Eigenvalues Of Matrix Research Articles

Asymptotics of Solutions to a Singularly Perturbed Problem

The work examines the asymptotics of solutions to singularly perturbed differential equations. The zeros of the matrix eigenvalues lie on the real axis. Moving on to the complex plane, we define the negative region in which the research is carried out. Level lines completely cover this area. One of the level lines divides the area into four parts. In each of these parts of the region we choose integration paths. The integration paths should be as decreasing from the starting point to the last point. Carrying out calculations along the chosen integration path, we obtain asymptotic estimates for solutions of singularly perturbed differential equations.

Computing the Tracy-Widom Distribution for Arbitrary $\beta>0$

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

Low Complexity Radio Frequency Interference Mitigation for Radio Astronomy Using Large Antenna Array

With the ongoing growth in radio communications, there is an increased contamination of radio astronomical source data, which hinders the study of celestial radio sources. In many cases, fast mitigation of strong radio frequency interference (RFI) is valuable for studying short lived radio transients so that the astronomers can perform detailed observations of celestial radio sources. The standard method to manually excise contaminated blocks in time and frequency makes the removed data useless for radio astronomy analyses. This motivates the need for better RFI mitigation techniques for array of size M antennas. Although many solutions for mitigating strong RFI improves the quality of the final celestial source signal, many standard approaches require all the eigenvalues of the spatial covariance matrix ([Formula: see text]) of the received signal, which has [Formula: see text] computation complexity for removing RFI of size d where [Formula: see text]. In this work, we investigate two approaches for RFI mitigation, (1) the computationally efficient Lanczos method based on the Quadratic Mean to Arithmetic Mean (QMAM) approach using information from previously-collected data under similar radio-sky-conditions, and (2) an approach using a celestial source as a reference for RFI mitigation. QMAM uses the Lanczos method for finding the Rayleigh–Ritz values of the covariance matrix [Formula: see text], thus, reducing the computational complexity of the overall approach to [Formula: see text]. Our numerical results, using data from the radio observatory Long Wavelength Array (LWA-1), demonstrate the effectiveness of both proposed approaches to remove strong RFI, with the QMAM-based approach still being computationally efficient.

Data distillation in computational pathology by choosing few representants of the original variance: A use case in ovarian cancer

In computational pathology, a typical Whole Slide Image may easily reach a size of 3–4 Gigabytes, while a database, with hundred of cases, might reach Terabytes. In this scenario, training any model is expensive and therefore the possibility of reducing the size of the training data is appealing. This paper presents a method to summarize the information of a data set, a function of the variance, by selecting the most informative samples. The first step of the whole strategy consists in projecting small image patches (100 × 100) to a feature space and discretize it with a simple k-means to obtain the feature space vocabulary. The groups obtained by the k-means are the vocabulary words and therefore any small patch is represented by the centroid of the group to which such patch is projected. A second step, a probabilistic Latent Semantic Analysis constructs groups of words known as topics by computing frequencies of words in documents, which are larger patches containing between 450 and 500 small patches. A third step collects the documents representing 80 % of the topic variance, their patches are assembled and a Singular Value Decomposition (SVD) is applied to these patches. The Data Distillation process chooses only the small patches belonging to the topics showing higher variance in the matrix of eigenvalues S from the SVD decomposition. The method efficacy was evaluated by comparing the performance of models trained either with 40 % of an actual ovarian cancer dataset selected using this method and the entire dataset without any selection. Results show the F-score obtained with these two sets was similar, about 0.87, with different classifiers, namely Support Vector Machine and a Multilayer Perceptron.

Computation of the zeros of a quaternionic polynomial using matrix methods

In a recent paper, Ishfaq Dar (2024), worked on the problem of locating the zeros of quaternion polynomials by introducing various matrix techniques. In this paper, we use those newly developed matrix methods to locate the left eigenvalues of the quaternion companion matrix of the polynomial, which in turn yield various results concerning the location of the zeros of quaternionic polynomials including extensions of the results of Rubinstein and A. Aziz to the quaternionic setting as well.

