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Although Selberg-type single positive definite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6]. [ABSTRACT FROM AUTHOR]

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]

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]

In this paper, we analytically derive closed-form expressions for the tangency portfolio weights: the fully invested portfolio that maximizes the expected return over the risk-free rate, relative to the volatility of the portfolio return. We explicitly derive this portfolio from a range of underlying return models and show examples where it coincides with different well-known smart beta products. Specifically, we find the closed-form expression for the tangency portfolio weights for a return model with compound symmetric correlation matrix. We also deduce the tangency portfolio weights for the CAPM return model and illustrate in a case study that the estimated tangency portfolio weights may distinctly deviate from the market value weighted portfolio. Furthermore, we show that depending on the return model, the tangency portfolio weights may take a diverse set of shapes; from very diversified to highly concentrated portfolios. [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]

With the abundance of data, machine learning applications engaged increased attention in the last decade. An attractive approach to robustify the statistical analysis is to preprocess the data through clustering. This paper develops a low-complexity Riemannian optimization framework for solving optimization problems on the set of positive semidefinite stochastic matrices. The low-complexity feature of the proposed algorithms stems from the factorization of the optimization variable $\mathbf {X}=\mathbf {Y}\mathbf {Y}^{\mathrm{T}}$ X = Y Y T and deriving conditions on the number of columns of $\mathbf {Y}$ Y under which the factorization yields a satisfactory solution. The paper further investigates the embedded and quotient geometries of the resulting Riemannian manifolds. In particular, the paper explicitly derives the tangent space, Riemannian gradients and Hessians, and a retraction operator allowing the design of efficient first and second-order optimization methods for the graph-based clustering applications of interest. The numerical results reveal that the resulting algorithms present a clear complexity advantage as compared with state-of-the-art euclidean and Riemannian approaches for graph clustering applications. [ABSTRACT FROM AUTHOR]

In this paper, we construct and prove the existence of theoretical solutions to non-isentropic Euler equations with a time-dependent linear damping and Coriolis force in Cartesian form. New exact solutions can be acquired based on this form with examples presented in this paper. By constructing appropriate matrices , and vectors , special cases of exact solutions, where entropy , are obtained. This is the first matrix form solution of non-isentropic Euler equations to the best of the authors' knowledge. [ABSTRACT FROM AUTHOR]

In this paper, based on the Takagi–Sugeno fuzzy model approach, a novel output feedback control strategy is developed for a class of continuous-time nonlinear time delay systems via modified repetitive control scheme. The main objective of this paper is to design a fuzzy rule-based modified repetitive controller such that the system output can track any given periodic reference signal even in the presence of periodic disturbances. For this purpose, by employing fuzzy-dependent Lyapunov–Krasovskii functional together with the free-weighting matrix technique, a new set of sufficient conditions is presented in terms of linear matrix inequalities to guarantee the robust stability of the resulting closed-loop system. Based on these conditions and using internal model principle, it is shown that the system output tracks the given reference input within the desired error. Finally, two application-oriented examples are presented to validate the obtained theoretical results in which the proposed control design scheme gives better performance over the existing control scheme. [ABSTRACT FROM AUTHOR]

