameters of the coronavirus vectors, as well as their variation on the time and image pixel scales, was established. [ABSTRACT FROM AUTHOR]

Symmetric weighing matrices are an important class of combinatorial designs and their constructions remains unknown even for some small orders. The existence of symmetric weighing matrices becomes an even more interesting challenge as these matrices possess both beautiful mathematical properties and many applications. In this paper, we present two original construction methods for symmetric weighing matrices. The suggested methods lead to two infinite families of symmetric weighing matrices. The first consists of symmetric weighing matrices W (2 p × 15 , 2 q × 25) for all q < p and p = 1 , 2 , ... while the second is an infinite family of symmetric weighing matrices W (2 p + 1 × 15 , 2 + 2 q × 25) for all q ≤ p and p = 1 , 2 , ... . These matrices are constructed by combining together a circulant and a negacyclic matrix with identical first row. The first of the infinite families includes a symmetric weighing matrix of order 30 and weight 25. The results presented here are new and have never been reported. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear controller for tracking a desired trajectory for an underactuated hovercraft is considered. It is a modification of a method known from the literature. However, the control algorithm considered here has two important features that differ from the mentioned control strategy. First, it concerns the case when the center of mass does not coincide with the geometric center, which results in additional forces and moments of force. The lack of symmetry causes the original trajectory tracking method not to take this fact into account, while the proposed approach is a generalization of the known concept. Here, a diagonalization of the inertia matrix has been applied, by means of a velocity transformation, which made it possible to reduce the symmetric matrix to a diagonal form. Secondly, the transformed quasi-velocity equations of motion allow some insight into the dynamics of the vehicle as it moves, which was not shown in the source work. The offered approach was verified by numerical tests for a hovercraft model with three DOF and for two desired trajectories. The method can be useful in preliminary simulation studies at the controller selection stage without experimental validation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the Cartan decomposition of the Lie group S O (2 , 1) . This class depends on two real parameters as components of the cylinder S 1 × R and we use a deformation inspired by Finsler indicatrices in order to obtain proper ellipses. A complex approach is also included. [ABSTRACT FROM AUTHOR]

The role of experts in decision-making is highly significant. The soft expert set (SES) is yet another milestone, as it plays a vital role in generalizing soft set theory. Soft expert set is creative and original because it involves more than one expert. This served as a driving force behind the creation of the current work. The present paper generalizes the framework of soft experts set to the inverse soft setting, namely the inverse soft expert set (ISES). Some captivating characteristic features of the soft expert set (SES) are established. On the inverse soft expert sets, numerous striking characteristics of symmetric matrices are presented. The ground-breaking idea of symmetric matrices on inverse soft expert sets (SMISES) is introduced. A MCDM problem for SMISES is described with an algorithm and interesting comparative analysis. [ABSTRACT FROM AUTHOR]

In the 3-dimensional Euclidean space E 3 , a quadric surface is either ruled or of one of the following two kinds z 2 = a s 2 + b t 2 + c , a b c ≠ 0 or z = a 2 s 2 + b 2 t 2 , a > 0 , b > 0 . In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property Δ I I N = Λ N , where Λ is a square symmetric matrix of order 3, and Δ I I denotes the Laplace operator of the second fundamental form I I of the surface. We prove that spheres with the nonzero symmetric matrix Λ , and helicoids with Λ as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property. [ABSTRACT FROM AUTHOR]

Recently, public-key cryptography based on tropical semi-rings have been proposed. However, the majority of them are damaged. The main reason is that they use a public matrix to construct commutative matrix semi-rings. New public-key cryptosystems are proposed in this paper. They are based on tropical congruent transformation of symmetric matrix by circular matrix. The NP-hard problem of solving a tropical system of nonlinear equations underlies the cryptosystem's security. Since a known matrix cannot express the used commutative subsemi-rings of circular matrices and there is no tropical matrix addition operation and power of matrix, the cryptosystems can withstand known attacks, including the KU attack, RM attack, and IK attack. The length of the public key and private key of the new cryptosystems is half that of those described in the literature. [ABSTRACT FROM AUTHOR]

The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method. [ABSTRACT FROM AUTHOR]

