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The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions { S (i,j)} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of { S (i,j)} can be derived. Some identities of { S (i,j)} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics. [ABSTRACT FROM AUTHOR]

A generalization of the lattice Kadomtsev–Petviashvili system associated with an elliptic curve, that is referred to as an elliptic integrable system, has been revisited by means of the Cauchy matrix scheme. Various types of explicit solutions are obtained, some of which offer new insights of both mathematical and physical significance. The construction of exact solutions to the elliptic lattice Kadomtsev–Petviashvili system is closely connected to that of a special Sylvester-type matrix equation. [ABSTRACT FROM AUTHOR]

Clean water is a necessity for many organisms, especially human life. Due to many factors, there is a significant shortage of potable water. This has led to efforts involving recovering water from wastewater or the sea through different technologies. Recently, the desalination of seawater via the reverse osmosis system has shown to be a promising method for drinking water treatment and recovery. Such a technique relies on mathematical models based on many parameters, resulting in special PDEs to model the reverse osmosis system. This paper develops a numerical method to solve a reverse osmosis model. The governing PDE is converted into a Sylvester equation that is proved to be uniquely solvable, stable, consistent, and convergent. The numerical scheme developed is validated with experimental data from the literature, and some numerical simulations. [ABSTRACT FROM AUTHOR]

Clean water is a necessity for many organisms, especially human life. Due to many factors, there is a significant shortage of potable water. This has led to efforts involving recovering water from wastewater or the sea through different technologies. Recently, the desalination of seawater via the reverse osmosis system has shown to be a promising method for drinking water treatment and recovery. Such a technique relies on mathematical models based on many parameters, resulting in special PDEs to model the reverse osmosis system. This paper develops a numerical method to solve a reverse osmosis model. The governing PDE is converted into a Sylvester equation that is proved to be uniquely solvable, stable, consistent, and convergent. The numerical scheme developed is validated with experimental data from the literature, and some numerical simulations.

In this paper, a new fuzzy Functional observer is developed for nonlinear neutral systems. Also, the existence conditions for it are studied. The delay-dependent stability of this observer is guaranteed by the combination of the solution of the Sylvester equation and the Lyapunov–Krasovskii stability approach. The parameters of the studied observer are obtained by solving linear matrix inequalities (LMIs). The performance of the approach developed in this paper is demonstrated at the end of the paper by numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses the dynamic quaternion-valued Sylvester equation (DQSE) using the quaternion real representation and the neural network method. To transform the Sylvester equation in the quaternion field into an equivalent equation in the real field, three different real representation modes for the quaternion are adopted by considering the non-commutativity of quaternion multiplication. Based on the equivalent Sylvester equation in the real field, a novel recurrent neural network model with an integral design formula is proposed to solve the DQSE. The proposed model, referred to as the fixed-time error-monitoring neural network (FTEMNN), achieves fixed-time convergence through the action of a state-of-the-art nonlinear activation function. The fixed-time convergence of the FTEMNN model is theoretically analyzed. Two examples are presented to verify the performance of the FTEMNN model with a specific focus on fixed-time convergence. Furthermore, the chattering phenomenon of the FTEMNN model is discussed, and a saturation function scheme is designed. Finally, the practical value of the FTEMNN model is demonstrated through its application to image fusion denoising. [ABSTRACT FROM AUTHOR]

We consider the Alternating Direction Implicit (ADI) method to compute the numerical solution of a continuous Sylvester equation A X + X B = C , based on the recently developed inexact ADI iteration, and we propose classical acceleration techniques to enhance its convergence rate. An extrapolated variant (EADI) and a block successive overrelaxation variant (block SOR-ADI) of the ADI iterative method are described. These relaxation approaches are similar to what is used in Gauss-Seidel and Jacobi methods for linear systems, and, to our knowledge, novel, especially the block SOR-ADI scheme. Convergence properties of these two relaxed variants are analyzed when the matrix A is positive definite and the matrix B is positive semi-definite (not necessarily Hermitian matrices), or conversely. Our numerical experiments suggest that these new schemes are computationally attractive. The convergence rate of the ADI method is usually increased, particularly with the block SOR-ADI variant. A comparison with the well-known Hermitian and skew-Hermitian splitting (HSS) method emphasizes the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