Restricted maximum of non-intersecting Brownian bridges

Consider a system of N non-intersecting Brownian bridges in [0,1], and let ℳN(p) be the maximal height attained by the top path in the interval [0, p], p ∈ [0, 1]. It is known that, under a suitable rescaling, the distribution of ℳN(p) converges, as N → ∞, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to p → 1 and p → 0). It is also known that, for fixed N, ℳN(1) is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for ℳN(p) for fixed N, showing that ℳN(p) / √p converges in distribution, as p → 0, to the rightmost charge in a generalized Laguerre Unitary Ensemble, which coincides with the top eigenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble.

Investigation of fluid added mass matrix during hydroelastic slamming of wedges

The investigation of the fluid added mass (FAM) matrix during the water entry of an elastic structure is crucial for understanding the mechanism of hydroelastic slamming, as it characterizes the fluid–structural interaction process. In this paper, we excavate a model for directly estimating the FAM matrix during hydroelastic slamming and employ this model to study the limitations and optimized values of the relaxation factor for a partitioned coupling solver. Our simulation of hydroelastic slamming couples the Wagner theory with the modal superposition method (MSM), and the FAM matrix is computed using a monolithic coupling scheme. We conduct a series of parametric studies to analyze how the modal number, hydroelasticity, deadrise angle, and structural boundary condition influence the eigenvalues of the FAM matrix. Based on these studies, we then propose an analytical model to directly express the eigenvalues of the FAM matrix, demonstrating their linear relationship with the wetted ratio. Furthermore, we apply this analytical model to analyze the relaxation factor in the partitioned coupling scheme, theoretically explaining its strong connection to the FAM effect. The limitations and optimized values of the relaxation factor are theoretically presented, and these results are verified through our numerical tests.

Rates of Bootstrap Approximation for Eigenvalues in High-Dimensional PCA

In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $\Sigma$. However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix $\hat\Sigma$, and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve the dimension-free rate of ${\tt{r}}(\Sigma)/\sqrt n$ up to logarithmic factors, where ${\tt{r}}(\Sigma)$ is the effective rank of $\Sigma$, and $n$ is the sample size. From a methodological standpoint, our work also illustrates that applying a transformation to the eigenvalues of $\hat\Sigma$ before bootstrapping is an important consideration in high-dimensional settings.

АСИМПТОТИЧНІ ВЛАСТИВОСТІ ВИПАДКОВИХ МАТРИЦЬ

The work explores the asymptotic properties of large-dimensional stochastic matrices N under the condition of independence of matrix elements or rows (columns). An analysis of the main properties of eigenvalues of stochastic matrices is conducted. The work is dedicated to investigating the asymptotic characteristics of random matrices under the absence of the second moment and also considers the presence of "heavy tails" in the corresponding transitions in the adjacency matrices of the respective graph. The main result of the work is formulated in terms of the transition matrix of a discrete Markov chain and its eigenvalues. In proving the theorem, a non-degenerate Markov chain is considered, describing a mathematical model of random processes that do not change over time with known transition probabilities between states and a corresponding stochastic matrix P, one of whose eigenvalues is unity, and all eigenvalues of the stochastic matrix do not exceed it in absolute value. The proof of this fact follows from the Perron–Frobenius theorem, which concerns the properties of positive matrices and their eigenvalues. Thus, the theorem considered manages to expand the class of random matrices A for which convergence of eigenvalues of the matrix can be applied under the conditions imposed on the elements of the adjacency matrix. Moreover, the imposed conditions are relaxed compared to classical results, where the existence of a finite second moment for the elements of the adjacency matrix is required. This result generalizes both classical results for the normal distribution and similar results of other authors. The obtained result can be used in graph clustering problems to choose the optimal number of clusters, namely, they can be applied to determine the optimal number of clusters in a Grid system, complex networks, in investigations of the structure of the web space, etc.

Crafting a Digital Divide Index for Assessing Agricultural Disparities Among Farmers: An Application of AHP Process

a new measurement tool for assessing the digital divide disparity among farmers in e agriculture was devised during 2023-24. The index comprises four dimensions, namely ICT ownership, awareness, accessibility, and usage, respectively. For dimension selection, the researchers employed the Analytic Hierarchy Process (AHP). The maximum eigen value (ë max) of the pairwise comparison matrix was 4.0049, serving as the overall weight for the decision-making process. In the item selection process, Likert-type scale procedures were used to assess statements for final indicator selection. An interview schedule with 45 indicators was administered to 60 farmers in a non-sampling area, with five ambiguous indicators subsequently removed. The digital divide index customized for farmers demonstrated strong reliability, with a split-half reliability coefficient of 0.846 (Spearman-Brown formula) and a Pearson correlation coefficient of 0.733. Internal consistency was good, as indicated by a Cronbach’s Alpha of 0.813, and the test displayed strong validity with a score of 0.919. Overall, these findings confirm the reliability and validity of the test in measuring the intended construct. Ultimately, the 38 items were consolidated into a standardized farmers’ digital divide index for assessing agricultural disparities in the farming community.