The paper deals with the new design procedure for PID-D robust controller with state/output derivative feedback to stabilize the closed-loop system using linear matrix inequality (LMI) and LMI regions with common in the frame of $H_2$ performance quadratic cost functions known as $(Q,S,R)$. The designed robust controller ensures robust properties of the closed loop system for linear time invariant and linear time variant gain scheduled plant descriptor systems. The robust controller design procedure is based on the extended original derivative of Lyapunov function and to obtain the convex conditions with respect to uncertainties, gain scheduled variables, and ensures that proposed new method guarantees less conservativeness to the design procedure we have introduced using the auxiliary matrices. The obtained results should be utilized to design power system stabilizer for synchronous generators and so on. The obtained robust PI-D controller with output and state/output derivative feedback for uncertain polytopic systems ensures that all closed-loop eigenvalues are lying in the defined LMI region. Two examples illustrate the effectiveness of the new design procedure. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a switching for 2-designs, which defines a type of trade. We illustrate this method by applying it to some symmetric (64, 28, 12) designs, showing that the switching introduced in this paper in some cases can be applied directly to orbit matrices. In that way we obtain six new symmetric (64, 28, 12) designs. Further, we show that this type of switching (of trades) can be applied to any symmetric design related to a Bush-type Hadamard matrix and construct symmetric designs with parameters (36, 15, 6) leading to new Bush-type Hadamard matrices of order 36, and symmetric (100, 45, 20) designs yielding Bush-type Hadamard matrices of order 100. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear random matrix approach is used to analyze the management of swimming place waters, and in this way, an intelligent management model is designed and applied to the actual swimming place waters management. Firstly, some basic small deviation results of the random matrix are presented. Then, several types of small deviation inequalities are obtained for the maximum eigenvalues of independent stochastic semi-positive definite (PSD) matrices. These small deviation inequalities are independent of the matrix dimension, and the results apply to high-dimensional and even infinite-dimensional matrices. For the inverse eigenvalue problem of higher-order doubly random matrices, the conclusion that two symmetric doubly random matrices are combined into a higher-order symmetric doubly random matrix is proved. In other words, the method of constructing new symmetric random matrices using smaller matrices with known spectra is applied to the inverse eigenvalue problem of higher-order symmetric random matrices, and sufficient conditions for the existence of the solution of the inverse eigenvalue problem of higher-order random matrices are derived, avoiding the discussion of the parity of the number of eigenvalues. This paper focuses on realistic management, to explore the standardization of safety management of swimming venues, and is based on modern management theory, risk management theory and methods, using research integration method, fieldwork method, questionnaire survey method, expert interview method, and other research methods, detailing the methods and conclusions of risk identification in safety management of swimming venues, and further exploring the safety management standards of swimming venues on this basis. And, combined with risk management theory, depending on the probability of occurrence and the degree of harm of risk, four basic disposal methods such as risk avoidance, risk control, risk transfer, and risk self-retaining are proposed. [ABSTRACT FROM AUTHOR]

The article discusses papers published within the issue, including "On the Block Triangular Form of Symmetric Matrices," by Iain Duff and Bora Ucar and "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization," by Benjamin Recht, Maryann Fazel and Pablo Parrilo.

This paper is concerned with the design of fault detection for sampled-data systems, which are based on multirate sampling, with frequency specifications. A general multirate system is considered in this paper, where not only the inputs and outputs but also their different channels have different sampling rates. The purpose of this paper is to make this residual system with multirate sampling satisfy a given disturbance attenuation level over a restricted frequency range. With the use of the lifting technique, this paper reformulates a single-rate linear time-invariant system, which is equivalent to the multirate time-varying system. For a given restricted frequency range, convex conditions are obtained in designing a required fault detection filter. Then, the restricted frequency ranges problem are also solved specifically via the generalized Kalman–Yakubovic–Popov lemma. Finally, this paper uses a continuous-stirred tank reactor system to illustrate the effectiveness and advantages of the fault detection filter design method. [ABSTRACT FROM AUTHOR]

This paper deals with the issue of reliable sliding mode control for descriptor systems with semi-Markov parameters using the Takagi–Sugeno fuzzy model, in which an average dwell time approach is utilized to tackle generic uncertain transition rates (TRs). From the analysis of the inner mechanism of switching singular system, a continuous sliding surface function is proposed. Different from continuity with probability one for stochastic systems, the absolute continuity of system solution is a key point in this paper being ensured for application of the average dwell time approach. Then, the mean-square exponential stability of the obtained sliding mode is analyzed based on two types of uncertain TRs. Moreover, a sliding mode controller is constructed to ensure the reachability condition in finite time. Lastly, the method is verified numerically by a single-link robot arm model. [ABSTRACT FROM AUTHOR]

The adaptive sliding mode control (SMC) problem is addressed for singular semi-Markovian jump systems (S-MJSs) against actuator attacks, in which the transition rates rely on the random sojourn time and are not constant, and the system states are unavailable. Moreover, the vulnerability of control signals transmitted via communication network means that the actuators may receive the attacked control signals. For the sake of reducing the effect of actuator attacks, the neural network technique is used to approximate the false information injected by adversaries. Meanwhile, a sliding mode observer is introduced to estimate the unmeasured states. An adaptive SMC law is proposed to guarantee that the estimation states and errors can reach to the sliding surfaces, and the stochastic admissibility of the singular S-MJSs can be ensured. In the end, an example is applied to illustrate the method in this paper. [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]