It is shown in a recently published work that the GNN (gradient-based neural network) model activated by the msbp (modified sign-bi-power) function exhibits superior fixed-time convergence for solving the Sylvester equation in the noise-free case. Encouraged by this point, in this paper, we study its robust performance when it is disturbed by three kinds of noises, i.e., dynamic bounded vanishing noise, dynamic bounded non-vanishing noise, and constant noise. Detailed mathematical analyses are conducted to show the robustness properties (e.g., convergence property, the upper bound of steady-state solution error) of the GNN model, which are further verified by two simulation examples. Ultimately, the GNN model is applied to the path tracking of a planar four-link redundant robot arm.

Hyperspectral imaging is able to provide a finer delivery of various material properties than conventional imaging systems. Yet in reality, an optical system can only generate data with high spatial resolution but low spectral one, or vice versa, at video rates. As a result, an issue that fuses low-resolution hyperspectral and high-resolution multispectral images has gained great attention. However, most fusion approaches depend purely on hand-crafted regularizers or data-driven priors, leading to the issues of tricky parameter selection or poor interpretability. In this work, a subspace-based deep prior regularization is proposed to tackle these problems, which takes both hand-crafted regularizer and data-driven prior into account. Specifically, we leverage the spectral correlation of the images and transfer them from the original space to the subspace domain, within which a modified U-net-based deep prior learning network (SDPL-net) is designed for the fusion issue. Moreover, instead of taking the output of SDPL-net directly as the result, we further feed the output back to the model-based optimization. Under such prior regularization, the recovered high-resolution hyperspectral image holds a high consistency to its inherent structure and hence tends to present enhanced reliability and accuracy. Experimental results on simulated and real data reveal that the proposed method excels other state-of-the-art methods in both quantitative and qualitative metrics.

For a linear matrix function f in X ∈ R m × n we consider inhomogeneous linear matrix equations f (X) = E for E ≠ 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution. [ABSTRACT FROM AUTHOR]

Given the matrix equation A X + X B + f (X) C = D in the unknown n × m matrix X , we analyze existence and uniqueness conditions, together with computational solution strategies for f : R n × m → R being a linear or nonlinear function. We characterize different properties of the matrix equation and of its solution, depending on the considered classes of functions f. Our analysis mainly concerns small dimensional problems, though several considerations also apply to large scale matrix equations. [ABSTRACT FROM AUTHOR]

Dynamical systems that involve non-linearity of the dynamics is a major challenge encountered in learning these systems. Similarly, the lack of adequate models for phenomena that reflect the governing physics can be an obstacle to an appropriate analysis. Nonetheless, some numerically or experimentally measured data can be found. Based on this data, and using a data-driven method such as the Loewner framework, it is possible to manage this data to derive a high fidelity reduced dynamical system that mimics the behaviour of the original data. In this paper, we tackle the issue of large amount of data presented by samples of transfer functions in a frequency-domain. The main step in this framework consists in computing singular value decomposition (SVD) of the Loewner matrix which provides accurate reduced systems. However, the large amount of data prevents this decomposition from being computed properly. We exploit the fact that the Loewner and shifted Loewner matrices, the key tools of Loewner framework, satisfy certain large scale Sylvester matrix equations. Using an extended block Krylov subspace method, a good approximation in a factored form of the Loewner and shifted Loewner matrices can be obtained and a minimal computation cost of the SVD is ensured. This method facilitates the process of a large amount of data and guarantees a good quality of the inferred model at the end of the process. Accuracy and efficiency of our method are assessed in the final section. [ABSTRACT FROM AUTHOR]

