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From the microscopic electroporation to the irregular distribution of cell populations, the inactivation efficacy of pulsed electric fields (PEFs) from in vitro experiments has lacked a unified physical model due to its cross-scale complexity. Inspired by a coarse-grained approach from the percolation theory, the inactivation process is simulated from a simple yet robust lattice model, where the spatiotemporal heterogeneity of the collective structure and the stochastic PEF strike are portrayed as random matrices, while also accounting for the rules of single-cell electroporation and subsequent death. Beyond successfully simulating the inactivation of monolayer adherent cells and suspended cells, which are in good agreement with in vitro results, our model reveals that (1) macroscopically three-staged inactivation pattern originates from the "accelerate–uniform–decelerate" transition of inactivation velocity, and (2) the inactivation patterns obey a universal scaling law under varied field strength, which is not satisfied under varied pulsed widths. The simulation not only sheds light on the PEF inactivation of the macroscopic cell collectives but also provides a simple and generalized numerical method for predicting PEF efficacy in experiments or engineering. [ABSTRACT FROM AUTHOR]

We prove the Eigenstate Thermalization Hypothesis for general Wigner-type matrices in the bulk of the self-consistent spectrum, with optimal control on the fluctuations for obs ervables of arbitrary rank. As the main technical ingredient, we prove rank-uniform optimal local laws for one and two resolvents of a Wigner-type matrix with regular observables. Our results hold under very general conditions on the variance profile, even allowing many vanishing entries, demonstrating that Eigenstate Thermalization occurs robustly across a diverse class of random matrix ensembles, for which the underlying quantum system has a non-trivial spatial structure. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 \frac{1}{2}<\alpha<1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our results is also provided. [ABSTRACT FROM AUTHOR]

Community detection is a crucial task in network analysis that can be significantly improved by incorporating subject-level information, ie, covariates. Existing methods have shown the effectiveness of using covariates on the low-degree nodes, but rarely discuss the case where communities have significantly different density levels, ie, multiscale networks. In this paper, we introduce a novel method that addresses this challenge by constructing network-adjusted covariates, which leverage the network connections and covariates with a node-specific weight for each node. This weight can be calculated without tuning parameters. We present novel theoretical results on the strong consistency of our method under degree-corrected stochastic blockmodels with covariates, even in the presence of misspecification and multiple sparse communities. Additionally, we establish a general lower bound for the community detection problem when both the network and covariates are present, and it shows that our method is optimal for connection intensity up to a constant factor. Our method outperforms existing approaches in simulations and a LastFM app user network. We then compare our method with others on a statistics publication citation network where 30 % of nodes are isolated, and our method produces reasonable and balanced results. Our method is implemented in the R package NAC. [ABSTRACT FROM AUTHOR]

The canonical equation K 107 is founded for random zebra-structured matrices. [ABSTRACT FROM AUTHOR]

The limit theorem for the estimator G 59 for the regularized solution of the Kolmogorov–Wiener filter is proved. [ABSTRACT FROM AUTHOR]

This work considers the low‐rank approximation of a matrix A(t)$$ A(t) $$ depending on a parameter t$$ t $$ in a compact set D⊂ℝd$$ D\subset {\mathbb{R}}^d $$. Application areas that give rise to such problems include computational statistics and dynamical systems. Randomized algorithms are an increasingly popular approach for performing low‐rank approximation and they usually proceed by multiplying the matrix with random dimension reduction matrices (DRMs). Applying such algorithms directly to A(t)$$ A(t) $$ would involve different, independent DRMs for every t$$ t $$, which is not only expensive but also leads to inherently non‐smooth approximations. In this work, we propose to use constant DRMs, that is, A(t)$$ A(t) $$ is multiplied with the same DRM for every t$$ t $$. The resulting parameter‐dependent extensions of two popular randomized algorithms, the randomized singular value decomposition and the generalized Nyström method, are computationally attractive, especially when A(t)$$ A(t) $$ admits an affine linear decomposition with respect to t$$ t $$. We perform a probabilistic analysis for both algorithms, deriving bounds on the expected value as well as failure probabilities for the approximation error when using Gaussian random DRMs. Both, the theoretical results and numerical experiments, show that the use of constant DRMs does not impair their effectiveness; our methods reliably return quasi‐best low‐rank approximations. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-N asymptotics of generalized Bessel functions, as well as a large deviations principle for the more general family of radial Heckman–Opdam processes. As an application, we prove a theorem on the asymptotic behavior of weight multiplicities of irreducible representations of compact or complex simple Lie algebras in the limit of large rank. The theorems in this paper generalize several known results describing analogous asymptotics for Dyson Brownian motion, spherical matrix integrals, and Kostka numbers. [ABSTRACT FROM AUTHOR]

