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Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce the readers, especially the researchers, to some topics connected with functions of matrix argument, scaling models, distributions of products and ratios, Bayesian structures, symmetric products and symmetric ratios of matrices, scalar and matrix-variate fractional integrals, functions of matrix argument through entropy optimization, singular matrix-variate gamma and beta functions etc which are currently active so that interested readers can get into these classes of problems for their current research or teaching. Let X be a p × q, p = q matrix of rank p in the real domain. If the function f(X), associated with X, is a function of XX', where a prime denotes the transpose, then such a function appears in a number of different disciplines. This paper examines the recent developments in such matrix-variate functions when X is in the real or complex domain. Connections to Bayes procedures, quantum physics, scalar and matrix texture models in communication and engineering problems, fractional integrals, distributions of symmetric products and symmetric ratios of matrices, singular matrix-variate gamma and beta functions and other related areas are pointed out. Only an overview of the current research in these topics with some illustrative examples are given in this paper. Since the material is summarized from the author's own works, most of the references are author's own papers, and hence similarity index, similarity with author's own works, may be high. The materials also cover some current results which are being published. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [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]

Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative 'decimation' procedure that was recently introduced by Camilli and Mézard, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. The main advantage of decimation is that its theoretical performance can be studied using statistical physics techniques. In this paper we provide the general analysis applying to a large class of compactly supported priors and we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. We then study the decimation phase diagrams in two concrete cases, the sparse Ising prior and the uniform prior, for which we show that matrix factorization is theoretically possible when the ration P / N of rank to variable is below a certain threshold. This suggest that a possible route to solving algorithmically the matrix factorization problem could be to find efficient decimation algorithms; in this respect, we show that simple simulated annealing, though being effective for limited signal sizes, is not scalable. [ABSTRACT FROM AUTHOR]

We propose a robust randomized indicator method for the reliable detection of eigenvalue existence within an interval for symmetric matrices A. An indicator tells the eigenvalue existence based on some statistical norm estimators for a spectral projector. Previous work on eigenvalue indicators relies on a threshold which is empirically chosen, thus often resulting in under or over detection. In this paper, we use rigorous statistical analysis to guide the design of a robust indicator. Multiple randomized estimators for a contour integral operator in terms of A are analyzed. In particular, when A has eigenvalues inside a given interval, we show that the failure probability (for the estimators to return very small estimates) is extremely low. This enables to design a robust rejection indicator based on the control of the failure probability. We also give a prototype framework to illustrate how the indicator method may be applied numerically for eigenvalue detection and may potentially serve as a new way to design randomized symmetric eigenvalue solvers. Unlike previous indicator methods that only detect eigenvalue existence, the framework also provides a way to find eigenvectors with little extra cost by reusing computations from indicator evaluations. Extensive numerical tests show the reliability of the eigenvalue detection in multiple aspects. [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]

Symmetric non-negative matrix factorization (NMF) can naturally capture the embedded clustering structure in the graph representation. It is an important method for linear and nonlinear data clustering applications. However, it is sensitive to the initialization of variables, and the quality of the initialization matrix greatly affects the clustering performance. In semi-supervised clustering, it faces the challenge of learning a more discriminative representation from limited labeled data. This paper introduces a constrained propagation self-adaptive self-supervised non-negative matrix factorization clustering algorithm (CPS³NMF) to solve the above problems. The algorithm propagates finite constraints to unconstrained data points, constructing a similarity matrix imbued with constraint information. The resultant similarity matrix serves the role of a non-negative symmetric matrix decomposition in SNMF and functions as graph regularization for the assignment matrix, fully utilizing the limited constraint information to preserve the geometrical structure of data space. Concurrently, leveraging the sensitivity of initial features in SNMF, the algorithm employs adaptively learned weights to rank the quality of multiple initial matrices. By integrating results from multiple clustering attempts, it progressively enhances the performance of semi-supervised clustering. Experiments on 6 public datasets show that the proposed CPS³NMF algorithm outperforms other state-of-the-art algorithms, proving its effectiveness in semi-supervised clustering. [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]

In this paper, we present a statistical approach to evaluate the relationship between variables observed in a two-factors experiment. We consider a three-level model with covariance structure Σ ⊗ Ψ 1 ⊗ Ψ 2 , where Σ is an arbitrary positive definite covariance matrix, and Ψ 1 and Ψ 2 are both correlation matrices with a compound symmetric structure corresponding to two different factors. The Rao's score test is used to test the hypotheses that observations grouped by one or two factors are uncorrelated. We analyze a fermentation process to illustrate the results. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper, we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given integer m ∈ N , we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in C m (R d) and have linearly independent integer shifts. As by-products, our results resolve convergence, smoothness, and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property. [ABSTRACT FROM AUTHOR]

The aim of the current paper is to study a partially described inverse eigenvalue problem of a specific symmetric matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by the minimal eigenvalue of all leading principal submatrices and eigenpair (λ (n) 2, x) such that λ (n) 2 is the maximal eigenvalue of the required matrix. We investigate conditions for the solvability of the problem, and finally an algorithm and its numerical results are presented. [ABSTRACT FROM AUTHOR]