Biclustering is a powerful data mining technique that allows simultaneously clustering rows (observations) and columns (features) in a matrix-format data set, which can provide results in a checkerboard-like pattern for visualization and exploratory analysis in a wide array of domains. Multiple biclustering algorithms have been developed in the past two decades, among which the convex biclustering can guarantee a global optimum by formulating in as a convex optimization problem. On the other hand, the application of biclustering has not progressed in parallel with the algorithm techniques. For example, biclustering for increasingly popular microbiome research data is under-applied possibly due to its compositional constraints for each sample. In this manuscript, we propose a new convex biclustering algorithm, called the bi-ADMM, under general setups based on the ADMM algorithm, which is free of extra smoothing steps to visualize informative biclusters required by existing convex biclustering algorithms. Furthermore, we tailor it to the algorithm named biC-ADMM specifically to tackle compositional constraints confronted in microbiome data. The key step of our methods is to utilize the Sylvester Equation to derive the ADMM algorithm, which is new to the clustering research. The effectiveness of the proposed methods is examined through a variety of numerical experiments and a microbiome data application. [ABSTRACT FROM AUTHOR]

The problem of having an identical speed when dealing with multiple motors always exists in the industry. There are several methods to address the problem, but all the methodologies have two common drawbacks. Firstly, the control design requires continuous information on the desired speed and the actual speed of motors; secondly, it is sometimes difficult to directly measure the speed variables. In the proposed study, both of these drawbacks are addressed by designing an observer-based event-triggered networked multi-agent system. The proposed method uses the leader following consensus approach with a centralized event triggering control design so that whenever a follower's speed diverges from that of the leader, an event is triggered, which communicates and resets all the agents to the leader's speed. Moreover, an observer is designed such that the ith agent uses its jth neighbor agent and observer speed information to estimate the leader's speed. The stability of the proposed design is formulated by Lyapunov stability, while the simulation results endorse the design concepts and energy saving.

The Sylvester equation plays an important role in many branches of mathematical physics. The goal of this paper is to show that a special case of the Sylvester equation can be related to the lattice B-type Kadomtsev–Petviashvili (BKP) system via the generalized Cauchy matrix method. We use the variables given in the Sylvester equation to define the function and several scalar functions that are closely related to the lattice BKP equation. After rederiving the lattice BKP equation, we make it clear that besides its multisoliton solutions, various other types of exact solutions also exist. Furthermore, Lax pairs for the lattice BKP equation are obtained in different ways. [ABSTRACT FROM AUTHOR]

The Haar wavelet collocation with iteration technique is applied for solving a class of time-fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with iteration technique are compared with those obtained by analytical methods such as Adomian decomposition method (ADM) and variational iteration method (VIM). The results show that the present scheme is effective and appropriate for obtaining the numerical solution of the timefractional Modified Camassa-Holm equation and Time fractional Modified Degasperis-Procesi equation. [ABSTRACT FROM AUTHOR]

In many fields, the issue of solving the time-varying Sylvester equation (TVSE) is commonly encountered. Consequently, finding its exact solution has become a research hotspot. In general, the ZNN and IEZNN models are the most useful algorithms that are frequently utilized to solve the TVSE problem. However, the ZNN model is borned with noise susceptibility and the IEZNN model loses the adaptive performance due to its constant coefficient in solving the TVSE problem. In this paper, a residual error feedback zeroing neural network (REFZNN) is proposed to adaptively solve the TVSE problem. The REFZNN model feeds back the residual error to the solustion system, which forms a feedback regulation to reduce the residual error between the system output and the system target. Then, the convergence and noise patience of the REFZNN model are proved by theoretical analyses. Finally, the validity of the proposed model is verified by designing computer simulation experiments and its superiority is confirmed by the performance comparisons with the ZNN and IEZNN models.