Associated to two given sequences of eigenvalues λ 1 ≥ ⋯ ≥ λ n and μ 1 ≥ ⋯ ≥ μ n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n → ∞ . Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. [ABSTRACT FROM AUTHOR]

We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let $\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ $\in \mathbb{C} ^{m \times n}$ be an $m \times n$ data matrix after self-normalization (n samples and m features), where $\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*})^{*}$. Specifically, we generate a permutation matrix $\textbf X_\pi$ by permuting the entries of $\textbf x_j$ $(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of $\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where $\textbf{U} _n$ is a sequence of $p \times m$ non-random complex matrices. The conditions we require are $p/n \to c >0$ and $m/n \to \gamma > 0$. [ABSTRACT FROM AUTHOR]

We study the rank of the adjacency matrix A$$ A $$ of a random Erdős‐Rényi graph G∼픾(n,p). It is well known that when p=(log(n)−ω(1))/n$$ p=\left(\log (n)-\omega (1)\right)/n $$, with high probability, A$$ A $$ is singular. We prove that when p=ω(1/n)$$ p=\omega \left(1/n\right) $$, with high probability, the corank of A$$ A $$ is equal to the number of isolated vertices remaining in G$$ G $$ after the Karp‐Sipser leaf‐removal process, which removes vertices of degree one and their unique neighbor. We prove a similar result for the random matrix B$$ B $$, where all entries are independent Bernoulli random variables with parameter p$$ p $$. Namely, we show that if H$$ H $$ is the bipartite graph with bi‐adjacency matrix B$$ B $$, then the corank of B$$ B $$ is with high probability equal to the max of the number of left isolated vertices and the number of right isolated vertices remaining after the Karp‐Sipser leaf‐removal process on H$$ H $$. Additionally, we show that with high probability, the k$$ k $$‐core of 픾(n,p) is full rank for any k≥3$$ k\ge 3 $$ and p=ω(1/n)$$ p=\omega \left(1/n\right) $$. This partially resolves a conjecture of Van Vu for p=ω(1/n)$$ p=\omega \left(1/n\right) $$. Finally, we give an application of the techniques in this paper to gradient coding, a problem in distributed computing. [ABSTRACT FROM AUTHOR]

Compressed sensing (CS) is an innovative signal acquisition and reconstruction technology that has broken through the limit of the Nyquist sampling theory. It is widely employed to optimize the measurement processes in various applications. One of the core challenges of CS is the construction of a measurement matrix. However, traditional random measurement matrices are often impractical. Additionally, many existing deterministic binary measurement matrices fail to provide the required flexibility for practical applications. In this study, inspired by the observation that pseudo-random sequences share similar properties with random sequences, we constructed a deterministic sparse measurement matrix with a flexible measurement number based on an pseudo-random sequence—the Legendre sequence. Empirical analysis of the phase transition and an assessment of the practical features of the proposed measurement matrix were conducted. We validated the effectiveness of the proposed measurement matrix on randomly synthesized signals and images. The results of our simulations reveal that our proposed measurement matrix performs better than several other measurement matrices, particularly in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In [H. H. Nguyen and R. Van Peski, Universality for cokernels of random matrix products, Adv. Math. 438 2024, Paper No. 109451], Nguyen and Van Peski raised the question of whether the surjective flag of ℤ p {{\mathbb{Z}}_{p}} -modules modeled by cok ⁢ ( M 1 ⁢ … ⁢ M k ) ↠ ⋯ ↠ cok ⁢ ( M 1 ) \mathrm{cok}(M_{1}\dots M_{k})\twoheadrightarrow\cdots\twoheadrightarrow% \mathrm{cok}(M_{1}) for independent random matrices M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p}) satisfies the Cohen–Lenstra heuristic. We answer the question affirmatively when M 1 , … , M k {M_{1},\dots,M_{k}} follow the Haar measure, and our proof demonstrates how classical ideas in Cohen–Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices. [ABSTRACT FROM AUTHOR]

Hierarchical count histograms involve the publication of count statistics at various granularities based on a predefined hierarchy within a dimension table in a data warehouse. This task finds extensive applications in on-line analytical processing (OLAP) scenarios. This paper focuses on the rigorous privacy-preserving constraint when dealing with an untrusted server. We conduct a systematic investigation of this task and uncover the limitations of the straightforward baseline approach using local differential privacy, as it fails to strike an optimal balance between privacy and utility. We are thus motivated to propose DP-HORUS, a novel crypto-assisted Differentially Private framework for Hierarchical cOunt histogRams under Untrusted Server. DP-HORUS consists of a series of novel designs, including (1) Encrypted hierarchical tree (EHT) structure, which maintains the concept hierarchy in the input data; (2) Random matrix (RM), which reduces communication and computational cost; (3) To further boosted the utility, we propose DP-HORUS+ encompassing two additional modules of histograms structure (HS) and hierarchical consistency (HC), which are respectively introduced to reduce the noise caused by data sparsity and to ensure the hierarchy consistency; (4) To further boost the robust performance, we propose a series of schemes for workload queries based on DP-HORUS. We provide both theoretical analysis and extensive empirical study on both real-world and synthetic datasets, which demonstrates the superior utility of the proposed methods over the state-of-the-art solutions while ensuring strict privacy guarantee. [ABSTRACT FROM AUTHOR]