In this paper, we investigate bounds on eigenvalues of real symmetric interval matrices. We present a method that computes bounds on eigenvalues of real symmetric interval matrices. It outperforms many methods developed in the literature and produces as sharp as possible bounds on eigenvalues of real symmetric interval matrices. The aim is to apply the proposed method to compute lower bounds on eigenvalues of a symmetric interval hessian matrix of a nonconvex function in the αBB method and use them to produce a tighter underestimator that improves the αBB algorithm for solving global optimization problems. In the end, we illustrate by example, the comparison of various approaches of bounding eigenvalues of real symmetric interval matrices. Moreover, a set of test problems found in the literature are solved efficiently and the performances of the proposed method are compared with those of other methods. [ABSTRACT FROM AUTHOR]

In this paper we expressed the eigenvalues of a sort of heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From the prescribed eigenvalues, we computed eigenvectors for these types of matrices, giving also a formula not dependent on any unknown parameter for its determinant and inverse. Potential applications of the results are still provided. [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]

In this paper, we have studied various types of Monte Carlo methods along with the Power method to evaluate the maximum and minimum eigenvalue of a linear system of equations. We have studied how the accuracy of the maximum eigenvalue depends on the parameters, (moves in Markova chain), (no of Markova chain), (accelerating parameter), and a parameter m (the power applied on the resolving matrix). We have applied these methods to the randomly chosen symmetric matrices. We have also made comparisons for the different matrices of different orders depending on the parameters by using the Monte Carlo methods. We are using Matlab 2020R for the calculation. [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]

Aiming at the problem of long average delay of traditional UAV emergency communication, a performance optimization method of UAV communication link based on emergency communication environment is proposed. The target tasks in the emergency communication channel are divided into different target task groups. According to the distance between the target tasks in the target task group, the symmetric matrix is constructed, all elements in the symmetric matrix are classified, and the classification results are calculated by ant colony algorithm to construct the emergency communication task scheduling model of UAV. Combined with the emergency communication task scheduling model and the demand, priority, time window, and communication ability of UAV emergency scheduling task, the time delay control of UAV emergency communication is realized with the goal of maximizing the efficiency of UAV task execution, maximizing the completion rate of UAV task execution and minimizing the risk of UAV delay. The experimental results show that the average delay of the three methods also increases. When the system load is 50%, the average delay of common methods is 23 ms, the average delay of traditional methods is 76 ms, the average delay of the method proposed in this paper is 26 ms, and the common average delay is the shortest. When the system load is 99%, the common methods cannot realize emergency communication scheduling due to system load overshoot, and the corresponding average delay cannot be considered, indicating that the common methods are unstable. The average delay of the method proposed in this paper is stable under different system loads and always remains below 40 ms. It is proved that compared with the traditional methods, the method proposed in this paper takes less time for UAV emergency communication and has less average delay under different system loads, so it has better application value. [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]

This paper is concerned with the event-triggered finite-time control problem for networked switched linear systems by using an asynchronous switching scheme. Not only the problem of finite-time boundedness, but also the problem of input-output finite-time stability is considered in this paper. Compared with the existing event-triggered results of the switched systems, a new type of event-triggered condition is proposed. Sufficient conditions are established to guarantee the event-based asynchronous closed-loop systems are both finite-time bounded and input-output finite-time stable. A set of event-triggered finite-time bounded and input-output finite-time stabilizing controllers are designed under the asynchronous control scheme. It is revealed that the triggered thresholds determine the number of sampling points transmitted to the controller, and the smaller triggered parameters indicate the less-sampled data needed to be transmitted to the controller under the event-triggered scheme. Finally, a boost converter circuit is applied to bring out the advantages of the proposed control scheme. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of reducing the conservatism for stabilization of discrete-time Takagi–Sugeno fuzzy systems via a new extended representation approach. Different from previous slack variable methods, an advanced slack variable method without limitations is proposed for further making good use of available fuzzy weighting functions. Owing to the introduction of extended representations for homogenous polynomials, those strong limitations in recent literature are relaxed and thus the primary and secondary sequences of all normalized fuzzy weighting functions can be mapped to one augmented multi-indexed matrix for each homogeneous polynomial. Therefore, less conservative stabilization conditions can be provided as a result of these effective measures and thus a part of the online computational cost is transferred to the offline implementation. Finally, the benefits of the proposed method are illustrated by means of numerical experiments. [ABSTRACT FROM AUTHOR]