This research work is devoted to boundary control of a catalytic cracking reactor using state and error feedback regulators. The process mathematical model is governed by a set of partial differential equations (PDEs), for which infinite-dimensional representation and spectral properties of the system generator are employed to solve the regulation problems. The primary objective is to track a desired output reference trajectory in the presence of disturbances that are generated by a distributed parameter exosystem. Initially, a state feedback stabilizing regulator is designed to drive the process output towards the reference trajectory. The second objective is to develop a dynamic controller that employs the tracking error as input. The closed-loop plant is shown to be exponentially stable and the tracking error asymptotically approaches zero. The performances of the designed regulators are shown through numerical simulations. • The state and error-feedback regulators have been designed for a cracking reactor PDEs model. • The system is under disturbances that are generated by infinite-dimensional exosystem. • The spectral approach is implemented to solve the associated Sylvester operator. • Robustness of the error-feedback regulator has been investigated. [ABSTRACT FROM AUTHOR]

This work is devoted to solve the state and error-feedback control problems for a catalytic reverse flow reactor (CFRR), which is modeled by nonlinear partial differential equations (PDEs). These two regulation problems will be solved based on the linearized infinite-dimensional representation. The objective is to track a desired output reference under the presence of disturbances. Both the reference trajectory and the disturbance profiles are generated by a distributed parameter exosystem. First, a state feedback stabilizing regulator is designed which drives the process output to a reference trajectory. The second main aim is to develop a dynamical controller that uses the tracking error as an input. Furthermore, it has been demonstrated that the closed-loop plant is exponentially stable and the tracking error asymptotically goes to zero. The developed regulators are evaluated through numerical simulations for the case study of methane combustion. • Description of the reverse flow reactor by PDEs model with its infinite-dimensional representation. • Design of a state-feedback regulator for the CFRR model under the presence of distributed disturbances. • Design of an error-feedback regulator for the CFRR model under the presence of distributed disturbances. • Test the performances of the designed regulators through numerical simulations for methane combustion. [ABSTRACT FROM AUTHOR]

This paper aims at finding a fixed-time solution to the Sylvester equation by using a gradient neural network (GNN). To reach this goal, a modified sign-bi-power (msbp) function is presented and applied on a linear GNN as an activation function. Accordingly, a fixed-time convergent GNN (FTC-GNN) model is developed for solving the Sylvester equation. The upper bound of the convergence time of such an FTC-GNN model can be predetermined if parameters are given regardless of the initial conditions. This point is corroborated by a detailed theoretical analysis. In addition, the convergence time is also estimated utilizing the Lyapunov stability theory. Two examples are then simulated to demonstrate the validation of the theoretical analysis, as well as the superior convergence performance of the presented FTC-GNN model as compared to the existing GNN models. [ABSTRACT FROM AUTHOR]

In this paper, some necessary and sufficient solvability conditions are established for the generalized Sylvester operator equations AXB-CXD = E, AXB-CYD = E and AX + YB+CZ = E on Hilbert spaces, respectively. Moreover, we give a solvability condition for the *-Sylvester operator equation AX -X*B = C, which holds for finite matrices due to Wimmer (1994). [ABSTRACT FROM AUTHOR]

Exponential time integrators are well‐established discretization methods for time semilinear systems of ordinary differential equations. These methods use φ−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the φ−function action over vectors for two‐dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential‐related matrix function that we express using Kronecker products of one‐dimensional matrices. We exploit state‐of‐the‐art implementations of φ−functions to compute this auxiliary function's action and then recover the original φ−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

Hyperspectral image (HSI) super-resolution is commonly used to overcome the hardware limitations of existing hyperspectral imaging systems on spatial resolution. It fuses a low-resolution (LR) HSI and a high-resolution (HR) conventional image of the same scene to obtain an HR HSI. In this work, we propose a method that integrates a physical model and deep prior information. Specifically, a novel, yet effective two-stream fusion network is designed to serve as a regularizer for the fusion problem. This fusion problem is formulated as an optimization problem whose solution can be obtained by solving a Sylvester equation. Furthermore, the regularization parameter is simultaneously estimated to automatically adjust contribution of the physical model and the learned prior to reconstruct the final HR HSI. Experimental results on both simulated and real data demonstrate the superiority of the proposed method over other state-of-the-art methods on both quantitative and qualitative comparisons. [ABSTRACT FROM AUTHOR]