Explicit and unconditionally stable finite-difference time-domain subgridding algorithm based on hanging variables

Due to the limitation of the stability condition, the finite-difference time-domain (FDTD) method cannot efficiently deal with electromagnetic problems containing fine structures. The explicit and unconditionally stable (EUS) FDTD method can eliminate the constraint of the stability condition and improve the simulation efficiency of fine structures by filtering out the unstable modes for the system matrix. However, the EUS-FDTD method needs to solve the eigenvalues of the numerical system matrix, and the symmetry of the numerical system matrix needs to be ensured when the subgridding scheme is used to discretize targets containing fine structures. The existing EUS-FDTD subgridding method encounters some problems such as complex implementation and insufficient accuracy. In order to solve the above problems, in this work, the hanging variables subgridding (HVS) algorithm is applied to the EUS-FDTD algorithm. Starting from the symmetry of the system matrix, the stability of the hanging variables subgridding algorithm is proven, and a high-precision, stable, and easy-to-implement HVS-EUS-FDTD scheme is proposed. Numerical examples of the radiation of linear magnetic currents in free space, electromagnetic scattering of multiple dielectric objects, and a three-dimensional cavity containing a medium demonstrate the stability, high accuracy and efficiency of the proposed method. Numerical experiments show that the computational efficiency of the HVS-EUS-FDTD algorithm can be improved hundreds of times compared with that of the uniform fine grid FDTD algorithm, and the highest computational efficiency can be improved up to ratio (the size ratio of coarse grid to fine grid) times compared with that of the HVS-FDTD algorithm.

Dynamic analysis and comparison of the performance of linear and nonlinear controllers applied to a nonlinear non-interactive and interactive process

<p>This article presents an in-depth dynamic analysis and comparative evaluation of three distinct control strategies—proportional-integral (PI) compensator, linear quadratic regulator (LQR), and sliding mode control (SMC)—applied to a nonlinear process in two configurations: non-interactive system (NIS) and interactive system (IS). The primary objective was to optimize the regulation of fluid levels in a dual-tank system subject to external disturbances and varying operational conditions. The process dynamics were initially modeled using nonlinear differential equations, which were subsequently linearized to facilitate the design of the PI and LQR controllers. The PI compensator design was rooted in state-space representation and was tuned using the Ziegler-Nichols method to achieve the desired transient and steady-state performance. The LQR design employed optimal control theory, minimizing a quadratic cost function to derive the state feedback gain matrix, ensuring system stability by shifting the eigenvalues of the closed-loop system matrix into the left half of the complex plane. In contrast, the SMC leveraged the full nonlinear dynamics of the process, establishing a sliding surface to drive the system states toward a desired trajectory with robustness against model uncertainties and external disturbances. The SMC's performance was evaluated by analyzing the existence and stability of the sliding mode using the derived switching laws for the actuation signal. The comparative study was conducted through simulations in MATLAB/Simulink environments, where each controller's performance was assessed based on transient response, robustness to disturbances, and computational complexity. The results indicate that while the PI compensator and LQR provide satisfactory performance under linearized assumptions, the SMC demonstrates superior robustness and precision in managing the nonlinearities inherent in the IS configuration. This comprehensive analysis underscores the critical trade-offs between simplicity, computational overhead, and control efficacy when selecting appropriate control strategies for nonlinear, multi-variable processes.</p>

Trial-by-trial Variability of TMS-EEG in Healthy Controls and Patients with Depressive Disorder.