The current high‐dimensional linear factor models fail to account for the different types of variables, while high‐dimensional nonlinear factor models often overlook the overdispersion present in mixed‐type data. However, overdispersion is prevalent in practical applications, particularly in fields like biomedical and genomics studies. To address this practical demand, we propose an overdispersed generalized factor model (OverGFM) for performing high‐dimensional nonlinear factor analysis on overdispersed mixed‐type data. Our approach incorporates an additional error term to capture the overdispersion that cannot be accounted for by factors alone. However, this introduces significant computational challenges due to the involvement of two high‐dimensional latent random matrices in the nonlinear model. To overcome these challenges, we propose a novel variational EM algorithm that integrates Laplace and Taylor approximations. This algorithm provides iterative explicit solutions for the complex variational parameters and is proven to possess excellent convergence properties. We also develop a criterion based on the singular value ratio to determine the optimal number of factors. Numerical results demonstrate the effectiveness of this criterion. Through comprehensive simulation studies, we show that OverGFM outperforms state‐of‐the‐art methods in terms of estimation accuracy and computational efficiency. Furthermore, we demonstrate the practical merit of our method through its application to two datasets from genomics. To facilitate its usage, we have integrated the implementation of OverGFM into the R package GFM. [ABSTRACT FROM AUTHOR]

Few-shot learning (FSL) uses prior knowledge and supervised experience to effectively classify hyperspectral images (HSIs), thereby reducing the cost of large numbers of labeled samples. However, existing few-shot methods ignore the correlation between cross-domain feature channels, and the feature representation ability is insufficient. To address above issue, this paper proposes a novel Residual Channel Attention Based Sample Adaptation Few-Shot Learning for Hyperspectral Image Classification(RCASA-FSL) for hyperspectral image classification (HSIC), which can capture and enhance cross-domain dependencies through multi-layer residual connection and random-based feature recalibration. Specifically, a Deep Residual Feature Channel Attention Mechanism (DRFCAM) is designed to obtain cross-domain dependencies by residual concatenation, and further the residual structure is stacked for mining depth discrimination information. Furthermore, a new Random-based Feature Recalibration Module (RFRM) is proposed to reassign the feature weights via random matrix, which fully explore feature weight relationships to guide the sample adaptation process. Besides, we design a joint loss function with combining the FSL loss and domain adaptive loss for further optimization model. Experiments conducted on several standard hyperspectral datasets demonstrate that the proposed RCASA-FSL is superior to other FSL techniques in both quantitative and qualitative aspects. [ABSTRACT FROM AUTHOR]

In this paper, we propose two random matrix based extended target tracking models, which apply to the trajectory‐aligned and drifting target motions. The trajectory‐aligned model is specifically designed to handle targets moving along the direction of their extent orientations, while the drift model is tailored to targets whose trajectories deviate from their orientations in time. We utilise the well‐known variational Bayes method to perform inference and obtain posterior densities via computationally efficient, analytical, iterative steps. Through comprehensive experiments conducted on simulated and real data, our methods have demonstrated superior performance compared to previous approaches in scenarios involving both drifting and trajectory‐aligned targets. These results highlight the efficacy of our proposed models in accurately tracking targets and estimating their extent. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach that combines symbolic data analysis with matrix theory through the concept of interval-valued random matrices. This framework is designed to address the complexities of real-world data, offering enhanced statistical modeling techniques particularly suited for large and complex datasets where traditional methods may be inadequate. We develop both frequentist and Bayesian methods for the statistical inference of interval-valued random matrices, providing a comprehensive analytical framework. We conduct extensive simulations to compare the performance of these methods, demonstrating that Bayesian estimators outperform maximum likelihood estimators under the Frobenius norm loss function. The practical utility of our approach is further illustrated through an application to climatology and temperature data, highlighting the advantages of interval-valued random matrices in real-world scenarios. [ABSTRACT FROM AUTHOR]

We consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a τ$\tau$‐function sequence of the Painlevé III′$\text{III}^{\prime }$ equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes G$G$‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we propose a new test for testing the equality of two population covariance matrices in the ultra-high dimensional setting that the dimension is much larger than the sizes of both of the two samples. Our proposed methodology relies on a data splitting procedure and a comparison of a set of well selected eigenvalues of the sample covariance matrices on the split datasets. Compared to the existing methods, our methodology is adaptive in the sense that (i). it does not require specific assumption (e.g., comparable or balancing, etc.) on the sizes of two samples; (ii). it does not need quantitative or structural assumptions of the population covariance matrices; (iii). it does not need the parametric distributions or the detailed knowledge of the moments of the two populations. Theoretically, we establish the asymptotic distributions of the statistics used in our method and conduct the power analysis. We justify that our method is powerful under weak alternatives. We conduct extensive numerical simulations and show that our method significantly outperforms the existing ones both in terms of size and power. Analysis of two real datasets is also carried out to demonstrate the usefulness and superior performance of our proposed methodology. An R package UHDtst is developed for easy implementation of our proposed methodology. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

In this article we investigate no-resonance conditions for quantum many body chaotic systems and random matrix models. No-resonance conditions are properties of the spectrum of a model, usually employed as a theoretical tool in the analysis of late time dynamics. The first order no-resonance condition holds when a spectrum is non-degenerate, while higher order no-resonance conditions imply sums of an equal number of energies are non-degenerate outside of permutations of the indices. This resonance condition is usually assumed to hold for quantum chaotic models. In this work we use several tests from random matrix theory to demonstrate that the statistics of sums of eigenvalues, that are of interest to due to the no-resonance conditions, have Poisson statistics, and lack level repulsion. This result is produced for both a quantum chaotic Hamiltonian as well as the Gaussian Unitary Ensemble and Gaussian Orthogonal Ensemble. This implies some models may have violations of the no-resonance condition or "near" violations. We finish the paper by generalizing important bounds in quantum equilibration theory to cases where the no-resonance conditions are violated, and to the case of random matrix models. [ABSTRACT FROM AUTHOR]

This article, using generalized inverses of matrices, studies the spaces whose elements are matrices. Based on the results obtained, for linear models for random matrices, the article explores the role of ordinary least square estimators in identifying linear estimable functions, and the role of minimum norm ordinary least square estimators in creating linear unbiased estimators. With added conditions on the covariance matrix for vectorized response, it is shown that a linear unbiased estimator constructed from the minimum norm ordinary least square estimator is a best linear unbiased estimator with respect to the risk induced from squared distance loss. [ABSTRACT FROM AUTHOR]

We establish novel two-sided bounds for the tracial seminorm of multilinear Schur multipliers that tighten previously known bounds. The result is obtained by a newly developed method based on polynomial chaoses. [ABSTRACT FROM AUTHOR]

With the continuous development of composite materials for various application fields, the demand for their performance has increased substantially. Particle-reinforced composites are very promising owing to the unique comprehensive properties. However, the effects of NbC particles on the mechanical characteristics of steel matrix composites are still not fully clear. In this paper, a finite element model was established based on the real microstructure of NbC-Fe composites. The particle shape and shape combination design were studied by taking into consideration the plastic deformation and ductile cracking of the matrix, as well as the debonding of the matrix–particle interface. The influences of parameters on the damage mechanism of particle-reinforced metal matrix composites were also examined. The results revealed that the microstructure-based model effectively described the main material deformation and failure mechanisms. Stress concentrations mainly occurred on the aggregated particles, especially at particle sharp corners, particle–matrix interfaces, and edges along the loading direction. The stress in the matrix was much lower than that in the particles and spread at 45° along the loading direction concentration. The matrix–particle debonding was insignificant, and fracture took place near the particle sharp corners. Also, the matrix failure was initiated and propagated around the particle crack tip to connect microcracks caused by particle fracture, or through high-stress/strain-concentration areas, resulting in failure of the entire workpiece. The predicted numerical data provided a better understanding of the mechanical phenomena of NbC-Fe composites at the nanoscale, which is relevant for future research on the micromechanical mechanism of NbC-Fe composites. [ABSTRACT FROM AUTHOR]

With the rise of the Internet of things, the limitations of traditional PKI certificate authentication technology have progressively emerged. Hence, identity-based public key algorithms have been proposed. In this paper, we propose a new approach to generate an identity key, named identity public key (IPK). IPK identity key generation protocol is based on SM2 elliptic curve cryptography and random matrix theory. It builds upon the concept of combined public key (CPK) and resolves the linear collusion issue of CPK as well as the authenticity verification problem of the declared public key of the simplified TF-CPK by enhancing the identity mapping approach. Another main aim of this paper is to prove the security of the IPK identity key generation protocol. Roughly speaking, we verify the security of each part of the private key and the security of the composite private key. We also increase the function and performance comparison with other schemes, such as IBC(SM9) and TF-CPK. Our scheme has the lowest computation cost, which demonstrates its rationality. [ABSTRACT FROM AUTHOR]

