Function Matrix Research Articles

On the Lanczos Method for Computing Some Matrix Functions

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions f(A)β and βTf(A)β when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results.

STEWART PLATFORM MULTIDIMENSIONAL TRACKING CONTROL SYSTEM SYNTHESIS

Context. Creating guaranteed competitive motion control systems for complex multidimensional moving objects, including unstable ones, that operate under random controlled and uncontrolled disturbing factors, with minimal design costs, is one of the main requirements for achieving success in this class devices market. Additionally, to meet modern demands for the accuracy of motion control processes along a specified or programmed trajectory, it is essential to synthesize an optimal control system based on experimental data obtained under conditions closely approximating the real operating mode of the test object. Objective. The research presented in this article aims to synthesize an optimal tracking control system for the Stewart platform’s working surface motion, taking into account its multidimensional dynamic model. Method. The article employs a method of a multidimensional tracking control system structural transformation into an equivalent stabilization system for the motion of a multidimensional control object. It also utilizes an algorithm for synthesizing optimal stabilization systems for dynamic objects, whether stable or not, under stationary random external disturbances. The justified algorithm for synthesizing optimal stochastic stabilization systems is constructed using operations such as addition and multiplication of polynomial and fractional-rational matrices, Wiener factorization, Wiener separation of fractional-rational matrices, and the calculation of dispersion integrals. Results. As a result of the conducted research, the problem of defining the concept of analytical design for a Stewart platform’s optimal motion control system has been formalized. The results include the derived transformation equations from the tracking control system to the equivalent stabilization system of the Stewart platform’s working surface motion. Furthermore, the structure and parameters of the main controller transfer function matrix for of this control system have been determined. Conclusions. The justified use of the analytical design concept for the Stewart platform’s working surface optimal motion control system formalizes and significantly simplifies the solution to the problem of synthesizing complex dynamic systems, applying the developed technology presented in [1]. The obtained structure and parameters of the Stewart platform’s working surface motion control system main controller, which is divided into three components W1, W2, and W3, improve the tracking quality of the program signal vector, account for the cross-connections within the Stewart platform, and increase the accuracy of executing the specified trajectory by increasing the degrees of freedom in choosing the controller structure

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Stability analysis of aperiodic impulsive delay systems using sum of squares approach

AbstractThe stability problem of aperiodic impulsive delay systems with unstable continuous and discrete dynamics is addressed in this article. New delay‐dependent stability conditions are proposed by employing a clock‐dependent Lyapunov–Krasovskii‐like functional composed of a Lyapunov–Krasovskii functional and a high‐order polynomial matrix function. The stability conditions are then solved using sum of squares programming techniques, yielding less conservative results than the looped‐functionals approach. Finally, the simulation results demonstrate the effectiveness of the results.

Synthetic aperture image enhancement with near-coinciding Nonuniform sampling case

Multireceiver synthetic aperture sonar (SAS) uses multiple receivers to collect echoed signals, leading to nonuniform sampling in the azimuth dimension if the moving distance between adjacent pings is not half the receiver array length. The filter bank reconstruction (FBR) method, based on matrix inversion, is commonly used to address this by reconstructing uniform data from nonuniform signals. However, in practice, near-coinciding sampling can occur due to inaccuracies in the towed velocity of the SAS system, influenced by ocean conditions. The signal correlation of these near-coinciding receivers is very strong, and thus the steering vectors corresponding to these receivers are not completely independent. This further leads to an ill-conditioned system transfer function matrix made up of the receiver steering vectors, and results in inaccurate calculations of inverse matrix. Consequently, the FBR method suffers from significant signal-to-noise ratio loss or fails to reconstruct uniform signals accurately. This degradation in signal quality directly impacts the accuracy of target reconstruction, leading to errors in identifying and locating targets. This paper quantitatively defines nonuniform sampling with near-coinciding samples and discusses performance loss in various cases. To enhance imaging performance, we propose a method for reconstructing uniform signals. Simulation results demonstrate that the proposed method outperforms the conventional FBR method, providing better target reconstruction and higher efficiency.