The split-step Padé approach is an extremely efficient tool for the integration of pseudodifferential parabolic equations, which are widely used for the modeling of acoustic wave propagation. In this study, a generalization of this method to the case of parabolic equations with unknown vector functions is presented. Such generalization requires an algorithm for efficient numerical evaluation of a function of a matrix with differential operators as its elements. After a finite-difference discretization this algorithm reduces to the solution of several Sylvester-like problems. The generalized split-step Padé method presented here is an attractive tool for solving 3D problems of sound propagation in the ocean within the framework of coupled mode parabolic equations theory. • A method is proposed for the numerical solution of the one-way counterpart of a system of coupled Helmholtz equations. • The method is based on a generalization of the split-step Padé algorithm to the case of unknown vector functions. • The method is an attractive tool for handling many practical problems where the propagation of coupled guided modes takes place. [ABSTRACT FROM AUTHOR]

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation L M are bold, please chekc if bold neceaasry, if not, please remove all bold of equation − M K = r s T together with an evolution equation set of r and s , master function S (i , j) = s T K j C (I + M C) − 1 L i r is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ -function by τ = | I + M C |. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed. [ABSTRACT FROM AUTHOR]

This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method. [ABSTRACT FROM AUTHOR]

Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M ∈ A n × n , M ˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M ∈ C n × n , σ (M) ⊂ C denotes the set of eigenvalues of M. It is shown that if A ∈ A n × n and B ∈ A m × m are such that for all φ ∈ M A , σ (A ˆ (φ)) ∩ σ (B ˆ (φ)) = ∅ , then for all C ∈ A n × m , the Sylvester equation A X − X B = C has a unique solution X ∈ A n × m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra. [ABSTRACT FROM AUTHOR]

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case n = 4 , the proofs for the case n = 5 and the case of arbitrary dimension n. The results can be used in symbolic computation. [ABSTRACT FROM AUTHOR]

We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + ... + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, ... , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of large scale singular (continuous‐time) Lyapunov equations of the form AX + XA⊤ + BB⊤ = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low‐rank approximation to the least squares solution. Selected illustrative numerical examples are provided. [ABSTRACT FROM AUTHOR]

In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.

In this paper, a parametric approach to design a Luenberger functional observer for linear time-varying (LTV) systems with time-delay is investigated. Based on the solution to generalized Sylvester equation (GSE), the complete general parametric expressions for the functional observer gain matrices are established with the time-varying coefficient matrices, the time-varying closed-loop system and a group of arbitrary parameters. With the parametric method, the observation error system can be transformed into a linear system with the expected eigenstructure. Finally, a numerical simulation is provided to illustrate the effectiveness of the parametric approach.

We investigate the factorized solution of generalized stable Sylvester equations such as those arising in model reduction, image restoration, and observer design. Our algorithms, based on the matrix sign function, take advantage of the current trend to integrate high performance graphics accelerators (also known as GPUs) in computer systems. As a result, our realisations provide a valuable tool to solve large-scale problems on a variety of platforms. [ABSTRACT FROM AUTHOR]

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration. [ABSTRACT FROM AUTHOR]

This paper aims at finding a fixed-time solution to the Sylvester equation by using a gradient neural network (GNN). To reach this goal, a modified sign-bi-power (msbp) function is presented and applied on a linear GNN as an activation function. Accordingly, a fixed-time convergent GNN (FTC-GNN) model is developed for solving the Sylvester equation. The upper bound of the convergence time of such an FTC-GNN model can be predetermined if parameters are given regardless of the initial conditions. This point is corroborated by a detailed theoretical analysis. In addition, the convergence time is also estimated utilizing the Lyapunov stability theory. Two examples are then simulated to demonstrate the validation of the theoretical analysis, as well as the superior convergence performance of the presented FTC-GNN model as compared to the existing GNN models.

