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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]

The perturbation of the QR decompostion is analyzed from the probalistic point of view. The perturbation error is approximated by a first-order perturbation expansion with high probability where the perturbation is assumed to be random. Different from the previous normwise perturbation bounds using the Frobenius norm, our techniques are used to develop the spectral norm, as well as the entry-wise perturbation bounds for the stochastic perturbation of the QR decomposition. The statistics tends to be tighter (in the sense of the expectation) and more realistic than the classical worst-case perturbation bounds. The novel perturbation bounds are applicable to a wide range of problems in statistics and communications. In this paper, we consider the perturbation bound of the leverage scores under the Gaussian perturbation, the probability guarantees and the error bounds of the low rank matrix recovery, and the upper bound of the errors of the tensor CUR-type decomposition. We also apply our perturbation bounds to improve the robust design of the Tomlinson-Harashima precoding in the Multiple-Input Multiple-Output (MIMO) system. [ABSTRACT FROM AUTHOR]

The limit theorem for the estimator G 59 {G_{59}} for the regularized solution of the Kolmogorov–Wiener filter is proved. [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]

The assembly and persistence of ecological communities can be understood as the result of the interaction and migration of species. Here we study a single community subject to migration from a species pool in which inter-specific interactions are organised according to a bipartite network. Considering the dynamics of species abundances to be governed by generalised Lotka–Volterra equations, we extend work on unipartite networks to we derive exact results for the phase diagram of this model. Focusing on antagonistic interactions, we describe factors that influence the persistence of the two guilds, locate transitions to multiple-attractor and unbounded phases, as well as identifying a region of parameter space in which consumers are essentially absent in the local community. [ABSTRACT FROM AUTHOR]

Epilepsy, as a serious neurological disorder, can be detected by analyzing the brain signals produced by neurons. Electroencephalogram (EEG) signals are the most important data source for monitoring these brain signals. However, these complex, noisy, nonlinear and nonstationary signals make detecting seizures become a challenging task. Feature-based seizure detection algorithms have become a dominant approach for automatic seizure detection. This study presents an algorithm for automatic seizure detection based on novel features with clinical and statistical significance. Our algorithms achieved the best results on two benchmark datasets, outperforming traditional feature-based methods and state-of-the-art deep learning algorithms. Accuracy exceeded 99.99% on both benchmark public datasets, with the 100% correct detection of all seizures on the second one. Due to the interpretability and robustness of our algorithm, combined with its minimal computational resource requirements and time consumption, it exhibited substantial potential value in the realm of clinical application. The coefficients of variation of datasets proposed by us makes the algorithm data-specific and can give theoretical guidance on the selection of appropriate random spectral features for different datasets. This will broaden the applicability scenario of our feature-based approach. [ABSTRACT FROM AUTHOR]

In natural ecosystems, species can be characterized by the nonlinear density-dependent self-regulation of their growth profile. Species of many taxa show a substantial density-dependent reduction for low population size. Nevertheless, many show the opposite trend; density regulation is minimal for small populations and increases significantly when the population size is near the carrying capacity. The theta-logistic growth equation can portray the intraspecific density regulation in the growth profile, theta being the density regulation parameter. In this study, we examine the role of these different growth profiles on the stability of a competitive ecological community with the help of a mathematical model of competitive species interactions. This manuscript deals with the random matrix theory to understand the stability of the classical theta-logistic models of competitive interactions. Our results suggest that having more species with strong density dependence, which self-regulate at low densities, leads to more stable communities. With this, stability also depends on the complexity of the ecological network. Species network connectance (link density) shows a consistent trend of increasing stability, whereas community size (species richness) shows a context-dependent effect. We also interpret our results from the aspect of two different life history strategies: r and K-selection. Our results show that the stability of a competitive network increases with the fraction of r-selected species in the community. Our result is robust, irrespective of different network architectures. [ABSTRACT FROM AUTHOR]

We study the spectra of random weighted bipartite graphs. We establish that, under specific assumptions on the edge probabilities, the symmetrized empirical spectral distribution function of the graph's adjacency matrix converges to the symmetrized Marchenko-Pastur distribution function. [ABSTRACT FROM AUTHOR]

In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in Zhou (JTP 36:1203–1226, 2023). Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in Zhou (JTP 36:1203-1226, 2023) to allow for all possible ratios of row to column of the underlying random matrix. [ABSTRACT FROM AUTHOR]

The late Professor Heinz Neudecker (1933–2017) made significant contributions to the development of matrix differential calculus and its applications to econometrics, psychometrics, statistics, and other areas. In this paper, we present an insightful overview of matrix-oriented findings and their consequential implications in statistics, drawn from a careful selection of works either authored by Professor Neudecker himself or closely aligned with his scientific pursuits. The topics covered include matrix derivatives, vectorisation operators, special matrices, matrix products, inequalities, generalised inverses, moments and asymptotics, and efficiency comparisons within the realm of multivariate linear modelling. Based on the contributions of Professor Neudecker, several results related to matrix derivatives, statistical moments and the multivariate linear model, which can literally be considered to be his top three areas of research enthusiasm, are particularly included. [ABSTRACT FROM AUTHOR]

Compared with the single or even multiple targets, the group targets exhibit complex and time-varying structure, making the group shape estimation and evaluation quite challenging. This paper proposes a data-driven multi-sensor target group shape modeling and recognition approach to arbitrary shape estimation for group targets, and a group target shape fitting evaluation metric. The proposed approach consists of three parts. Firstly, the information flooding method is used to realize the collection and dissemination of the target information in the field of view by strongly connected sensors. Secondly, a density peak clustering method is utilized to cluster the data set. Finally, an improved Delaunay triangular network algorithm is used to fit the shape of group targets. The proposed group shape fitting evaluation metric can quantitatively evaluate the accuracy of any group target shape estimate. The effectiveness and reliability of the proposed algorithm are verified in comparison with the classic target shape fitting methods such as the hypersurface and random matrices.