Enhanced Looped Lyapunov Functional for Sampled-Data Control for T-S Fuzzy Systems with Time Delay

This paper addresses the problem of sampled-data control in T-S fuzzy systems with time delays. Initially, a negative-definite criterion is developed for the matrix containing membership functions with the help of the idea of a switching mechanism. Building upon this criterion, a membership-function-based looped Lyapunov functional is introduced. Furthermore, a sampling-time-dependent looped Lyapunov functional is constructed, incorporating crucial information about the sampling instance. Notably, this functional matrix is not constrained to satisfy positive-definiteness, and it varies with each sampling point. By utilizing the enhanced looped Lyapunov functional, less conservative stability conditions and a new controller criterion for the T-S fuzzy system with time delays are derived. This demonstrates that our proposed stability conditions allow for a larger sampling interval compared to existing work. Finally, the proposed method’s effectiveness is demonstrated through the validation of two simulation examples.

κ-Fractional Derivative Operator and Solution of Fractional Kinetic Equations Involving (κ,𝔍)- Second Appell Hypergeometric Matrix Functions with the two Variables

κ-Fractional Derivative Operator and Solution of Fractional Kinetic Equations Involving (κ,𝔍)- Second Appell Hypergeometric Matrix Functions with the two Variables

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.

Prediction of aviation safety incident risk level based on ECSDNN

Prediction of aviation safety incident risk level is an important means of active risk management. Considering the high-dimensional complexity and class imbalance of massive aviation safety incident data, a prediction method for aviation safety incident risk level based on integrated cost-sensitive deep neural network (ECSDNN) is proposed. The feature representation of aviation safety incident data is realized by using the method of embedded coding of categorical attributes and splicing of numerical attributes; the cost-sensitive matrix and cost-sensitive loss function are designed by comprehensively considering the misclassification ratio and fixed cost, and the base classifier model based on cost-sensitive deep neural network (CSDNN) is constructed; the hard voting method is used to integrate multiple base classifiers with different parameters and performances to construct the integrated prediction model ECSDNN for aviation safety incident risk level. The experimental results on the Aviation Safety Incident Reporting System (ASRS) dataset show that compared with the optimal prediction ability of the baseline algorithm, the prediction accuracy of the ECSDNN model is improved 4.51%; compared with the single CSDNN base classifier, the prediction accuracy is improved 3.17%.

Investigating the optical bistability of pure spheroidal nanoinclusions in passive and active host matrices

The study examined the effects of the depolarization factor (L) and the real and imaginary parts of the dielectric function of the host matrix (εh ) on the local field enhancement factor and optical bistability of pure spheroidal nanoinclusions with both passive and active host matrices. By solving the Laplace equation in the quasi-static limit, we derived expressions for the electric potentials of the pure spheroidal nanoinclusions. We then incorporated L and the Lorentz-Drude model into these expressions to derive the equation for the enhancement factor in the core of the spheroidal nanoinclusions. The results show that, regardless of whether L varies or remains constant, the pure spheroidal nanoinclusions exhibit only one set of enhancement factor peaks, independent of whether the host matrix is passive or active. However, for the same increase in ε h , the enhancement factor intensities of the pure metal spheroidal nanoinclusions are higher when the host matrix is active compared to when it is passive. The number and intensities of the enhancement factor peaks, as well as the optical bistability of the pure spheroidal nanoinclusions, vary significantly depending on whether the core is made of a passive or active dielectric material. Furthermore, by adjusting parameters such as L and the real and imaginary parts of the host matrices, we were able to achieve tunable enhancement factors and optical bistability, which could be useful for applications in optical sensing, nonlinear optics, and quantum optics.