Turbulence in natural environments affects the reliability of communication. In this paper, we propose a symmetric matrix-assisted multiple-input multiple-output (MIMO) maritime visible light communication (VLC) system to confront turbulence and improve reliability. In our design, by exploiting the repeatability of elements in the symmetric matrix, the matrix can expand the Euclidean distance between signals and effectively restrains the interference between signals, thus improving the reliability of maritime VLC systems. In addition, we derive the theoretical symbol error rate (SER) of the proposed scheme. Then, simulation results are provided to validate the theoretical SER. In addition, the SER performances of the proposed system are compared with the three benchmark schemes of repetitive coding (RC), spatial modulation (SM), and spatial multiplexing (SMP), and the transmission performances in VLC systems with different link distances, water qualities, and wind speeds are also investigated. The proposed symmetric matrix-assisted MIMO maritime VLC system can combat interferences effectively and enhance reliability performance. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration and a significant modification when calculating severely clustered eigenvectors. The numerical results show that this method is competitive with other existing methods, especially when computing part eigenvectors or severely clustered ones. [ABSTRACT FROM AUTHOR]

The reciprocal distance Laplacian matrix of a connected graph G is defined as R D L (G) = R T (G) − R D (G) , where R T (G) is the diagonal matrix of reciprocal distance degrees and R D (G) is the Harary matrix. Clearly, R D L (G) is a real symmetric matrix, and we denote its eigenvalues as λ 1 (R D L (G)) ≥ λ 2 (R D L (G)) ≥ ... ≥ λ n (R D L (G)) . The largest eigenvalue λ 1 (R D L (G)) of R D L (G) , denoted by λ (G) , is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree R T m a x , minimum reciprocal distance degree R T m i n , and Harary index H (G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ (G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread R D L S (G) = λ 1 (R D L (G)) − λ n − 1 (R D L (G)) in terms of various graph parameters. We determine the extremal graphs in many cases. [ABSTRACT FROM AUTHOR]

This paper studies the consensus control problem of multiple high-speed trains with nonlinear running resistance based on a distributed event-triggered control protocol. Firstly, the multiple high-speed trains are established as a multi-agent model with a leader–follower. For the system whose leader has an unknown input, the effective state observers are introduced to observe the state of the train. Then, an observer-based event-triggered strategy is designed to determine the time when the train state is transmitted to the local controller and adjacent trains. The proposed control scheme is based on local information, and the solutions of the state observer and the gain symmetric matrices parameters of the distributed event-triggered control strategy are constructed in the form of linear matrix inequalities (LMI). Through the Lyapunov stability theory, sufficient conditions for the stable and safe operation of multiple high-speed trains systems are obtained. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the trajectory tracking control problem for underactuated underwater vehicles, for which a model is expressed in terms of quasi-velocities arising from the inertia matrix decomposition. The control approach takes into account non-modeled dynamics and external disturbances and is suitable for symmetric vehicles. It is shown that such systems can be diagonalized using inertial quasi-velocities (IQVs). The strategy consists of the velocity controller and two adaptive integral sliding mode control algorithms. The proposed approach, introducing velocity transformation and using backstepping methods and integral sliding mode control, allows trajectory tracking for vehicles in described models with symmetric inertia matrix. Proof of the stability of the closed system was carried out using IQV. The proposed scheme has been verified on two 3 DOF models of underwater vehicles with thruster limitations. A brief discussion of the results is also given. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the construction of certain symmetric and nonsymmetric pentadiagonal matrices from particular spectral information. The construction of the symmetric pentadiagonal matrix considers the extreme eigenvalues of its leading principal submatrices and a prescribed entry, and the construction of the nonsymmetric pentadiagonal matrix also considers an eigenvector and two prescribed entries. [ABSTRACT FROM AUTHOR]

The irreducibility of nonnegative matrices is an important condition for the diagonal transformation algorithm to succeed. In this paper, we introduce zero symmetry to replace the irreducibility of nonnegative matrices and propose an improved diagonal transformation algorithm for finding the maximum eigenvalue without any partitioning. The improved algorithm retains all of the benefits of the diagonal transformation algorithm while having fewer computations. Numerical examples are reported to show the efficiency of the proposed algorithm. As an application, the improved algorithm is used to check whether a zero symmetric matrix is an H-matrix. [ABSTRACT FROM AUTHOR]

The embarrassingly parallel nature of the Bisection Algorithm makes it easy and efficient to program on a parallel computer, but with an expensive time cost when all symmetric tridiagonal eigenvalues are wanted. In addition, few methods can calculate a single eigenvalue in parallel for now, especially in a specific order. This paper solves the issue with a new approach that can parallelize the Bisection iteration. Some pseudocodes and numerical results are presented. It shows our algorithm reduces the time cost by more than 35–70% compared to the Bisection algorithm while maintaining its accuracy and flexibility. [ABSTRACT FROM AUTHOR]

