Matrix Inequalities Research Articles

A family of subfactors arising from a pair of complex Hadamard matrices

In this paper, we define a new equivalence relation ‘[Formula: see text]’ on the set of all Hadamard inequivalent complex Hadamard matrices of order [Formula: see text] and show that pairs [Formula: see text] of equivalent matrices [Formula: see text] produce an infinite family of potentially new subfactors of the hyperfinite type [Formula: see text] factor [Formula: see text]. All these subfactors are irreducible with the Jones index [Formula: see text], including all possibilities. We also show that this family contains infinitely many infinite-depth subfactors. As an application, we compute the Connes–Størmer relative entropy and the angle between the pair [Formula: see text] of spin model subfactors arising from the pair [Formula: see text] of equivalent matrices. On the other hand, pairs [Formula: see text] of inequivalent matrices [Formula: see text] lead to subalgebras of [Formula: see text] with infinite Pimsner–Popa index.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

Constrained solutions of generalized coupled discrete-time periodic matrix equations with application in state observer design for linear periodic systems

PurposeIt is desired to provide a diversified iterative scheme for solving the constrained solutions of the generalized coupled discrete-time periodic (GCDTP) matrix equations from the perspective of optimization.Design/methodology/approachThe paper considers generalized reflexive solutions of the GCDTP matrix equations by applying the Jacobi gradient-based iterative (JGI) algorithm, which is an extended variant of the gradient-based iterative (GI) algorithm.FindingsThrough numerical simulation, it is verified that the efficiency and accuracy of the JGI algorithm are better than some existing algorithms, such as the GI algorithm in Hajarian, the RGI algorithm in Sheng and the AGI algorithm in Xie and Ma.Originality/valueIt is the first instance in which the GCDTP matrix equations are solved applying the JGI algorithm.

Lateral Response Analysis of a Large‐Diameter Pile Under Combined Horizontal Dynamic and Axial Static Loads in Nonhomogeneous Soil

ABSTRACTThe large diameter piles are widely used in structures such as offshore wind turbines due to their superior lateral load‐bearing capacity. To explore the lateral response of a large‐diameter pile under combined horizontal dynamic and axial static loads in nonhomogeneous soil, a simplified analytical model of the pile–soil interaction is developed. This model represents the pile as a Timoshenko beam resting on the Pasternak foundation, incorporating the double‐shear effect by considering both pile and soil shear. The governing matrix equations for the pile elements are derived from the principle of virtual work. Further, the pile's lateral deformations and internal forces are obtained using the modified finite beam element method (FBEM) and then validated through existing analytical solutions. Finally, the contribution of various properties of pile, soil, and applied load to the pile's lateral vibration response are performed. It is found that both pile and soil shear effects significantly impact the lateral dynamic response of a large‐diameter pile. Additionally, in nonhomogeneous soil, decreasing surface soil strength and dimensionless frequency lead to increased lateral displacements and bending moments of the pile, which are significantly affected by the P‐Δ effect under increasing axial load.

Research on dynamic solution analysis and applications based on the Rydberg atom superheterodyne structure

This paper investigates the dynamic solution of the density matrix equation based on the Rydberg atom superheterodyne structure. Compared to the current analytical method relying on the steady-state solution, the dynamic solution is related to the Rabi frequency and the frequency of the signal to be measured. Therefore, it can comprehensively describe the instantaneous bandwidth and gain characteristics of the receiver and is in good agreement with experimental results. Additionally, we propose an atomic all-heterodyne receiver architecture that combines electric-field heterodyne and optical heterodyne techniques and demonstrates the reception and recovery of modulated signals under this architecture with linear frequency modulation (LFM) signals as an example. Our research offers interesting theoretical insights that can be applied to the performance analysis and system optimization of atomic receivers.

