Matrix Theory Research Articles

On some pre-orders and partial orders of linear operators on infinite dimensional vector spaces

AbstractThis article is devoted to the generalization of the Drazin pre-order and the G-Drazin partial order to Core-Nilpotent endomorphisms over arbitrary k-vector spaces, namely, infinite dimensional ones. The main properties of this orders are described, such as their respective characterizations and the relations between these orders and other existing ones, generalizing the existing theory for finite matrices. In order to do so, G-Drazin inverses are also studied in this framework. Also, it includes a generalization of the space pre-order to linear operators over arbitrary k-vector spaces.

Origin of regularity in different nuclei from the viewpoint of the effective quadrupole deformation and triaxiality

Abstract This paper presents a theoretical investigation of the possible relation between the regularity of nuclei and the quadrupole deformation of different levels (and also triaxiality). The present paper aims to uncover the underlying physical reasons for this regularity. To this aim, we calculated the effective quadrupole deformation of different levels using the interacting boson model. Also, different ratios between the quadrupole deformations of the energy levels in the ground, beta, and gamma bands are defined. The results for the energy levels confirm the correctness of labeling these states by the quantum numbers of U(5) and SU(3) dynamical symmetry limits (and also the extraction processes). Also, we observed a repetition pattern for these ratios for regular nuclei. Of course, the regularity and sameness of repetition patterns for the levels of the rotational bands are more than the levels of the ground band. For further study, we analyzed the effective quadrupole deformation values of different levels of regular nuclei using random matrix theory. The results show a strong statistical correlation for these quantities and confirm the observed repetition pattern. Also, the results of our studies showed that regular nuclei have triaxial properties.

Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles & L-functions

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painlevé equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over [Formula: see text] in various works before. We obtain the asymptotics and the leading order coefficient explicitly.

On the number of closed gaps of discrete periodic one-dimensional operators

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.

Dynamically selected steady states and criticality in non-reciprocal networks

We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are random Gaussian variables with non-zero mean and arbitrary degree of reciprocity. Using a path-integral approach, we analyse the dynamics, averaged over the network ensemble, in the thermodynamic limit. Our results show that for any degree of reciprocity in the couplings, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The critical lines separating the disordered from the ordered phases is consistent with spectral properties of the coupling matrix, as derived from random matrix theory. As the non-reciprocity (or asymmetry) in the couplings increases, both ordered phases diminish in size, ultimately resulting in the disappearance of the spin-glass phase when the couplings become anti-symmetric. We investigate non-fixed point steady-state solutions for uncorrelated interactions. For such solutions the time-lagged correlation function evolves according to a gradient-descent dynamics on a potential, which depends on the stationary variance. Our analysis shows that in the spin-glass region, the variance dynamically selected by the system leads the correlation function to evolve on the separatrix curve, limiting different realizable steady states, whereas in the ferromagnetic region, a fixed point solution is selected as the only realizable steady state. In the spin-glass region, stationary solutions are unstable against perturbations that break time-translation invariance, indicating chaotic behaviour in large single network instances. Numerical analysis of Lyapunov exponents confirms that chaotic behaviour emerges throughout the spin-glass region, for any value of the coupling correlations. While negative correlations increase the strength of chaos, positive ones reduce it, with chaos disappearing for reciprocal (i.e. symmetric) couplings, where marginal stability is attained. On the other hand, in finite size non-reciprocal networks, fixed points and limit cycles can arise in the spin-glass region, especially close to the critical line. Finally, we show that chaos is suppressed when the strength of external noise exceeds a certain threshold. Intriguing analogies between chaotic phases in non-equilibrium systems and spin-glass phases in equilibrium are put forward.

Duality in non-Hermitian random matrix theory

We consider 9 Gaussian matrix ensembles characterized by single symmetry among the 38-fold symmetry classification classes of non-Hermitian random matrices, and establish exact duality formulae of certain observables between them. Particularly, averaged products of K characteristic polynomials in an N×N matrix ensemble can be expressed in terms of another K×K matrix ensemble. Our method is to combine matrix-valued heat equations and differential identities for determinants and Pfaffians and has more possible applications.

Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices

We consider large non-Hermitian N×N matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [34] and proven for Wigner matrices in [23], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type An−1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F=R,ℂ,H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

Microbial perspective on restoration of degraded urban soil using ornamental plants

The urban soil where abandoned buildings are demolished is barren and structurally poor, and this degraded soil requires restoration. Ornamental plants enhance the urban environment, increase biodiversity, and affect soil physicochemical properties, microbial diversity; however, their effects remain unclear. Thus, in this study, a mixed-planting meadow consisting of 14 perennial ornamental flower species, including Iris tectorum, Iris lacteal, and Patrinia scabiosaefolia, etc. Was planted at a demolition site with sewage-contaminated soil in Beijing. Simultaneously, a single-planting lawn of I. tectorum was established in a nearby park. We aimed to examine soil physicochemical properties, sequence soil bacterial 16S rRNA and fungal ITS amplicons, and analyze soil microbial diversity and community structure at both sites at five time points in the year after planting, To explore the effect of herbaceous ornamental plants on degraded urban soil, we used FAPROTAX and FUNGuild to predict bacterial and fungal functions, the bin-based null model to evaluate the soil microbial community, and random matrix theory to construct soil microbial molecular networks. The mixed-planting meadow produced a visually appealing landscape and dynamic seasonal enrichment, significantly increasing soil total nitrogen (TN) and organic matter (SOM) contents by 1.99 and 1.21 times, respectively. TN had a positive correlation with soil microbial α diversity and community structure. Dominant phyla at both sites included Proteobacteria, Actinobacteria, and Ascomycota. Although soil microorganisms were primarily influenced by stochastic processes, stochasticity was notably higher in the mixed-planting meadow than in the single-planting lawn. The mixed-planting meadow significantly increased the relative abundance of beneficial microorganisms, improving nitrification and aerobic ammonium oxidation of soil bacteria, as well as symbiotroph of fungi. No significant changes were observed in the single-planting lawn. The mixed-planting meadow established a complex soil microbial molecular network, enhancing the correlation between bacteria and fungi and increasing the number of key microorganisms. Our findings suggest the potential of mixed-planting meadow in restoring degraded urban soils by influencing the soil microbial community and enhancing the ecological service function. Our study provides theoretical support for applying mixed-planting meadow communities to improve the soil environment of urban green spaces.

A Spectral Investigation of Criticality and Crossover Effects in Two and Three Dimensions: Short Timescales with Small Systems in Minute Random Matrices

Random matrix theory, particularly using matrices akin to the Wishart ensemble, has proven successful in elucidating the thermodynamic characteristics of critical behavior in spin systems across varying interaction ranges. This paper explores the applicability of such methods in investigating critical phenomena and the crossover to tricritical points within the Blume–Capel model. Through an analysis of eigenvalue mean, dispersion, and extrema statistics, we demonstrate the efficacy of these spectral techniques in characterizing critical points in both two and three dimensions. Crucially, we propose a significant modification to this spectral approach, which emerges as a versatile tool for studying critical phenomena. Unlike traditional methods that eschew diagonalization, our method excels in handling short timescales and small system sizes, widening the scope of inquiry into critical behavior.

Real-time prediction of transit origin–destination flows during underground incidents

Efficient transportation planning and management are critical for ensuring the smooth operation of rail transit systems, particularly in urban areas with high passenger demand. Real-time prediction of transit flow during underground incidents is vital for ensuring passenger safety, minimizing disruptions, and optimizing resource allocation. This study presents a comprehensive framework to address the considerable challenges associated with short-term origin–destination (OD) flow prediction, particularly during incidents. We propose integrating convolutional long short-term memory cells with self-attention cells to learn the long-range spatiotemporal information of historical OD flows. To overcome the absence of a real-time OD matrix, we introduce an improved graph convolution operation, considering the local connection of ridership, to capture vital spatiotemporal interaction patterns. Quantitative and qualitative analyses were conducted to evaluate the performance of the framework, providing insights into the role of each module. Experimental results suggest that incorporating incident matrix theory enhances the accuracy of capturing declining OD flows during incidents by up to approximately 25%. Furthermore, integrating external features like weather and time comprehensively assesses the impact of erratic change factors, especially during off-peak periods. Additionally, introducing more than one cycle of historical OD flow as an input may introduce noise, and increased attention is needed for OD flows with average daily flows exceeding 30 passengers. Designed with a high degree of modularity and adaptability, the proposed framework serves as a valid tool for transportation professionals and researchers to address evolving challenges in the management and operation of urban transportation systems.