By making use of prestarlike functions, we introduce in this paper a certain family of normalized holomorphic functions defined in the open unit disk, and we establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (2) and T 3 (1) for the functions belonging to this family. We also mention some known and new results that follow as special cases of our results. [ABSTRACT FROM AUTHOR]

Motor faults, especially mechanical faults, reflect eminently faint characteristic amplitudes in the stator current. In order to solve the issue of the motor current lacking effective and direct signal representation, this paper introduces a visual fault detection method for an induction motor based on zero-sequence current and an improved symmetric dot matrix pattern. Empirical mode decomposition (EMD) is used to eliminate the power frequency in the zero-sequence current derived from the original signal. A local symmetrized dot pattern (LSDP) method is proposed to solve the adaptive problem of classical symmetric lattice patterns with outliers. The LSDP approach maps the zero-sequence current to the ultimate coordinate and obtains a more intuitive two-dimensional image representation than the time–frequency image. Kernel density estimation (KDE) is used to complete the information about the density distribution of the image further to enhance the visual difference between the normal and fault samples. This method mines fault features in the current signals, which avoids the need to deploy additional sensors to collect vibration signals. The test results show that the fault detection accuracy of the LSDP can reach 96.85%, indicating that two-dimensional image representation can be effectively applied to current-based motor fault detection. [ABSTRACT FROM AUTHOR]

Many practical systems can be modeled in terms of uncertainties, which refer to the differences or errors between actual data and mathematical simulations. However, systems including slight uncertainties and exogenous disturbances may lead to the instability of those systems. Besides, the behavior of systems is preferable to investigate within a prescribed bound over a fixed time interval. Therefore, in this paper, we study a robust finite-time control of discrete-time linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance. A distinctive feature of this research is that the considered systems consist of finite-time bounded subsystems and finite-time unbounded subsystems. A class of quasi-alternative switching signals is validly designed to analyze the mechanism and switching behaviors of the systems among their subsystems. By utilizing a copositive Lyapunov–Krasovskii functional method combined with the slow mode-dependent average dwell time and the fast mode-dependent average dwell time switching techniques, new sufficient conditions containing several symmetric negative-definite matrices are derived to guarantee robust finite-time control of the systems. These results are applied to a water-quality controllability model in streams to the standard level. Finally, the consistent results between the theoretical analysis and the corresponding numerical simulations are shown. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetric matrix optimization problem arising in the process of unsupervised feature selection. By relaxing the orthogonal constraint, this problem is transformed into a constrained symmetric nonnegative matrix optimization problem, and an efficient algorithm is designed to solve it. The convergence theorem of the new algorithm is derived. Finally, some numerical examples show that the new method is feasible. Notably, some simulation experiments in unsupervised feature selection illustrate that our algorithm is more effective than the existing algorithms. [ABSTRACT FROM AUTHOR]

With a focus on the multivariable coupling characteristics in a cross-directional (CD) basis weight control system, we study the coupling characteristics of a CD control system and decoupling control, and we propose a novel multivariable interpolation decoupling control strategy and a real-time decomposition algorithm in this paper. Based on a model of the CD basis weight profile, a system non-square interaction matrix of high-dimensional data is analyzed by experimental studies and numerical simulation. Along the diagonal of the interaction matrix, the matrix block method is adopted to reduce the system dimension and convert it into a square system. A multivariable control system with high dimensionality is divided into several subsystems. For the high-dimensional Toeplitz symmetric subsystem with small-scale coupling characteristics, an interpolation decoupling algorithm is proposed. Then, a decoupling compensator with the structure of a symmetric Toeplitz matrix was obtained. Compared with the conventional diagonal decoupling matrix, the branch number of the new decoupling network is reduced from 2408 to 186, which realizes the fast decoupling of multivariable systems. The results were even better when we used a double size interaction matrix obtained by interpolation between actual values. By designing the diagonalized controller for the new decoupled system, a decouped CD control system for corrugated paper with a basis weight of 133 g/m2 was implemented in an actual project in a paper mill. [ABSTRACT FROM AUTHOR]