Exact finite-time correlation functions for multiterminal setups: Connecting theoretical frameworks for quantum transport and thermodynamics

Transport in open quantum systems can be explored through various theoretical frameworks, including the quantum master equation, scattering matrix, and Heisenberg equation of motion. The choice of framework depends on factors such as the presence of interactions, the coupling strength between the system and environment, and whether the focus is on steady-state or transient regimes. Existing literature mainly treats these frameworks independently. Our work establishes connections between them by clarifying the role and status of these approaches using two paradigmatic models for single and multipartite quantum systems in a two-terminal setup under voltage and temperature biases. We derive analytical solutions in both steady-state and transient regimes for the populations, currents, and current correlation functions. Exact results from the Heisenberg equation are shown to align with scattering matrix and master equation approaches within their respective validity regimes. Crucially, we establish a protocol for the weak-coupling limit, bridging the applicability of master equations at weak-coupling with Heisenberg or scattering matrix approaches at arbitrary coupling strength. Published by the American Physical Society 2024

Some Numerical Results of the Albedo Problem in Spherical Geometry with Linearly Anisotropic Scattering

We examine the effect of linear anisotropic scattering on the solution of spherical geometry transport problems. We consider the albedo problem of a solid homogeneous sphere with linearly anisotropic scattering and write two matrix equations to solve the albedo problem with the H N and SVD methods in pseudo slab geometry. These methods are based on Case method of singular eigenfunctions. The numerical results of both methods are in agreement with each other.

Impulse synchronization strategy for supply chains considering combined effects and demand saturation

Supply chains often face sudden changes in production, distribution, and consumption as a result of factors such as pandemics, Black Friday, and mass production by manufacturers; this is called the ”impulse phenomenon.” The effect of the impulse phenomenon on the supply chain system can be positive or negative. Understanding how to use the impulse phenomenon to synchronize supply chain systems and avoid chaos is therefore important for supply chain management. Existing studies mainly use continuous synchronization strategies as a means to avoid chaos in supply chain systems; however, a discrete impulse synchronization strategy is easier to implement. Considering combined effects of consumption,distribution,production and demand saturation, this study constructs a supply chain system composed of a manufacturer, distributor, and retailer. We then construct a supply chain system with uncertain parameters and input disturbance. Using Lyapunov stability theory and matrix inequality, we obtain the impulse synchronization strategies of two supply chain systems. Finally, taking China’s supply chain system of corn, wheat, and rice as an example, we compare goodness of fit. The production, distribution, and consumption of rice are controlled using a temporary storage purchase policy, batch ordering, and promotion activities, and the effectiveness of the impulse synchronization strategy is verified. Our results show the following: (1) Considering combined effects and demand saturation, the supply chain system has better goodness of fit and better impulse synchronization. (2) When a chaotic supply chain system deviates from its expected state, if impulse intensity and the time interval satisfy Theorem 1 or 2, the supply chain system can be restored to its expected state; if not, the synchronization effect might not be achieved. (3) Generally, when impulse intensity is low, the time interval is short, if impulse intensity is high, the time interval should be longer.

Assessing the influence of nonlinear mooring restoring forces in the optimization and design of FOWT

Abstract The design of mooring systems plays a pivotal role in the success of Floating Offshore Wind Turbine (FOWT) projects. This process naturally involves iterations, as design and analysis are intricately linked. Despite the inherent nonlinearity of mooring system restoring forces, many designers opt for employing an equivalent linear mooring system stifness matrix to expedite the optimization process in early design stages. This article aims to underscore the limitations associated with relying on the equivalent linear model when compared to accounting for nonlinear restoring forces during optimization processes. To illustrate this point, a case study was conducted using the reference semi-submersible platform VolturnUS-S. The study considered intermediate to large water depths and addressed three different mooring confgurations: catenary, semi-taut, and taut lines. The analysis focused on several critical aspects, including ofset watch circles, line tensions (including those at the anchors), and a cost estimate based on the different models. The findings indicate that, for catenary-based mooring systems, the linear model remains a good approximation, leading to no significant loss of accuracy in the context of early design stages. As the lines become tauter, however, the nonlinearities become more pronounced and the errors involved in the linear model can reach unacceptable levels.

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for solving tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES−BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES−BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D−BTF(m,k) -in which m is the restart parameter and k is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D−BTF(m,k) method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D−BTF(m) method. It is also important to note that FAd-SGMRES-D−BTF(m,k) represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES−BTF(m). Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES−BTF and FAd-SGMRES-D−BTF(m) to solve Sylvester tensor equations.