This paper considers the $\mathscr{L}_{2}$ string stability analysis and controller synthesis for vehicle platooning under jamming attacks and interferences from other communication channels. Jamming attacks and interferences for vehicle platooning systems are modeled as periodic denial of service attacks. Further, the vehicle platooning with denial of service attacks is modeled as a distributed switching system where an average dwell time approach is adopted for the stability analysis. Under the distributed switching system framework, this paper presents a co-design paradigm for the vehicle platooning. The distributed event-triggered method (ETM) and controllers are presented to guarantee $\mathscr{L}_{2}$ string stability, which can ensure that the output cannot be amplified along the vehicle platooning. Sufficient conditions for $\mathscr{L}_{2}$ string stability are then provided to guide the co-design of the ETM and controllers, which can be efficiently obtained by solving LMIs. Simulation results are presented to demonstrate the efficacy of the proposed co-design approach in addressing the denial of service attacks for vehicle platooning systems. [ABSTRACT FROM AUTHOR]

Non-Euclidean data is increasingly used in practical applications. As a typical representative, Symmetric Positive Definite (SPD) matrices can form a Riemannian manifold rather than a flat linear space. Hence, constructing a dimensionality reduction (DR) algorithm for SPD data directly on manifolds will encounter difficulties in modeling and solving. This paper proposes a novel DR technique based on Riemannian Manifold Tangent Spaces and Local Affinity for SPD data (RMTSLA-SPDDR). Our primary contributions are listed below: (1) We transfer the data from SPD manifolds to tangent spaces, which are all Euclidean spaces. Besides, the tangent spaces are symmetric matrix spaces, which indicates the proposed method greatly preserves the data form and properties; (2) Under the Affine-Invariant Riemannian Metric (AIRM), we incorporate tangent spaces and log transformation, so as to keep the geodesic distance between SPD data and the identity matrix equal to the corresponding Euclidean distance between the transformed data and the origin of the tangent space; (3) The method adopts the local affinity criterion to determine the bilinear transformation between tangent spaces, and take it as the transformation for SPD data consequently. There are no similar reports as the technique presented in this paper before, which means it is a new attempt. On five benchmark datasets, abundant experimental results indicate RMTSLA-SPDDR outperforms the other five state-of-the-art DR methods. [ABSTRACT FROM AUTHOR]

We investigate the matrix function optimization under the orthonormal constraint on the matrix variable. By introducing an index-notation-arrangement-based chain rule (I-Chain rule), we obtain the gradient of the cost function and propose a revisited orthonormal-constraint-based projected gradient method to locate a minimum of an objective/cost function of matrix variables iteratively subject to orthonormal constraint. To guarantee the convergence the proposed method, existing schemes require the gradient can be represented by the multiplication of a symmetrical matrix and the matrix variable itself. This condition has been relaxed in this paper. New techniques are proposed to establish the convergence property of the iterative algorithm. Simulation results show the effectiveness of our framework. This paper allows more extensive applications of matrix function optimization problems in science and engineering. [ABSTRACT FROM AUTHOR]

Transformations between abc, stationary dq0 ($\alpha \beta 0$) and rotating dq0 reference-frames are used extensively in the analysis and control of three-phase technologies such as machines and inverters. Previous work on deriving the matrices describing these transformations follows one of two approaches. The first approach derives Clarke's matrix by modifying symmetrical components. Park's matrix can be subsequently found from a rotation matrix. The second approach derives Park's matrix using trigonometric projection by interpreting the transformation as a rotation in the plane of the cross-section of a machine. Then, Clarke's matrix can be found trivially using a reference angle of zero in Park's matrix. This paper presents a third approach to deriving the Clarke and Park transformation matrices: a geometric interpretation. The approach exploits properties of the linear transformation using the Cartesian representation. We introduce the locus diagram of a three-phase quantity and show how these diagrams have applications in power quality. We show that, unlike a phasor diagram, a single locus diagram can fully represent a three-phase system with harmonics. [ABSTRACT FROM AUTHOR]

This paper deals with the impulsive formation control of spacecraft in the presence of constraints on the position vector and time. Determining a set of path constraints can increase the safety and reliability in an impulsive relative motion of spacecraft. Specially, the feasibility problem of the position norm constraints is considered in this paper. Under assumptions, it is proved that if a position vector be reachable, then the reach time and the corresponding time of impulses are unique. The trajectory boundedness of the spacecraft between adjacent impulses are analyzed using the Gerschgorin and the Rayleigh–Ritz theorems as well as a finite form of the Jensen's inequality. Some boundaries are introduced regarding the Jordan–Brouwer separation theorem which are useful in checking the satisfaction of a constraint. Two numerical examples (approximate circular formation keeping and collision-free maneuver) are solved in order to show the applications and visualize the results. [ABSTRACT FROM AUTHOR]

Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a more general set of convex programs known as semi-definite and second-order cone programs. However, these methods are excessively slow for high dimensions. On the other hand, optimization algorithms on manifolds have shown great abilities in finding solutions to non-convex problems in a reasonable time. This paper suggests using a Riemannian optimization approach to solve a subset of convex optimization problems wherein the optimization variable is a doubly stochastic matrix. Optimization over the set of doubly stochastic matrices is crucial for multiple communications and signal processing applications, especially graph-based clustering. The paper introduces and investigates the geometries of three convex manifolds, namely the doubly stochastic, the symmetric, and the definite multinomial manifolds which generalize the simplex, also known as the multinomial manifold. Theoretical complexity analysis and numerical simulation results testify the efficiency of the proposed method over state-of-the-art algorithms. In particular, they reveal that the proposed framework outperforms conventional generic and specialized approaches, especially in high dimensions. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve the problems on simultaneous actuator and sensor fault estimations as well as the nonfragile fault-tolerant control (FTC) for a kind of Markovian jump systems with faults and disturbances. First, the considered system is converted to an augmented system by putting the sensor fault into the new state. Then, an adaptive observer is designed for the descriptor system with the actuator fault adjusted by the designed adaptive law. Based on the estimated actuator faults, an output-feedback-based FTC strategy is proposed to stabilize the closed-loop system against actuator and sensor faults, and disturbances, while showing robustness for the control gain perturbations. Sufficient conditions for the existences of the observer and controller are provided in forms of linear matrix inequalities. Finally, a practical application is given to express the validation and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the distributed linearly separable computation problem, which is a generalization of many existing distributed computing problems such as distributed gradient coding and distributed linear transform. A master asks ${\mathsf {N}}$ distributed workers to compute a linearly separable function of ${\mathsf {K}}$ datasets, which is a set of ${\mathsf {K}}_{\mathrm{ c}}$ linear combinations of ${\mathsf {K}}$ equal-length messages (each message is a function of one dataset). We assign some datasets to each worker in an uncoded manner, who then computes the corresponding messages and returns some function of these messages, such that from the answers of any ${\mathsf {N}}_{\mathrm{ r}}$ out of ${\mathsf {N}}$ workers the master can recover the task function with high probability. In the literature, the specific case where ${\mathsf {K}}_{\mathrm{ c}}=1$ or where the computation cost is minimum has been considered. In this paper, we focus on the general case (i.e., general ${\mathsf {K}}_{\mathrm{ c}} $ and general computation cost) and aim to find the minimum communication cost. We first propose a novel converse bound on the communication cost under the constraint of the popular cyclic assignment (widely considered in the literature), which assigns the datasets to the workers in a cyclic way. Motivated by the observation that existing strategies for distributed computing fall short of achieving the converse bound, we propose a novel distributed computing scheme for some system parameters. The proposed computing scheme is optimal for any assignment when ${\mathsf {K}}_{\mathrm{ c}}$ is large and is optimal under the cyclic assignment when the numbers of workers and datasets are equal or ${\mathsf {K}}_{\mathrm{ c}}$ is small. In addition, it is order optimal within a factor of 2 under the cyclic assignment for the remaining cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better understanding of the strain constraint space, which is important to the Navier–Stokes regularity problem. In particular, we give a full characterization the orthogonal complement of the strain constraint space and investigate the geometry of the eigenvalue distribution of matrices in the strain constraint space. [ABSTRACT FROM AUTHOR]

Link prediction aims to predict missing links or eliminate spurious links by employing known complex network information. As an unsupervised linear feature representation method, matrix factorization (MF)-based autoencoder (AE) can project the high-dimensional data matrix into the low-dimensional latent space. However, most of the traditional link prediction methods based on MF or AE adopt shallow models and single adjacency matrices, which cannot adequately learn and represent network features and are susceptible to noise. In addition, because some methods require the input of symmetric data matrix, they can only be used in undirected networks. Therefore, we propose a deep manifold matrix factorization autoencoder model using global connectivity matrix, called DM-MFAE-G. The model utilizes PageRank algorithm to get the global connectivity matrix between nodes for the complex network. DM-MFAE-G performs deep matrix factorization on the local adjacency matrix and global connectivity matrix, respectively, to obtain global and local multi-layer feature representations, which contains the rich structural information. In this paper, the model is solved by alternating iterative optimization method, and the convergence of the algorithm is proved. Comprehensive experiments on different real networks demonstrate that the global connectivity matrix and manifold constraints introduced by DM-MFAE-G significantly improve the link prediction performance on directed and undirected networks. [ABSTRACT FROM AUTHOR]