High dimensionality and noise have limited the new biological insights that can be discovered in scRNA-seq data. While dimensionality reduction tools have been developed to extract biological signals from the data, they often require manual determination of signal dimension, introducing user bias. Furthermore, a common data preprocessing method, log normalization, can unintentionally distort signals in the data. Here, we develop scLENS, a dimensionality reduction tool that circumvents the long-standing issues of signal distortion and manual input. Specifically, we identify the primary cause of signal distortion during log normalization and effectively address it by uniformizing cell vector lengths with L2 normalization. Furthermore, we utilize random matrix theory-based noise filtering and a signal robustness test to enable data-driven determination of the threshold for signal dimensions. Our method outperforms 11 widely used dimensionality reduction tools and performs particularly well for challenging scRNA-seq datasets with high sparsity and variability. To facilitate the use of scLENS, we provide a user-friendly package that automates accurate signal detection of scRNA-seq data without manual time-consuming tuning. Single-cell RNA sequencing data analysis is limited by noise and high dimensionality. Here, authors present scLENS, a tool that automates accurate signal detection without manual input, particularly in complex datasets. [ABSTRACT FROM AUTHOR]

The stability of ecological systems is a fundamental concept in ecology, which offers profound insights into species coexistence, biodiversity, and community persistence. In this article, we provide a systematic and comprehensive review on the theoretical frameworks for analyzing the stability of ecological systems. Notably, we survey various stability notions, including linear stability, sign stability, diagonal stability, D-stability, total stability, sector stability, and structural stability. For each of these stability notions, we examine necessary or sufficient conditions for achieving such stability and demonstrate the intricate interplay of these conditions on the network structures of ecological systems. We further discuss the stability of ecological systems with higher-order interactions. [ABSTRACT FROM AUTHOR]

Joint models for longitudinal and time‐to‐event data are receiving increasing attention owing to its capability of capturing the possible association between these two types of data. Typically, a joint model consists of a longitudinal submodel for longitudinal processes and a survival submodel for the time‐to‐event response, and links two submodels by common covariates that may carry both fixed and random effects. However, research gaps still remain on how to simultaneously select fixed and random effects from the two submodels under the joint modeling framework efficiently and effectively. In this article, we propose a novel block‐coordinate gradient descent (BCGD) algorithm to simultaneously select multiple longitudinal covariates that may carry fixed and random effects in the joint model. Specifically, for the multiple longitudinal processes, a linear mixed effect model is adopted where random intercepts and slopes serve as essential covariates of the trajectories, and for the survival submodel, the popular proportional hazard model is employed. A penalized likelihood estimation is used to control the dimensionality of covariates in the joint model and estimate the unknown parameters, especially when estimating the covariance matrix of random effects. The proposed BCGD method can successfully capture the useful covariates of both fixed and random effects with excellent selection power, and efficiently provide a relatively accurate estimate of fixed and random effects empirically. The simulation results show excellent performance of the proposed method and support its effectiveness. The proposed BCGD method is further applied on two real data sets, and we examine the risk factors for the effects of different heart valves, differing on type of tissue, implanted in the aortic position and the risk factors for the diagnosis of primary biliary cholangitis. [ABSTRACT FROM AUTHOR]

Multi-state transition model is typically used to analyze longitudinal data in medicine and sociology. Moreover, variables in longitudinal studies usually are error-prone, and random effects are heterogeneous, which will result in biased estimates of the interest parameters. This article is intended to estimate the parameters of the multi-state transition model for longitudinal data with measurement error and heterogeneous random effects and further consider the covariate related to the covariance matrix of random effects is also error-prone when the covariate in the transition model is error-prone. We model the covariance matrix of random effects through the modified Cholesky decomposition and propose a pseudo-likelihood method based on the Monte Carlo expectation-maximization algorithm and the Bayesian method based on Markov Chain Monte Carlo to infer and calculate the whole estimates. Meanwhile, we obtain the asymptotic properties and evaluate the finite sample performance of the proposed method by simulation, which is well in terms of Bias, RMSE, and coverage rate of confidence intervals. In addition, we apply the proposed method to the MFUS data. [ABSTRACT FROM AUTHOR]