Tunable enhancement of the cylindrical Luneburg lens focusing ability with the aid of a conformal graphene strip

We consider the plane wave focusing characteristics of the layered cylindrical Luneburg lens equipped with a conformal strip of graphene, in the H-polarization case. The angular width and location of the strip is arbitrary, and its surface impedance is characterized with the aid of the quantum-physics Kubo formalism. We use a mathematically accurate full-wave analytical regularization technique, which is based on the explicit inversion of the problem static part and yields a Fredholm second-kind matrix equation. This guarantees the convergence of the resulting meshless numerical algorithm. We compute the focusing ability of a microsize lens as a function of the frequency in the wide range up to 60 THz. This analysis shows that a graphene strip, placed into the focal area of the Luneburg lens, enhances its focusing ability at the resonance frequency of the strip plasmon mode proportionally to the quality factor. This frequency is defined by the strip width and is tunable with the aid of graphene’s chemical potential.

Modeling and control of dynamical biomechanical arm systems with elastic joints sensitive to the effect of an external load

In this study, a biomechanical model mimicking the human hand-arm system under heavy disk excitation is developed to define the stability threshold between the preload vibration and the stress of the human hand-arm system. The fully electromechanical system, consisting of a DC motor, a transmission system, and three discrete masses representing the upper arm, forearm, and hand lifting a heavy disk, is connected by shock absorbers and various springs. The challenge of mimicking the angular activity of the elbow joint in torsion and flexion was also included to assess its contribution to system stability and to study the absorption of mechanical energy in the human hand-arm system. Finally, the movements of the model were described by a differential matrix equation, and its analysis facilitates the understanding of the threshold of the chaotic behaviors observed in an oscillating arms system. Specifically, it explores how the motion of a DC motor can be regulated using a single controller parameter within the context of a multi-degree-of-freedom human hand-arm system. The study uses numerical integrations to analyze system behaviors, with an emphasis on the use of frequency spectra, Poincaré maps, and bifurcation diagrams for visualization and interpretation. The results indicate that the system exhibits chaotic behavior when subjected to certain conditions, particularly when the mass load exceeds 35 kg. Imposing motion on the mass block further intensifies the chaos, leading to manifestations such as strong oscillations and frequency-synchronization effects. These characteristics, exhibited by the idealized model, which imitates the human body through a mechanical model and computation of the motions of the body, play a vital role in various fields of ergonomics and sports biomechanics. Understanding the dynamic behaviors of such systems, especially in response to varying conditions, has significant implications for engineering applications.

Coherent control of dressing interaction of sequential-doubly-dressed four-wave mixing

Abstract We investigate the coherent control of dressing interaction in four-wave mixing (FWM) in a Y-type four-level atomic system based on the dressed states theory. We study the density matrix equation acted by two dressing fields simultaneously governing FWM signal intensity and the eigenvalues under the dressed state formalism. It is shown that the peak position and intensity of Autler–Townes splitting both originate from the dressing interaction as the detuning of the probe field is scanned. Moreover, it is observed that the interaction could be controlled by the dressing fields. Such flexibly controlled nonlinear signal has potential applications in optical communication and quantum information processing.

Closed-loop continuous-time Laguerre-based subspace identification via orthogonal projection approach

This paper presents a closed-loop Laguerre-based subspace identification method for continuous-time systems via orthogonal projection approach. A bank of Laguerre filters operating in the all-pass domain, which is capable of handling the non-equidistant data, is used to get the input–output matrix equation of the systems. We put the emphasis on the parity space opposed to the observable subspace. The instrumental variable method and principal component analysis estimate the system consistently, solving the issue of biased estimate for the closed-loop operation of the system caused by the feedback controller. The simulation results demonstrate the effectiveness of the proposed approach.

A Beneficial Numerical Approach to Solve Systems of Linear Integro-Differential Equations

The system of linear Fredholm-Volterra integro-differential equations (FVIDEs) has been solved in this paper by an improved approximation method. Generalised Bernstein polynomials and collocation points have been used to construct the theory of the method. The aim of the technique is to reduce systems of integro-differential equations into an algebraic matrix equation, which corresponds to a linear algebraic equation system, by means of Bernstein polynomials. In order to analyse the applicability of the method, some illustrative examples have also been considered. It has been shown that the proposed method is faster and more effective than the others when comparing the numerical results.