Burner and Flame Transfer Matrices of Jet-Stabilized Flames: Influence of Jet Velocity and Fuel Properties

Abstract To achieve the decarbonization of electrical power generation, gas turbines need to be upgraded to combust high-hydrogen content fuels reliably. One of the main challenges in this upgrade is the burner design. A promising burner concept for a high-hydrogen fuel mixture are jet burners, which are highly flashback resistant thanks to their high bulk velocity. Due to its nonacoustically compact extension and the presence of hydrogen in the fuel mixture, new challenges arise in assessing the (thermo)acoustic response of this burner design. A burner transfer matrix (BTM) and the flame transfer function (FTF) or transfer matrix (FTM) are typically measured with the multimicrophone method (MMM) to assess the performance of new burner types in relation to thermoacoustic stability. With the switch toward hydrogen, the fuel/air mixture is significantly altered in its properties regarding the speed of sound and density, which are of fundamental importance for acoustic waves propagation and their reconstruction via the MMM, as highlighted in recent work. In this work, we extend this discussion by studying the influence of the gas composition within the burner when measuring BTMs, and its indirect effect on the assessment of FTFs. Experimentally, we achieve this by adapting the preheating temperature during the measurement of the BTM with a nonreactive mixture in order to match the speed of sound of the hydrogen–air mixture required to flow in the burner under reactive conditions. Additionally, we present an analytical model for the jet burner transfer matrix, which is validated against the experimental data. Since the BTM is fundamental for the assessment of the FTM and FTF, the propagation of the error of changing fuel mixtures in the burner is evaluated. The influence of the variation in reactant composition of the BTM on the FTM assessment is noticeable, particularly in the gain of the FTFs. Furthermore, the influence of the total mass flow and, thus, the bulk flow velocity on the FTF is analyzed.

Unique Glycans in Synaptic Glycoproteins in Mouse Brain.

The synapse is an essential connection between neuronal cells in which the membrane and secreted glycoproteins regulate neurotransmission. The post-translational modifications of glycoproteins with carbohydrates, although essential for their functions as well as their specific localization, are not well understood. Oddly, whereas galactose addition to glycoproteins is required for neuronal functions, galactosylation is severely restricted for Asn-linked on N-glycans in the brain, and genetic evidence highlights the important roles of galactose in brain functions and development. To explore this novel glycosylation, we exploited an orthogonal technology in which a biotinylated sialic acid derivative (CMP-biotin-Sia) is transferred to terminally galactosylated proteins by a recombinant sialyltransferase (rST6Gal1). This approach allowed us to identify the carrier proteins as well as their localization on brain sections. Immunohistochemical analysis of the biotinylated glycoproteins in brain sections demonstrates that they are largely positioned in the pre- and postsynaptic membranes. Consistent with this positioning, glycoproteomic analyses of the labeled glycoproteins identified a number of them that are involved in synaptic function, cell adhesion, and extracellular matrix interactions. The discovery of these galactosylated N-glycoproteins and their relative confinement to synapses provide novel insights into the unusual and specific nature of protein glycosylation in the brain.

Generalized matrix functions and Hermitian matrices

Generalized matrix functions and Hermitian matrices

A new approach to MacMahon’s Partition Analysis and associated applications

We introduce a new elimination procedure for Partition Analysis to obtain generating functions free of Omega variables starting with a crude generating function containing or not factored polynomials. In contrast to Han’s approach (Ann Comb. 2003;7(4):467–480), we work with factored polynomials by judiciously constructing the quotient of a series modulo a certain prime ideal. Our results include diverse areas such as an Omega-based approach to compute the Hadamard product of power series and certain trigonometric integrals generalizing the main result of Kim (J Integer Seq. 2009;12:09.7.4) and recovering an integral identity of Boros and Moll (J Comput Appl Math. 1999;106(2):361–368), respectively. Furthermore, we construct a new Omega representation of analytic matrix functions which requires no spectral information and uses the matrix rank to simplify the calculations. By specializing to tridiagonal matrices, the main result of Kiliç (Appl Math Comput. 2008;197(1):345–357) is obtained.