A novel grey prediction model with four-parameter and its application to forecast natural gas production in China

Due to the non-homology problem and the simple structural characteristics, a grey prediction model will have defects in modeling. In this paper, the structure of the GM(1,1,x(1)) model is deformed, and additional parameters are added. A novel four-parameter grey prediction model NFGM(1,1) is established to avoid the non-homology problem. The accumulation order of the NFGM(1,1) model is optimized to enhance its performance. This paper first introduces a nonlinear term and a linear term into the GM(1,1,x(1)) model to compensate for its structural defects, which can enhance the accuracy of the model in modeling complex modeling sequences. Secondly, a simplified basic formula of the model is proposed to estimate its parameters and iteratively establish the model, which can avoid the problem of non-homologous errors during modeling. Then a novel four-parameter grey prediction model NFGM(1,1) is constructed. Thirdly, the unbiasedness of NFGM(1,1) is proved and verified by matrix theory. Fourthly, by optimizing the order of the NFGM(1,1) model, the model is more flexible and adjustable, and a novel fractional-order four-parameter grey prediction model FNFGM(1,1) can be obtained. Finally, the FNFGM(1,1) model is applied to the prediction of natural gas production in China. The model results show that the FNFGM(1,1) model exhibits superior performance compared to the NFGM(1,1), TWGM(1,1), TDGM(1,1), DGM(1,1), and GM(1,1) models, with the mean relative simulation/prediction/comprehensive percentage errors of 0.92%/1.42%/1.07%, respectively. According to the predicted results, China's natural gas production will reach 3542.9 × 108 m3 in 2027 and some relevant policy recommendations are put forwarded.

Testing the number of common factors by bootstrapped sample covariance matrix in high-dimensional factor models

This paper studies the impact of bootstrap procedure on the eigenvalue distributions of the sample covariance matrix under a high-dimensional factor structure. We provide asymptotic distributions for the top eigenvalues of bootstrapped sample covariance matrix under mild conditions. After bootstrap, the spiked eigenvalues which are driven by common factors will converge weakly to Gaussian limits after proper scaling and centralization. However, the largest non-spiked eigenvalue is mainly determined by the order statistics of the bootstrap resampling weights, and follows extreme value distribution. Based on the disparate behavior of the spiked and non-spiked eigenvalues, we propose innovative methods to test the number of common factors. Indicated by extensive numerical and empirical studies, the proposed methods perform reliably and convincingly under the existence of both weak factors and cross-sectionally correlated errors. Our technical details contribute to random matrix theory on spiked covariance model with convexly decaying density and unbounded support, or with general elliptical distributions.

Theoretical studies on electromagnetic induced transparency and exciton–exciton annihilation in dye aggregates

Electromagnetically induced transparency (EIT) and exciton–exciton annihilation (EEA) can be described by a common three level model, but the response to the external field is different. We present a theoretical study on EIT and EEA in dye molecular system under photoexcitation in the framework of reduced density matrix theory. It is found that EEA is more likely to occur under shorter pulse duration and stronger exciton coupling, while EIT is the opposite. The study of EIT in dye molecular system is of great significance for multi-signal processing in organic material based optical devices.

Communication topology for AUV formation consensus based on optimal cost control under time-varying communication and input delays

This study investigates the optimal communication topology for consensus in autonomous underwater vehicle (AUV) formations, considering both leader-following and leaderless scenarios. Initially, a second-order dynamic model of the AUV is established using the feedback linearization method. To address time-varying delays, a universal predictor-based consensus protocol is introduced, capable of operation with or without a designated leader. Subsequently, two global quadratic cost functions are formulated based on distinct consensus errors. Utilizing linear quadratic regulator (LQR) theory and matrix theory, the optimal communication topologies and associated gain matrices are deduced. Specifically, for leader-following AUV formations, the optimal communication topology is identified as an unevenly weighted star-shaped directed spanning tree. Conversely, for leaderless formations, the optimal topology adopts a complete digraph, which indicates high interaction requirements. To accommodate practical constraints, a suboptimal topology is devised, featuring a directed spanning tree. These findings offer a theoretical framework for selecting communication topologies in AUV formations, with numerical examples provided to validate the proposed approaches.

Towards high-performance polarimeters with large-area uniform chiral shells: a comparative study on the polarization detection precision enabled by the Mueller matrix and deep learning algorithm