Depressive disorder has been known to be associated with high variability in resting-state electroencephalography (EEG) signals. However, this phenomenon is often ignored in stimulus-related brain activities. This study proposed a new method to explore the EEG variability evoked by transcranial magnetic stimulation (TMS, TMS-EEG) in depressive disorder (DE) patients. The TMS-EEG data were collected from 34 DE patients and 36 healthy controls (HC). The maximum eigenvalue of the real binary correlation matrix, calculated between different trials using cross-correlation and surrogate methods, was extracted to assess trial-by-trial variability (TTV) of TMS-EEG. The new method was found to more sensitive and reliable than the standard deviation method. DE patients exhibited significantly smaller TTV in Gamma band and greater TTV in Delta band than HC. Furthermore, the HAMD-17 scores were negatively correlated with TTV values in Gamma band. This study represented the first investigation into the TTV in TMS-EEG data and revealed abnormal values in DE patients. Those findings enhance our understanding of TMS-EEG technology and provide valuable insights for studying the characteristics of DE.

REDUCIBILITY OF MULTIPERIODIC LINEAR SYSTEMS WITH A DIAGONAL DIFFERENTIATION OPERATOR AND ITS APPLICATION TO CONDITIONALLY PERIODIC SYSTEMS

Abstract. The main theorem is proven by substantiating the reducibility of an equivalent multiperiodic linear system, with a differentiation operator in the direction of diagonal of independent variable space. It is shown that helical lines with an elevation angle of  / 4, when applied to a P-circular cylindrical surface, are  -periodic characteristics of the differentiation operator.The reducibility of a multiperiodic linear system is investigated near the helix line which starts from the initial point of the phase circle, following the scheme of substantiating the reducibility of linear periodic systems. The concept of a monodromy matrix is introduced, which is constant along the first integrals of the characteristic equations of the differentiation operator and has the properties of multiperiodicity and smoothness. The existence of localised positive eigenvalues with all the properties of the monodromy matrix is proved. It is assumed that the monodromy matrix accepts the maximum number of different eigenvalues at the initial point of the phase circle compared to other points. In the neighbourhood of these eigenvalues, the localisation of eigenvalues of the monodromy matrix, defined on a cylindrical surface, is realized in its entirety. The Gershgorin method is employed for this purpose. The non-singularity of the monodromy matrix allows the existence of a curve of the complex plane that does not cover zero and surrounds the spectrum of the monodromy matrix. This proves the existence of the logarithm of the monodromy matrix, and therefore the reducibility of a multiperiodic linear system. Hence, a proof of the main theorem for a conditionally periodic system generating a multiperiodic system is obtained as a consequence.

Distributed Algorithms for Linear Equations Over General Directed Networks.

This article deals with linear equations of the form Ax = b . By reformulating the original problem as an unconstrained optimization problem, we first provide a gradient-based distributed continuous-time algorithm over weight-balanced directed graphs, in which each agent only knows partial rows of the augmented matrix (A b) . The algorithm is also applicable to time-varying networks. By estimating a right-eigenvector corresponding to 0 eigenvalue of the out-Laplacian matrix in finite time, we further propose a distributed algorithm over weight-unbalanced communication networks. It is proved that each solution of the designed algorithms converges exponentially to an equilibrium point. Moreover, the convergence rate is given out clearly. For linear equations without solution, these algorithms are used to obtain a least-squares solution in approximate sense. These theoretical results are illustrated by four numerical examples.

Stability Analysis of Equilibrium of DC Microgrid Under Distributed Control

Constant power loads (CPLs) often lead to system instability and voltage collapse. This paper analyzes the existence and stability of equilibrium of meshed DC microgrids with multiple CPLs under distributed control, which aims to realize current sharing and voltage regulation. Firstly, the power-flow equation of the system is derived. To analyze the solvability of power-flow equation, we transform the problem of the multidimensional quadratic equation solvability into the existence of a fixed-point for a contraction mapping. Then, the sufficient solvability condition is derived based on Banach's fixed-point theorem. Secondly, we build the small-signal model to determine the system qualitative behavior around equilibrium. By analyzing the eigenvalue of system Jacobian matrix based on inertia theorem, the analytical stability conditions are obtained. Further, under the load uncertainty, a robust stability condition which only depends on the maximum load information instead of the real-time load information is derived. Simulation results verify the feasibility of the proposed theorems.

Robustness and stability of doubly periodic patterns of the focusing nonlinear Schrödinger equation.

The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.

Sparse spectral graph analysis and its application to gastric cancer drug resistance-specific molecular interplays identification.