Non-homogeneous real-order systems often arise in the construction of equations arising in many inequality systems. This paper considers non-homogeneous linear systems attached to random initial time and associated with Caputo real-order operators and time-dependent elements in standard and block matrix forms. Is it possible to find the convergence and estimate bounds of such systems? We use the idea of comparison methodology along with establishing the nonnegativity of such systems when the coefficient matrix is Metzler and the non-homogeneous term becomes nonnegative. Using the hypothesis on the existence of a constant Metzler matrix via construction, we prove that non-trivial solutions to such non-homogeneous systems converge to 0 under some reasonable conditions. We use the generalized Laplace transform technique as a tool to establish some reasonable conditions. In short, we prove that, if every root of a characteristic equation associated with such a constant Metzler matrix lies in the open left half complex plane and the non-homogeneous term approaches 0 as time tends to ∞ , then the non-trivial solution to the original system must be globally asymptotically convergent to 0. This result is then extended to the non-homogeneous real-order systems associated with coefficient block matrix form. We do not know the bounds and rate decay of the solutions to such systems. As a typical result, we prove that, if there is a constant symmetric Metzler matrix Δ M + with maximum eigenvalue λ max negative and the matrix Δ M + - λ max I is negative semi-definite, then there exist different measures for bounds of solutions to such systems. A particular case of the above-mentioned result gives rise to the new concept of γ -Mittag–Leffler asymptotic stability in the absence of a non-homogeneous input term and it thus provides γ -Mittag–Leffler decay rate. We provide an application to an electrical circuit system that illustrates the novel significance of some proposed theoretical results. [ABSTRACT FROM AUTHOR]

The motto of the present paper is to study the generalized symmetric metric connection on f-Kenmotsu manifolds. We discuss the properties of f-Kenmotsu manifolds with generalized symmetric metric connection. Further, we investigate conservative pseudo-projective curvature tensor and conservative quasi-conformal curvature tensor on f-Kenmotsu manifolds. [ABSTRACT FROM AUTHOR]

The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. This paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features: 1) they are completely tractable, analytically, or numerically, when dealing with large covariance matrices; 2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e., Fréchet or geometric mean); and 3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. This paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models. [ABSTRACT FROM AUTHOR]

Spike sorting technologies support neuroscientists to access the neural activity with single-neuron or single-action-potential resolutions. However, conventional spike sorting technologies perform the feature extraction and the clustering separately after the spikes are well detected. It not only induces many redundant processes, but it also yields a lower accuracy and an unstable result especially when noises and/or overlapping spikes exist in the dataset. To address these issues, this paper proposes a unified optimization model integrating the feature extraction and the clustering for spike sorting. Unlike the widely used combination strategies, i.e., performing the principal component analysis (PCA) for spike feature extraction and the K-means (KM) for clustering in sequence, interestingly, this paper finds the solution of the proposed unified model by iteratively performing PCA and KM-like procedures. Subsequently, by embedding the K-means++ strategy in KM-like initializing and a comparison updating rule in the solving process, the proposed model can well handle the noises and overlapping interference as well as enjoy a high accuracy and a low computational complexity. Finally, an automatic spike sorting method is derived after taking the best of the clustering validity indices into the proposed model. The extensive numerical simulation results on both synthetic and real-world datasets confirm that our proposed method outperforms the related state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

Neural networks have achieved remarkable successes in machine learning tasks. This has recently been extended to graph learning using neural networks. However, there is limited theoretical work in understanding how and when they perform well, especially relative to established statistical learning techniques such as spectral embedding. In this short paper, we present a simple generative model where unsupervised graph convolutional network fails, while the adjacency spectral embedding succeeds. Specifically, unsupervised graph convolutional network is unable to look beyond the first eigenvector in certain approximately regular graphs, thus missing inference signals in non-leading eigenvectors. The phenomenon is demonstrated by visual illustrations and comprehensive simulations. [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 note revisits the theoretical foundation of the Daum–Huang particle filter concepts for solving Bayesian updating problems. We reexamine the defining necessary and sufficient condition, in the form of an operator equation, for a flow to generate a particular homotopy between the a priori and the a posteriori probability density functions. We show that two well-known flows indeed satisfy this sufficient condition in the linear-Gaussian case, restating some of the significant results in a recent series of papers coauthored by Fred Daum with our alternative proof, which we hope will provide a useful perspective for future flow-based filter developments. [ABSTRACT FROM AUTHOR]