Synchronization of M-sequences based on fast Нadamard transform

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

Two-layer surrogate-assisted collaborative framework for expensive constrained optimization problems involving mixed integer variables

Many surrogate-assisted evolutionary algorithms (SAEAs) have been shown exceptional abilities in solving expensive constrained optimization problems (ECOPs). However, few of them are designed for ECOPs involving mixed integer variables (ECOPs-MI). Therefore, a two-layer surrogate-assisted collaborative framework called TLSACF is designed, in which the radial basis function (RBF) and Gaussian Process (GP)-based mixed-integer collaborative frameworks are combined effectively. In the RBF-based collaborative framework, the combination between RBF-assisted prescreening and RBF-based local search achieves effective population update by balancing the exploration and exploitation on high potential parental information. For the GP-based collaborative framework, a variable type-based cooperative mutation is utilized to obtain potential candidate individuals by assigning targeted mutation for different variables, and a hypervolume-based complete expected improvement matrix function is derived to prescreen these candidate solutions from the perspective of balancing the optimization on constraints and the objective for the effective search on disconnected sub-feasible regions caused by integer variables. Experimental results on ten classical problems show that TLSACF performs excellent in solving ECOPs-MI.

A Kalman-filter based strategy for space-frequency force reconstruction

This contribution presents a Kalman filtering strategy for space-frequency reconstruction of mechanical sources. The proposed approach is based on the definition of an appropriate state-space model, where the state equation assumes that the force vector follows a random walk behavior, while the observation equation is the classical linear relationship between the force and the measured response through the transfer function matrix of the considered structure. One of the main challenges of the proposed approach is the fine-tuning of the covariance matrix associated with the process noise. In this work, it is estimated and adapted at each frequency during the filtering procedure. A numerical experiment is performed on a simply supported beam excited by a broadband point mechanical force to evaluate the reconstruction performance of the proposed approach. Further comparisons with other regularization strategies are also proposed to provide a fair overview of the results obtained.

Rotating and stationary sound sources separation method based on MCB

Traditional beamforming methods do not effectively separate the mixed field of sound sources for different rotational speeds. The coexistence of these sources often manifests out-of-focus sources (for instance, stationary sources appearing with a rotational focus), thereby blurring the beamforming outcomes. Most existing algorithms for segregating rotational different speed sources employ a virtual rotating array utilizing ring arrays. However, due to the constant microphone spacing, the acoustic imaging of the ring array suffers from the interference of high-level side lobes. In this work, a hybrid deconvolution method based on modal composition beamforming is used to explore the ability of a multi-arm spiral array to separate different speed sources. Considering the even energy distribution from the out-of-focus source along the circular trajectory, the beamforming result of out-of-focus sources exhibits a weaker amplitude than the source amplitude. Therefore, out-of-focus sources can be considered noise. First, the point spread function is derived based on the modal transfer function. Then, a system of linear equations is constructed based on an incomplete cross-point spread function matrix. Finally, the source separation is solved through sparse-constrained deconvolution. Simulation and experimental results demonstrate that this method can accurately locate and separate different rotational speed sound sources.

An effective criterion for a stable factorisation of strictly non-singular 2 × 2 matrix functions: use of the ExactMPF package

In this paper, we propose a method to factorise arbitrary strictly non-singular 2 × 2 matrix functions, allowing for stable factorisation. For this purpose, we use the E x a c t M P F package working within the Maple environment previously developed by the authors and perform an exact factorisation of a non-singular polynomial matrix function. A crucial point in the present analysis is the evaluation of a stability region of the canonical factorisation of the polynomial matrix functions. This, in turn, allows us to propose a sufficient condition for the given matrix function admitting stable factorisation.

Sex-biased human thymic architecture guides Tcell development through spatially defined niches.

Within the thymus, regulation of the cellular crosstalk directing Tcell development depends on spatial interactions within specialized niches. To create a spatially defined map of tissue niches guiding humanpostnatal Tcell development, we employed the multidimensional imaging platform co-detection by indexing (CODEX) as well as cellular indexing of transcriptomes and epitopes sequencing (CITE-seq) and assay for transposase accessible chromatin sequencing (ATAC-seq). We generated age-matched 4- to 5-month-old human postnatal thymus datasets for male and female donors, identifying significant sex differences in both Tcell and thymus biology. We demonstrate a possible role for JAG ligands in directing thymic-like dendritic cell development, identify important functions of a population ofextracellular matrix (ECM)- fibroblasts, and characterize the medullary niches surrounding Hassall's corpuscles. Together, these data represent an age-matched spatial multiomic resource to investigate how sex-based differences in thymus regulation and Tcell development arise, providing an essential resource to understand the mechanisms underlying immune function and dysfunction in males and females.