Uncovering acquired drug resistance mechanisms has garnered considerable attention as drug resistance leads to treatment failure and death in patients with cancer. Although several bioinformatics studies developed various computational methodologies to uncover the drug resistance mechanisms in cancer chemotherapy, most studies were based on individual or differential gene expression analysis. However the single gene-based analysis is not enough, because perturbations in complex molecular networks are involved in anti-cancer drug resistance mechanisms. The main goal of this study is to reveal crucial molecular interplay that plays key roles in mechanism underlying acquired gastric cancer drug resistance. To uncover the mechanism and molecular characteristics of drug resistance, we propose a novel computational strategy that identified the differentially regulated gene networks. Our method measures dissimilarity of networks based on the eigenvalues of the Laplacian matrix. Especially, our strategy determined the networks' eigenstructure based on sparse eigen loadings, thus, the only crucial features to describe the graph structure are involved in the eigenanalysis without noise disturbance. We incorporated the network biology knowledge into eigenanalysis based on the network-constrained regularization. Therefore, we can achieve a biologically reliable interpretation of the differentially regulated gene network identification. Monte Carlo simulations show the outstanding performances of the proposed methodology for differentially regulated gene network identification. We applied our strategy to gastric cancer drug-resistant-specific molecular interplays and related markers. The identified drug resistance markers are verified through the literature. Our results suggest that the suppression and/or induction of COL4A1, PXDN and TGFBI and their molecular interplays enriched in the Extracellular-related pathways may provide crucial clues to enhance the chemosensitivity of gastric cancer. The developed strategy will be a useful tool to identify phenotype-specific molecular characteristics that can provide essential clues to uncover the complex cancer mechanism.

Impact of Network Topology on Neural Synchrony in a Model of the Subthalamic Nucleus-Globus Pallidus Circuit.

Synchronous neural oscillations within the beta frequency range are observed across the parkinsonian basal ganglia network, including within the subthalamic nucleus (STN) - globus pallidus (GPe) subcircuit. The emergence of pathological synchrony in Parkinson's disease is often attributed to changes in neural properties or connection strength, and less often to the network topology, i.e. the structural arrangement of connections between neurons. This study investigates the relationship between network structure and neural synchrony in a model of the STN-GPe circuit comprised of conductance-based spiking neurons. Changes in net synaptic input were controlled for through a synaptic scaling rule, which facilitated separation of the effects of network structure from net synaptic input. Five topologies were examined as structures for the STN-GPe circuit: Watts-Strogatz, preferential attachment, spatial, stochastic block, k-regular random. Beta band synchrony generally increased as the number of connections increased, however the exact relationship was topology specific. Varying the wiring pattern while maintaining a constant number of connections caused network synchrony to be enhanced or suppressed, demonstrating the ability of purely structural changes to alter synchrony. This relationship was well-captured by the algebraic connectivity of the network, the second smallest eigenvalue of the network's Laplacian matrix. The structure-synchrony relationship was further investigated in a network model designed to emulate the action selection role of the STN-GPe circuit. It was found that increasing the number of connections and/or the overlap of action selection channels could lead to a rapid transition to synchrony, which was also predicted by the algebraic connectivity.

On random force correction for large time steps in semi-implicitly discretized overdamped Langevin equations

In this study, we focused on the treatment of random forces in a semi-implicitly discretized overdamped Langevin (OL) equation with large time steps. In the usual implicit approach for a nonstochastic mechanical equation, the product of the time interval and Hessian matrix was added to the friction matrix to construct the coefficient matrix for solution updates, which were performed using Newton iteration. When large time steps were used, the additional term, which could be regarded as an artificial friction term, prevented the amplification of oscillations associated with large eigenvalues of the Hessian matrix. In this case, the damping of the high-frequency terms did not cause any discrepancy because they were outside of our interest. However, in OL equations, the friction coefficient was coupled to the random force; therefore, excessive artificial friction may have obscured the effects caused by the stochastic properties of the fluctuations. Consequently, we modified the random force in the proposed semi-implicit scheme so that the total random force was consistent with the friction including the additional artificial term. By deriving a discrete Fokker-Planck (FP) equation from the discretized OL equation, we showed how our modification improved the distribution of the numerical solutions of discrete stochastic processes. Finally, we confirmed the validity of our approach in numerical simulations of a freely jointed chain.