Polarization detection and imaging technologies have attracted significant attention for their extensive applications in remote sensing, biological diagnosis, and beyond. However, previously reported polarimeters heavily relied on polarization-sensitive materials and pre- established mapping relationships between the Stokes parameters and detected light intensities. This dependence, along with fabrication and detection errors, severely constrain the working waveband and detection precision. In this work, we demonstrated a highly precise, stable, and broadband full-Stokes polarimeter based on large-area uniform chiral shells and a post-established mapping relationship. By precisely controlling the geometry through the deposition of Ag on a large-area microsphere monolayer with a uniform lattice, the optical chirality and anisotropy of chiral shells can reach about 0.15 (circular dichroism, CD) and 1.7, respectively. The post-established mapping relationship between the Stokes parameters and detected light intensities is established through training a deep learning algorithm (DLA) or fitting the derived mapping-relationship formula based on the Mueller matrix theory with a large dataset collected from our home-built polarization system. For the detection precision with DLA, the mean squared errors (MSEs) at 710 nm can reach 0.10% (S1), 0.41% (S2), and 0.24% (S3), while for the Mueller matrix theory, the corresponding values are 0.14% (S1), 0.46% (S2), and 0.48% (S3). The in-depth comparative studies indicate that the DLA outperforms the Mueller matrix theory in terms of detection precision and robustness, especially for weak illumination, small optical anisotropy and chirality. The averaged MSEs over a broad waveband ranging from 500 nm to 750 nm are 0.16% (S1), 0.46% (S2), and 0.61% (S3), which are significantly smaller than those derived from the Mueller matrix theory (0.45% (S1), 1% (S2), and 39.8% (S3)). The optical properties of chiral shells, the theory and DLA enabled mapping-relationships, the combination modes of chiral shells, and the MSE spectra have been systematically investigated.

Existence conditions for functional ODE observer design of descriptor systems revisited

This paper is devoted to the problem of designing functional observers for linear time-invariant (LTI) descriptor systems. The observers are realized by using state–space systems governed by ordinary differential equations (ODEs). Available existence results for functional ODE observers in the literature are extended by introducing new and milder sufficient conditions. These conditions are purely algebraic and provided directly in terms of the system coefficient matrices. The proposed observer has an order less than or equal to the dimension of the functional vector to be estimated. The observer parameter matrices are obtained by using simple matrix theory, and the design algorithm is illustrated by numerical examples.

Unification of Decoupling Limits in String and M Theory.

We study and extend the duality web unifying different decoupling limits of type II superstring theories and M theory. We systematically build connections to different corners, such as matrix theories, nonrelativistic string and M theory, tensionless (and ambitwistor) string theory, Carrollian string theory, and spin matrix limits of AdS/CFT. We discuss target space, world sheet, and worldvolume aspects of these limits in arbitrary curved backgrounds.

Data-driven modeling in digital twin for power system anomaly detection

Background: Power system anomaly detection is of great significance for realizing system situation awareness and early detection of system operating risks. In view of the complex operating conditions of the system, there are a large number of opaque links in the mechanism, and the anomaly detection approach based on physical mechanism modeling is prone to system errors due to assumptions, simplification, and transfer in the modeling process. This paper focuses on digital twin based data-driven approaches for power system anomaly detection to compensate for the limitation of physical methods in dynamical modeling. Methods: First of all, a digital twin framework for power system real-time analysis is constructed based on the concept of digital twin. Then, this paper conducts researches on the core of the designed framework, i.e., digital twin modeling. Considering the complexity of power system operating conditions, data-driven modeling is preferred and a random matrix and free probability theory based model for anomaly detection of system operating situation is constructed. Results: Simulation data with different spatiotemporal structure generated through a Monte Carlo experiment verified the sensitivity of the constructed model for data correlations. Meanwhile, the case on the system operating data generated through the IEEE 118-bus system validate the effectiveness of the proposed model for the system anomaly detection. Conclusions: The constructed data-driven model can accurately characterize the correlations among data elements, has good sensitivity to the variation of data spatial and temporal correlations, and can depict the data residuals better than the M-P law curve, which indicates the practicability and necessity of the constructed data-driven model for the digital twin modeling of power system anomaly detection.

Asymptotic performance of reconfigurable intelligent surface assisted MIMO communication for large systems using random matrix theory

AbstractReconfigurable intelligent surface (RIS) can provide unprecedented spectral efficiency gains and excellent ability to manipulate electromagnetic waves. This article considered a RIS‐assisted multiuser multiple‐input multiple‐output (MIMO) downlink system, where the beamforming at the base station and RIS are jointly designed to maximize the sum‐rate. For the large dimension scenario and high‐rank beamforming matrix, the accurate deterministic approximations from random matrix theory are then utilized to simplify the RIS‐assisted MIMO systems. The asymptotical signal‐to‐interference‐plus‐noise ratio values obtained through random matrix theory is infinitely close to the theoretical limits calculated by accurately iteration. And the performance of the proposed algorithm computed via the sharing second‐order channel statistics matches that of the RIS algorithm with sharing full channel state information asymptotically. The deterministic approximations are instrumental to get improvement into the structure of the optimal beamforming and to reduce the implementation complexity in large‐scale MIMO system. Numerical simulations results are provided to evaluate and verify the accuracy of the asymptotic results obtained from the proposed algorithm in the finite system regime. With the complex operation process of large dimension matrix reducing to the deterministic approximations, a lower computational complexity can be obtained compared with other methods.