In the paper, orthogonal transforms based on proposed symmetric, orthogonal matrices are created. These transforms can be considered as generalized Walsh–Hadamard Transforms. The simplicity of calculating the forward and inverse transforms is one of the important features of the presented approach. The conditions for creating symmetric, orthogonal matrices are defined. It is shown that for the selection of the elements of an orthogonal matrix that meets the given conditions, it is necessary to select only a limited number of elements. The general form of the orthogonal, symmetric matrix having an exponential form is also presented. Orthogonal basis functions based on the created matrices can be used for orthogonal expansion leading to signal approximation. An exponential form of orthogonal, sparse matrices with variable parameters is also created. Various versions of orthogonal transforms related to the created full and sparse matrices are proposed. Fast computation of the presented transforms in comparison to fast algorithms of selected orthogonal transforms is discussed. Possible applications for signal approximation and examples of image spectrum in the considered transform domains are presented. [ABSTRACT FROM AUTHOR]

The elements of the bidiagonal decomposition (BD) of a totally positive (TP) collocation matrix can be expressed in terms of symmetric functions of the nodes. Making use of this result, and studying the relation between Wronskian and collocation matrices of a given TP basis of functions, we can express the entries of the BD of Wronskian matrices as the values of certain symmetric functions evaluated at a single node. Moreover, in the case of polynomial bases, we obtain compact formulae for the entries of the BD of their Wronskian matrices. Interesting examples illustrate the applications of the obtained formulae. [ABSTRACT FROM AUTHOR]

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme's eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set. [ABSTRACT FROM AUTHOR]

Event extraction is a complex and challenging task in the field of information extraction. It aims to identify event types, triggers, and argument information from the text. In recent years, overlapping event extraction has attracted the attention of researchers because of its higher challenge and practicability, and some work has carried out in-depth research on overlapping event extraction and achieved remarkable results. But these works (1) ignore the role of ontology knowledge in event extraction; (2) use the same semantic encoding for multi-stage models, lacking consideration for the independent characteristics of extraction tasks such as event types, triggers, and arguments; and (3) face issues in the training process of multi-stage models, such as error cascading and slow convergence. To address the above issues, we propose an ontology-guided and scheduled-sampling approach for overlapping event extraction, termed as OGSS. First, we design a symmetric matrix for event ontology knowledge representation and integrate it into the semantic encoding process, infusing ontology knowledge into event extraction. Second, for extraction targets such as event types, triggers, and arguments, we process the semantic encoding according to the characteristics of each extraction target, obtaining semantic representations tailored for each subtask. Finally, we view multi-stage predictions as sequential outputs of a joint model, using a scheduled sampling strategy between subtasks to effectively mitigate the cascading propagation of errors during training and accelerate model convergence. We conduct extensive experiments on the FewFc event extraction benchmark dataset. The results show that OGSS achieves significant improvements in overlapping event extraction tasks compared to previous methods. [ABSTRACT FROM AUTHOR]

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Recently, community detection has emerged as a prominent research area in the analysis of complex network structures. Community detection models based on non-negative matrix factorization (NMF) are shallow and fail to fully discover the internal structure of complex networks. Thus, this article introduces a novel constrained symmetric non-negative matrix factorization with deep autoencoders (CSDNMF) as a solution to this issue. The model possesses the following advantages: (1) By integrating a deep autoencoder to discern the latent attributes bridging the original network and community assignments, it adeptly captures hierarchical information. (2) Introducing a graph regularizer facilitates a thorough comprehension of the community structure inherent within the target network. (3) By integrating a symmetry regularizer, the model's capacity to learn undirected networks is augmented, thereby facilitating the precise detection of symmetry within the target network. The proposed CSDNMF model exhibits superior performance in community detection when compared to state-of-the-art models, as demonstrated by eight experimental results conducted on real-world networks. [ABSTRACT FROM AUTHOR]

Tree-like structures, characterized by hierarchical relationships and power-law distributions, are prevalent in a multitude of real-world networks, ranging from social networks to citation networks and protein–protein interaction networks. Recently, there has been significant interest in utilizing hyperbolic space to model these structures, owing to its capability to represent them with diminished distortions compared to flat Euclidean space. However, real-world networks often display a blend of flat, tree-like, and circular substructures, resulting in heterophily. To address this diversity of substructures, this study aims to investigate the reconstruction of graph neural networks on the symmetric manifold, which offers a comprehensive geometric space for more effective modeling of tree-like heterophily. To achieve this objective, we propose a graph convolutional neural network operating on the symmetric positive-definite matrix manifold, leveraging Riemannian metrics to facilitate the scheme of information propagation. Extensive experiments conducted on semi-supervised node classification tasks validate the superiority of the proposed approach, demonstrating that it outperforms comparative models based on Euclidean and hyperbolic geometries. [ABSTRACT FROM AUTHOR]