Recent advances in light microscopy have enabled single molecules to be imaged and tracked within living cells and this approach is impacting our understanding of cell biology. Computer modeling and simulation are important adjuncts to the experimental cycle since they aid interpretation of experimental results and help refine, test and generate hypotheses. Object-oriented computer modeling is particularly well-suited for simulating random, thermal, movements of individual molecules as they interact with other molecules and subcellular structures, but current models are often limited to idealized systems consisting of unit volumes or planar surfaces. Here, a simulation tool is described that combines a 3-dimensional, voxelated, representation of the cell consisting of subcellular structures (e.g. nucleus, endoplasmic reticulum, cytoskeleton, vesicles, and filopodia) combined with numerical floating-point precision simulation of thousands of individual molecules moving and interacting within the 3-dimensional space. Simulations produce realistic time-series video sequences comprising single fluorophore intensities and realistic background noise which can be directly compared to experimental fluorescence video microscopy data sets. [ABSTRACT FROM AUTHOR]

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability. [ABSTRACT FROM AUTHOR]

With the widespread adoption of metagenomic sequencing, new perspectives have emerged for studying microbial ecological networks, yielding metabolic evidence of interspecies interactions that traditional co‐occurrence networks cannot infer. This protocol introduces the integrated Network Analysis Pipeline 2.0 (iNAP 2.0), which features an innovative metabolic complementarity network for microbial studies from metagenomics sequencing data. iNAP 2.0 sets up a four‐module process for metabolic interaction analysis, namely: (I) Prepare genome‐scale metabolic models; (II) Infer pairwise interactions of genome‐scale metabolic models; (III) Construct metabolic interaction networks; and (IV) Analyze metabolic interaction networks. Starting from metagenome‐assembled or complete genomes, iNAP 2.0 offers a variety of methods to quantify the potential and trends of metabolic complementarity between models, including the PhyloMint pipeline based on phylogenetic distance‐adjusted metabolic complementarity, the SMETANA (species metabolic interaction analysis) approach based on cross‐feeding substrate exchange prediction, and metabolic distance calculation based on parsimonious flux balance analysis (pFBA). Notably, iNAP 2.0 integrates the random matrix theory (RMT) approach to find the suitable threshold for metabolic interaction network construction. Finally, the metabolic interaction networks can proceed to analysis using topological feature analysis such as hub node determination. In addition, a key feature of iNAP 2.0 is the identification of potentially transferable metabolites between species, presented as intermediate nodes that connect microbial nodes in the metabolic complementarity network. To illustrate these new features, we use a set of metagenome‐assembled genomes as an example to comprehensively document the usage of the tools. iNAP 2.0 is available at https://inap.denglab.org.cn for all users to register and use for free. Highlights: The updated integrated Network Analysis Pipeline (iNAP 2.0) is an advanced platform that facilitates the calculation of various metabolic complementarity and competition indices, including the PhyloMint index, SMETANA (species metabolic interaction analysis), and metabolic distance, as well as the construction and analysis of metabolic interaction networks.Innovatively, iNAP 2.0 uses the random matrix theory (RMT) approach to determine the threshold for metabolic interaction networks.PhyloMint PTM in iNAP 2.0 allows users to view potentially transferable metabolites between microbial interactions and integrate them as network nodes to construct a microbe‐metabolite bipartite network alongside the microbial interaction network. [ABSTRACT FROM AUTHOR]

We consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a 3×3$3\times 3$ Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by |x/t−1/(12α)|t2/3≤C$|x/t-1/(12\alpha)|t^{2/3}\le C$, where C>0$C>0$ is a constant, it turns out that the leading‐order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a 3×3$3\times 3$ matrix RH problem and appear in a variety of random matrix models. [ABSTRACT FROM AUTHOR]

Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector x 0 ∈ S n − 1 , where S n − 1 denotes the n -dimensional unit sphere, ‖ x 0 ‖ ℓ 0 = k < n , from m quadratic measurements of the form (1 − λ) 〈 A i , x 0 x 0 ⊺ 〉 + λ 〈 c i , x 0 〉 where A i , c i have i.i.d. Gaussian entries. This can be related to a constrained version of the 2-spin Hamiltonian with external field for which it was shown (in the absence of any structural constraint and in the asymptotic regime) in [1] that the geometry of the energy landscape becomes trivial above a certain threshold λ > λ c ∈ (0 , 1). Building on this idea, we characterize the recovery of x 0 as a function of λ ∈ [ 0 , 1 ]. We show that recovery of the vector x 0 can be guaranteed as soon as m ≳ k 2 (1 − λ) 2 / λ 2 ∨ k , λ > 1 / 2 provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field (1 − λ) / λ ≲ k − 1 / 2. A similar result (with a slightly deteriorating sample complexity) can be shown for weaker fields. Our proof relies on an interpolation between the linear and quadratic settings, as well as on standard convex geometry arguments. [ABSTRACT FROM AUTHOR]