Analysis of multi-phase lag gradients for a spherical cavity due to prescribed internal pressure

A refined multi-phase lag model is proposed here for a homogeneous unbounded thermoelastic spherical cavity in a curvilinear coordinate system. The associated solutions are obtained by forming non-dimensional vector matrix equations and applying Eigen value approach in the transform domain. The results have been verified using the valid boundary conditions associated with physical and field variables. Mathematica and MATLAB platforms are used for numerical analysis and graphical representation.

Reachable Set Estimation for Singular Discrete‐Time Nonlinear Systems With Time‐Varying Delays

ABSTRACTIn this article, the reachable set estimation problems for a class of singular discrete‐time nonlinear systems (SDNS) with time‐varying delay (TVD) are presented. First, we employ a Lyapunov function scheme and utilize zero‐type free matrix equations. Based on this, a criterion is derived in the form of linear matrix inequalities (LMIs) to guarantee that the considered system to be regular, causal, and the state trajectory of the system is bounded. Second, by developing the singular systems decomposition technique into slow and fast subsystems and using the introduced auxiliary lemma, we establish that the state trajectory of the fast subsystem is also bounded, and the reachable set of singular nonlinear system is bounded within the ellipsoid. Finally, Numerical examples are presented to illustrate the effectiveness of the proposed method.

Non-orthogonal unicast and multicast transmission with IRS-assisted Rate-Splitting Multiple Access in multiuser MISO systems

Abstract. In MISO systems, novel communication technologies such as NOMA and multicast transmission have attracted widespread attention to meet the diverse needs of users. However, traditional NOMA and multicast transmission schemes have certain limitations in dynamic interference management and resource allocation. To resolve this issue, this paper proposes a RSMA scheme assisted by phase quantized Intelligent Reflecting Surface (IRS) for downlink MISO systems with nonorthogonal transmission of unicast and multicast signals. In this scheme, through reconfiguring unicast and multicast signals and combining them into a super-common stream and partial unicast messages, RSMA achieves dynamic interference management of multi-user while effectively separating unicast and multicast streams. To maximize the weighted sum rate(WSR), we put forward an alternating optimization(AO) algorithm that utilizes Weighted Minimum Mean Square Error (WMMSE) estimation and convex set intersection representation, which jointly optimizes the equivalent beamforming matrix,rate allocation and phase shift matrix. Simulation results illustrate that the proposed single-layer RSMA exhibits performance closer to that of DPC encoding compared to traditional NOMA and MULP schemes. Furthermore, we extensively investigate the relationship between the WSR of RSMA and system parameters, providing important guidance for practical engineering applications.

Low‐rank approximations of frequency‐limited discrete‐time Gramians

AbstractWe consider the approximate computation of Gramians of large‐scale linear discrete‐time state‐space systems. Of particular interest are the Gramians which arise in frequency‐limited balanced truncation model order reduction. While there are many results for the continuous‐time situation, the discrete‐time scenario is usually considered to a much lesser extent. The Gramians are solutions of discrete‐time Lyapunov matrix equations (also called Stein matrix equations) which, if the reduction is restricted to smaller frequency intervals, also incorporate matrix functions in the form of matrix logarithms. We discuss the efficient numerical handling of these matrix function as well as the computation of low‐rank Gramian approximation by rational Krylov subspace and related iterative methods.

Multiscale Damage Identification Method of Beam-Type Structures Based on Node Curvature

This paper proposes a multiscale damage identification method for beam-type structures based on node curvature. Firstly, based on the assumption that micro-damage has little effect on stress redistribution and the basic relationship between structural bending moment and curvature, combined with the denoising function of wavelet analysis, the linear matrix equation before and after node curvature damage is solved using the singular value decomposition (SVD) method. Then, the theoretical feasibility of this method is verified with laboratory tests of a simply supported beam. Finally, the damage sensitivity and noise resistance of this method are verified using field measurements of a beam bridge. The results show that the nodal curvature serves as an indicator parameter for damage identification in beam-type structures, enabling the precise localization of damage within these structures. When utilizing a multiscale finite element model for analysis, the nodal curvature enhances the ability to identify both the location and severity of damage within small-scale elements. Furthermore, this method can provide a reference for the damage identification and health monitoring of other types of bridges.