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite. [ABSTRACT FROM AUTHOR]

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties. [ABSTRACT FROM AUTHOR]

The real world can be characterized as a complex network sto in symmetric matrix. Community discovery (or community detection) can effectively reveal the common features of network groups. The communities are overlapping since, in fact, one thing often belongs to multiple categories. Hence, overlapping community discovery has become a new research hotspot. Since the results of the existing community discovery algorithms are not robust enough, this paper proposes an effective algorithm, named Two Expansions of Seeds (TES). TES adopts the topological feature of network nodes to find the local maximum nodes as the seeds which are based on the gravitational degree, which makes the community discovery robust. Then, the seeds are expanded by the greedy strategy based on the fitness function, and the community cleaning strategy is employed to avoid the nodes with negative fitness so as to improve the accuracy of community discovery. After that, the gravitational degree is used to expand the communities for the second time. Thus, all nodes in the network belong to at least one community. Finally, we calculate the distance between the communities and merge similar communities to obtain a less- undant community structure. Experimental results demonstrate that our algorithm outperforms other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the orthogonal double covers (ODC) of K n , n as follows. The circular intensely orthogonal double cover design (CIODCD) of X = K n , n , ... , n ︸ m is defined as a collection T = { G 0 0 , G 1 0 , ... , G (n − 1) 0 } ∪ { G 0 1 , G 1 1 , ... , G (n − 1) 1 } of isomorphic spanning subgraphs of X such that every edge of X appears twice in the collection T , E (G i 0 ) ∩ E (G j 0 ) = E (G i 1 ) ∩ E (G j 1 ) = 0 , i ≠ j and   E (G i 0 ) ∩ E (G j 1 ) = λ = m 2 , i , j ∈ ℤ n. We define the half starters and the symmetric starters matrices as constructing methods for the CIODCD of X. Then, we introduce some results as a direct application to the construction of CIODCD of X by the symmetric starters matrices. [ABSTRACT FROM AUTHOR]

This article discusses the stability problem of sampled-data-based dynamic positioning ships (DPSs) using Takagi–Sugeno (T-S) fuzzy models. Firstly, dynamic equations for sampled-data DPSs are established. Simultaneously combining several symmetric matrices with new integral terms, a novel Lyapunov–Krasovskii function (LKF) is constructed, which allows the information of a sampling pattern to be fully captured. Next, via the constructed LKF, the positive definiteness requirements of a LKF are further relaxed, and the conservatism of the result can be reduced. Consequently, stability criteria are given, and fuzzy sampled-data controllers are designed in terms of linear matrix inequality (LMI). Finally, a simulation example is provided to verify the superiority and applicability of the developed methods. [ABSTRACT FROM AUTHOR]

We investigate a class of challenging general semidefinite programming problems with extra nonconvex constraints such as matrix rank constraints. This problem has extensive applications, including combinatorial graph problems, such as MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints. A customized approach based on the alternating direction method of multipliers (ADMM) is proposed to solve the general large-scale nonconvex semidefinite programming efficiently. We propose two reformulations: one using vector variables and constraints, and the other further reformulating the Burer–Monteiro form. Both formulations admit simple subproblems and can lead to significant improvement in scalability. Despite the nonconvex constraint, we prove that the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this matrix form, when the matrix factors are wide enough, the local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation. [ABSTRACT FROM AUTHOR]

This paper presents a unique integrated approach to meter placement and state estimation to ensure the network observability of active distribution systems. It includes observability checking, minimum measurement utilization, network state estimation, and trade-off evaluation between the number of real measurements used and the accuracy of the estimated state. In network parameter estimation, observability assessment is a preliminary task. It is handled by data analysis and filtering followed by calculation of the triangular factors of the singular, symmetric gain matrix using an algebraic method. Usually, to cover the deficiency of essential real measurements in distribution systems, huge numbers of virtual measurements are used. These pseudo measurements are calculated values, which are based on the network parameters, real measurements, and forecasted load/generation. Due to the application of a huge number of pseudo-measurements, large margins of error exists in the calculation phase. Therefore, there is still a high possibility of having large errors in estimated states, even though the network is classified as being observable. Hence, an integrated approach supported by forecasting is introduced in this work to overcome this critical issue. Finally, estimation of the trade-off in accuracy with respect to the number of real measurements used has been evaluated in order to justify the method's practical application. The proposed method is applied to a Danish network, and the results are discussed. [ABSTRACT FROM AUTHOR]