Let A be drawn uniformly at random from the set of all n\times n symmetric matrices with entries in \{-1,1\}. We show that \[ \mathbb {P}(\det (A) = 0) \leqslant e^{-cn}, \] where c>0 is an absolute constant, thereby resolving a long-standing conjecture. [ABSTRACT FROM AUTHOR]

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted N\times N matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the N\to \infty behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix's conjecture. [ABSTRACT FROM AUTHOR]

AbstractReal networks often have severe degree heterogeneity, with maximum, average, and minimum node degrees differing significantly. This article examines the impact of degree heterogeneity on statistical limits of network data analysis. Introducing the heterogeneity distribution (HD) under a degree-corrected mixed membership model, we show that the optimal rate of mixed membership estimation is an explicit functional of the HD. This result confirms that severe degree heterogeneity decelerates the error rate, even when the overall sparsity remains unchanged. To obtain a rate-optimal method, we modify an existing spectral algorithm, Mixed-SCORE, by adding a pre-PCA normalization step. This step normalizes the adjacency matrix by a diagonal matrix consisting of the bth power of node degrees, for some b∈R . We discover that b = 1/2 is universally favorable. The resulting spectral algorithm is rate-optimal for networks with arbitrary degree heterogeneity. A technical component in our proofs is entry-wise eigenvector analysis of the normalized graph Laplacian. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

Building on the work of Iwaniec, Luo and Sarnak, we use the n-level density to bound the probability of vanishing to order at least r at the central point for families of cuspidal newforms of prime level N → ∞ , split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the n-level density studied. Previous work has determined the optimal test functions for the 1 and 2-level densities in certain support ranges, with the effectiveness of the bounds only marginally increasing by the optimized test functions over simpler ones, and thus this is not expected to be a productive avenue for further research. Similarly the support has been increased as far as possible, and further progress is shown to be related to delicate and difficult conjectures in number theory. Thus we concentrate on the third method, and study the higher centered moments (which are similar to the n-level densities but combinatorially easier). We find the level at each rank for which the upper bounds on the order of vanishing is the best, thus producing world-record bounds on the order of vanishing to rank at least r for every r > 2 (for example, our bounds for vanishing to order at least 5 or at least 6 are less than half the previous bounds, a significant improvement). Additionally, we calculate the bound using the optimal test function for the 1-level density from previous work and compare it to the naive test functions for higher levels. We find that the optimal test function for certain levels are not the optimal for other levels, and some test functions may outperform others for some levels but not in others. Finally, we calculate the integrals needed to determine the bounds, doing so by transforming an n-dimensional integral to a 1-dimensional integral and greatly reducing the computation cost in the process. [ABSTRACT FROM AUTHOR]

In this paper, we compute the limit distribution of partial transposes (when both the number and the size of blocks tends to infinity) for a large class of ensembles of unitarily invariant random matrices. Furthermore, the asymptotic freeness relation between the ensembles of random matrices, their transposes and their left and right partial transposes is shown. [ABSTRACT FROM AUTHOR]

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models. [ABSTRACT FROM AUTHOR]

This study investigates the impact of Al 2 O 3 particle volume fraction and distribution on the deformation and damage of particle-reinforced metal matrix composites, particularly in the context of functionally graded metal matrix composites. In this study, a two-dimensional nonlinear random microstructure-based finite element modeling approach implemented in ABAQUS/Explicit with a Python-generated script to analyze the deformation and damage mechanisms in AA 6061 - T 6 / Al 2 O 3 composites. The plastic deformation and ductile cracking of the matrix are captured using the Gurson–Tvergaard–Needleman model, whereas particle fracture is modelled using the Johnson–Holmquist II model. Matrix-particle interface decohesion is simulated using the surface-based cohesive zone method. The findings reveal that functionally graded metal matrix composites exhibit higher hardness values (HRB ) than traditional metal matrix composites. The results highlight the importance of functionally graded metal matrix composites. Functionally graded metal matrix composites with a Gaussian distribution and a particle volume fraction of 10% achieve HRB values comparable to particle-reinforced metal matrix composites with a particle volume fraction of 20%, with only a 2% difference in HRB . Thus, HRB can be improved significantly by employing a low particle volume fraction and incorporating a Gaussian distribution across the material thickness. Furthermore, functionally graded metal matrix composites with a Gaussian distribution exhibit higher HRB values and better agreement with experimental distribution functions when compared to those with a power-law distribution. [ABSTRACT FROM AUTHOR]