Sylvester equation is often applied to various fields, such as mathematics and control systems due to its importance. Zeroing neural network (ZNN), as a systematic design method for time-variant problems, has been proved to be effective on solving Sylvester equation in the ideal conditions. In this paper, in order to realize the predefined-time convergence of the ZNN model and modify its robustness, two new noise-tolerant ZNNs (NNTZNNs) are established by devising two novelly constructed nonlinear activation functions (AFs) to find the accurate solution of the time-variant Sylvester equation in the presence of various noises. Unlike the original ZNN models activated by known AFs, the proposed two NNTZNN models are activated by two novel AFs, therefore, possessing the excellent predefined-time convergence and strong robustness even in the presence of various noises. Besides, the detailed theoretical analyses of the predefined-time convergence and robustness ability for the NNTZNN models are given by considering different kinds of noises. Simulation comparative results further verify the excellent performance of the proposed NNTZNN models, when applied to online solution of the time-variant Sylvester equation. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the large scale singular Sylvester equation of the form A X − X D ⊤ = E (or A X B ⊤ − C X D ⊤ = E), where the spectra Λ (A) and Λ (D) (or, Λ (A − λ C) and Λ (D − λ B)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided. [ABSTRACT FROM AUTHOR]

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.

The aim of this paper is to study some features of slice semi-regular functions SEM (Ω) on a circular domain Ω contained in the skew-symmetric algebra of quaternions H via the analysis of a family of linear operators built from left and right ⁎-multiplication on SEM (Ω) ; this class of operators includes the family of Sylvester-type operators S f , g. Our goal is achieved by a strategy based on a matrix interpretation of these operators as we show that SEM (Ω) can be seen as a 4-dimensional vector space on the field SEM R (Ω). We then study the rank of S f , g and describe its kernel and image when it is not invertible, finding meaningful differences in the cases when the rank is either 2 or 3. By using these results, we are able to characterize when the functions f and g are either equivalent under ⁎-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. In this way, information about the operator obtained by linear algebra techniques give as a significant application the solution of a problem in an area of function theory which had a remarkable development in the last decade (see [16]). We also provide a complete classification of idempotents and zero divisors on product domains of H. [ABSTRACT FROM AUTHOR]

In this article, a neural dynamics model is constructed and investigated for solving time-dependent Sylvester equation with matrix inversion involved in the solving process. Besides, to eliminate the matrix inversion in the model, the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno method is leveraged to construct a new model. Moreover, the global convergence performance and the effectiveness of the two discrete computational models are testified by providing theoretical analyses and numerical experiments with comparisons to the existing solutions, respectively. Two applications to robotics and the multiple-input multiple-output system are given to elucidate the feasibility of the proposed models for solving time-dependent Sylvester equation. [ABSTRACT FROM AUTHOR]

For a division ring F , the polynomials f ∈ F [ z ] can be evaluated "on the left" and "on the right" giving rise to left and right Lagrange interpolation problems. The problems containing interpolation conditions of the same type were considered in [7] where the solvability criterion was given in terms of polynomial independence of interpolation nodes. We establish the solvability criterion and describe all solutions of low degree (less than the number of interpolation conditions imposed) for the problem containing both "left" and "right" conditions. [ABSTRACT FROM AUTHOR]

Recursive blocked algorithms have proven to be highly efficient at the numerical solution of the Sylvester matrix equation and its generalizations. In this work, we show that these algorithms extend in a seamless fashion to higher-dimensional variants of generalized Sylvester matrix equations, as they arise from the discretization of PDEs with separable coefficients or the approximation of certain models in macroeconomics. By combining recursions with a mechanism for merging dimensions, an efficient algorithm is derived that outperforms existing approaches based on Sylvester solvers. [ABSTRACT FROM AUTHOR]