For a DC microgrid with constant power loads (CPLs), the existence of a feasible power-flow solution, which is usually difficult to analyze, is a necessary condition for the correct operation of systems. In this paper, the solvability of power-flow equation of DC microgrid with CPLs is analyzed, where a majority of distributed generations (DGs) are under MPPT control while other DGs are under droop control. At first, this paper builds a power-flow mathematical model of the DC microgrid. Secondly, three solvability conditions are proposed with the first one being applied to ensure the power-flow equation has a solution, while the other two applying to ensure the solution within the given voltage deviation. The first analytical condition () is obtained by applying Brouwer fixed-point theorem. Compared with the existing results, this obtained sufficient condition is less conservative. Furthermore, the other two sufficient solvability conditions (and) can guarantee the equilibrium not only exists, but also the voltage deviation is within an acceptable range. Finally, case studies verify the correctness of the proposed theorems. The obtained conditions provide a guidance for establishing a dependable DC microgrid. [ABSTRACT FROM AUTHOR]

The parameterized block splitting (PBS) is a convergent and efficient iterative method to solve the large complex symmetric linear systems. In this paper, by using PBS iterative technique, the Newton equation is approximately solved, then we establish the modified Newton-PBS iterative method to solve the complex nonlinear systems whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, the local convergence analysis are explored under appropriate conditions. Ultimately, we apply the new method and several known methods to experimental numerical examples, and experimental results verify the superiority and efficiency of our new method. Especially, in terms of CPU time and iteration steps, our method is obviously better. [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]

In this paper, several existing results related to secondary transpose are critically reviewed and a result analogous to spectral decomposition theorem is obtained for a real secondary symmetric matrix. Noting that Moore–Penrose inverse with reference to secondary transpose involution, namely s -g inverse, need not always exist, we explore a few necessary sufficient conditions for the existence of such Moore–Penrose inverse. Further, we provide expressions and determinantal formula to compute the same. [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, we prove the existence of infinitely many solutions for the following nonperiodic fractional Hamiltonian system t D ∞ α (- ∞ D t α u) (t) + L (t) u (t) = ∇ W (t , u (t)) , t ∈ R u ∈ H α (R) , where - ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order 1 2 < α < 1 on the whole axis, respectively, L ∈ C (R , R N 2) is a symmetric matrix valued function unnecessary coercive and W (t , x) ∈ C 1 (R × R N , R) is only locally defined and superquadratic near the origin with respect to x. Our results extend and improve some existing results in the literature. [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]

Similar to cells, bacteria, and other micro-organisms, synthetic chemically active colloids can harness the energy from their environment through a surface chemical reaction and use the energy to self-propel in fluidic environments. In this paper, we study the chemo-mechanical coupling that leads to the self-propulsion of chemically active colloids. The coupling between chemical reactions and momentum transport is a consequence of Onsager reciprocal relations. They state that the velocity and the surface reaction rate are related to mechanical and chemical affinities through a symmetric matrix. A consequence of Onsager reciprocal relations is that if a chemical reaction drives the motion of the colloid, then an external force generates a reaction rate. Here, we investigate Onsager reciprocal relations for a spherical active colloid that catalyzes a reversible surface chemical reaction between two species. We solve the relevant transport equations using a perturbation expansion and numerical simulations to demonstrate the validity of reciprocal relations around the equilibrium. Our results are consistent with previous studies and highlight the key role of solute advection in preserving the symmetry of the Onsager matrix. Finally, we show that Onsager reciprocal relations break down around a nonequilibrium steady state, which has implications for the thermal fluctuations of the active colloids used in experiments. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [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]

Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is e-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm O (epsilon). We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data. [ABSTRACT FROM AUTHOR]

Estimation of variance components is essential in data analysis based on a linear mixed model. Among different estimation methods, the most commonly used in practice are likelihood-based approaches: maximum likelihood and restricted maximum likelihood. These methods do not guarantee positive-unbiased estimators, however. The method of moments ensures unbiasedness but may yield negative estimates, which can cause problems when estimating fixed effects by generalized least squares. The paper develops an algorithm to derive strictly positive unbiased estimators with minimum variance in a well-defined class. The key idea is to take the method-of-moments estimator, given by a quadratic form of a symmetric matrix A , as a starting point and modify it using the class of square non-singular regularization matrices Q while preserving unbiasedness in addition to ensuring positivity. Different subclasses of structured Q are possible for convenience instead of all possible Q matrices. A search algorithm then finds a local or global optimal matrix A depending on Q and the corresponding optimal variance component estimator by minimizing the variance, a function of the unknown variance components and kurtosis parameters in the model. Availability of an optimal matrix A means that the theoretical and numerical properties of the estimator can be conveniently studied, also in small samples. The proposed method further allows finding matrices A leading to quadratic forms closely approximating the corresponding numerical values of the likelihood-based estimates. The paper investigates the dependence of variance functions on the unknown model parameters. Using two illustrative examples, Examples I and II, the paper also illustrates the use of the matrix Q and the determination of the kurtosis parameters in the search for the optimal variance component estimators by keeping the bias zero and the variance trim for the bias–variance trade-offs. [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 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]

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 픤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras). [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]

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]

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied. However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f , which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies. [ABSTRACT FROM AUTHOR]

The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In the paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model rk=-pk(|rk|)rk-∑mak,mrm, where rk∈Rd, k=1,...,n, and A={ak,m} is a symmetric positive-semidefinie matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than 1/z, that is, pk(z)z→∞ as z→∞. We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when ker(A) is trivial, the positions of the particles are also ultimately bounded. For systems with ker(A)≠{0}, we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto ker(A). We also discuss the ultimate boundedness for systems with bounded coupling, including the Morse potential systems, and systems governed by power-law potentials with strong repulsion properties. We show that the former systems are ultimately bounded in the velocity-acceleration domain, whereas the models based on the power-law potentials are not. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem of a graph G is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of G. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al. in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form M = A ⊕ B and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices. [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]

In this paper, we give an algorithm that finds an ϵ -approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices. [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]