In this paper, we derive new probabilistic bounds on the sensitivity of invariant subspaces, deflation subspaces and singular subspaces of matrices. The analysis exploits a unified method for deriving asymptotic perturbation bounds of the subspaces under interest and utilizes probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. As a result of the analysis, we determine with a prescribed probability asymptotic perturbation bounds on the angles between the corresponding perturbed and unperturbed subspaces. It is shown that the probabilistic asymptotic bounds proposed are significantly less conservative than the corresponding deterministic perturbation bounds. The results obtained are illustrated by examples comparing the known deterministic perturbation bounds with the new probabilistic bounds. [ABSTRACT FROM AUTHOR]

We give an integer linear program formulation for the star bicolouring of bipartite graphs. We develop a column generation method to solve the linear programming relaxation to obtain a lower bound for the minimum number of colours needed. We determine the star bicolouring using the iterative rounding method. We give computational results on arrowhead matrices, sparse random matrices, complete bipartite graphs, and matrices from the Harwell–Boeing collection. The findings demonstrate that the proposed method effectively establishes lower and upper bounds for the minimum number of colours needed for a star bicolouring of bipartite graphs, particularly for sparse bipartite graphs. [ABSTRACT FROM AUTHOR]

Two-term asymptotic formulæ for the probability distribution functions for the smallest eigenvalue of the Jacobi β-Ensembles are derived for matrices of large size in the régime where β > 0 is arbitrary and one of the model parameters α1 is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases β = 2 and/or small values of α1, explicit formulæ involving more familiar functions, such as the modified Bessel function of the first kind, are presented. [ABSTRACT FROM AUTHOR]

We study a class of Approximate Message Passing (AMP) algorithms for symmetric and rectangular spiked random matrix models with orthogonally invariant noise. The AMP iterates have fixed dimension |$K \geq 1$|⁠ , a multivariate non-linearity is applied in each AMP iteration, and the algorithm is spectrally initialized with |$K$| super-critical sample eigenvectors. We derive the forms of the Onsager debiasing coefficients and corresponding AMP state evolution, which depend on the free cumulants of the noise spectral distribution. This extends previous results for such models with |$K=1$| and an independent initialization. Applying this approach to Bayesian principal components analysis, we introduce a Bayes-OAMP algorithm that uses as its non-linearity the posterior mean conditional on all preceding AMP iterates. We describe a practical implementation of this algorithm, where all debiasing and state evolution parameters are estimated from the observed data, and we illustrate the accuracy and stability of this approach in simulations. [ABSTRACT FROM AUTHOR]

Given pairwise comparisons between multiple items, how to rank them so that the ranking matches the observations? This problem, known as rank aggregation, has found many applications in sports, recommendation systems and other web applications. We focus on the ranking problem under the Erdös–Rényi outliers model: only a subset of pairwise comparisons is observed, being either clean or corrupted copies of the true score differences. We investigate the spectral ranking algorithms that are based on unnormalized and normalized data matrices. The key is to understand their performance in recovering the underlying scores of each item from the observed data. This reduces to deriving an entry-wise perturbation error bound between the top eigenvectors of the unnormalized/normalized data matrix and its population counterpart. By using the leave-one-out technique, we provide a sharper |$\ell _{\infty }$| -norm perturbation bound of the eigenvectors and derive an error bound on the maximum displacement for each item, with only |$O(n\log n)$| samples. In addition, we also derive the sample complexity to perform top- |$K$| ranking under mild assumptions. Our theoretical analysis improves upon the state-of-the-art results in terms of sample complexity, and our numerical experiments confirm these theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents two modifications for Loidreau's cryptosystem, a rank metric-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break this cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in Modification I. In Modification II, we choose a random matrix of low column rank to mix with the secret matrix. Our analysis shows that these two modifications can both resist the existing structural attacks. Furthermore, these modifications have a much more compact representation of public keys compared to Classic McEliece, which has been selected into the fourth round of the NIST-PQC project. [ABSTRACT FROM AUTHOR]

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ -weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution. [ABSTRACT FROM AUTHOR]

The performance of adaptive beamforming will deteriorate severely under small sample support, especially when the number of snapshots is smaller than the number of sensors. In this paper, we propose an effective algorithm for robust adaptive beamforming under small sample. Firstly, we utilize standard Guassian random matrices to construct projection matrices for dimension reduction of sample covariance matrix (SCM) and steering vector (SV). Subsequently, the dimensionality-reduced SCM and SV are used to obtain more accurate Capon power spectrum in the case of small sample. By integrating the corresponding Capon power spectrum over the angular sector without desired signal, the interference-plus-noise covariance matrix (INCM) is then reconstructed. Moreover, the SV of desired signal is estimated by solving a quadratic programming problem. Finally, the weight vector of the beamformer is calculated based on the reconstructed INCM and the estimated SV. Simulation results demonstrate the effectiveness and robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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