This paper examines the distribution of eigenvalues for a 2 × 2 random confusion matrix used in machine learning evaluation. We also analyze the distributions of the matrix's trace and the difference between the traces of random confusion matrices. Furthermore, we demonstrate how these distributions can be applied to calculate the superiority probability of machine learning models. By way of example, we use the superiority probability to compare the accuracy of four disease outcomes machine learning prediction tasks. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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Copyright of Aero Weaponry is the property of Aero Weaponry Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Sylvester index of a random Hermitian matrix in the Gaussian ensemble has been considered by Dean and Majumdar. We consider this Sylvester index for a matrix ensemble of random Hermitian matrices defined by a probability density of the form exp (- tr Q (x))) , where Q is a convex polynomial. The main result is the determination of the statistical distribution of the eigenvalues under the condition of a prescribed Sylvester index. We revisit some known results, giving complete proofs, for which we use logarithmic potential theory and complex analysis. [ABSTRACT FROM AUTHOR]

High-voltage cables, as applied in the electro-mobility, are highly complex structures regarding their vibration behaviour. The high complexity leads to considerable uncertainty in models for a finite element method (FEM) simulation, which is shown, for example, in the contact modelling between the strands of the cable. To handle this uncertainty and model the structural dynamic, a nonparametric probabilistic approach (NPPA) with random matrices is used for the first time on high-voltage cables. This novel application of NPPA has an advantage over typical FEM analysis by using a more manageable simulation model and eliminating the need for a complex deterministic simulation model. Initially, the NPPA is analysed and enhanced, with an optimization for the dispersion parameter and a frequency shift introduced as methodological improvements. These enhancements result in a comparable scatter band of the frequency response. Following preliminary studies, the cable's dynamic behaviour is examined through experimental modal analysis, after which the dispersion parameters are computed. The NPPA is then applied to the simplified deterministic model with the calculated dispersion parameters, and a Monte Carlo simulation is done. As a result of this simulation, a scatter band is given. The results from the simulation are then compared to the results of an experiment. It is shown that the frequency response from the experiment is almost always in the inner area of the scatter band. Consequently, this innovative method can be used for a risk evaluation according to the path of the frequency response function and an evaluation of the structural behaviour.

We model backreaction in AdS2 JT gravity via a proposed boundary dual Sachdev-Ye-Kitaev quantum dot coupled to Dirac fermion matter and study it from the perspective of quantum entanglement and chaos. The boundary effective action accounts for the backreaction through a linear coupling of the Dirac fermions to the Gaussian-random two-body Majorana interaction term in the low-energy limit. We calculate the time evolution of the entanglement entropy between graviton and Dirac fermion fields for a separable initial state and find that it initially increases and then saturates to a finite value. Moreover, in the limit of a large number of fermions, we find a maximally entangled state between the Majorana and Dirac fields in the saturation region, implying a transition of the von Neumann algebra of observables from type I to type II. This transition in turn indicates a loss of information in the holographically dual emergent spacetime. We corroborate these observations with a detailed numerical computation of the averaged nearest-neighbor gap ratio of the boundary spectrum and provide a useful complement to quantum entanglement studies of holography.

Purposes Users’ abnormal power consumption behaviors need to be distinguished quickly and accurately. Methods An abnormal state detection model is proposed on the basis of smart meter data and data decomposition and random matrix theory, realizing the identification of users’ abnormal power consumption behaviors. The variational mode decomposition (VMD) algorithm is used to eliminate the noise of power data and the influence of noise data. The Random Matrix Theory (RMT) is combined with the Auto-Regressive Moving Average Model (ARMA) to improve the applicability of RMT to time series and realize the judgment of abnormal state of electricity consumption. Findings Taking the actual power consumption data of a certain area as an example, the method conveniency and efficiency for the case of large data samples and non-Gaussian distribution have been verified, which provides a new direction for the identification of abnormal power consumption behavior.

We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polynomials. We give a backwards stability result which can be upgraded to a forward stability result when the reference measure has a density supported on a single interval with square root behavior at the endpoints. Our analysis implies the Lanczos algorithm run on many large random matrix models is in fact forward stable, and hence nearly deterministic, even when computations are carried out in finite precision arithmetic. Since the Lanczos algorithm is not forward stable in general, this provides yet another example of the fact that random matrices are far from "any old matrix", and care must be taken when using them to test numerical algorithms. [ABSTRACT FROM AUTHOR]

The paper examines methods for assessing the distribution of elements in a stochastic matrix assuming the exponential distribution of elements in the corresponding adjacency matrix of a graph. Two cases are considered: the first assumes the homogeneity of all the graph vertices, while the second assumes the heterogeneity in the distribution of vertices with the corresponding density calculations. Hypothesis tests are formulated for the respective distributions to determine the membership of two graph vertices in the same cluster. [ABSTRACT FROM AUTHOR]

The paper examines methods for assessing the distribution of elements in a stochastic matrix assuming an exponential distribution of elements in the corresponding adjacency matrix of a graph. Two cases are considered: the first assumes homogeneity of all graph vertices, while the second assumes heterogeneity in the distribution of vertices with corresponding density calculations. Hypothesis testing tests are formulated for the respective distributions to determine the membership of two graph vertices in the same cluster. [ABSTRACT FROM AUTHOR]

For an n × n independent-entry random matrix Xn with eigenvalues λ1,. . ., λn, the seminal work of Rider and Silverstein [Ann. Probab., 34 (2006), pp. 2118-2143] asserts that the fluctuations of the linear eigenvalue statistics Pn i=1 f(λi) converge to a Gaussian distribution for sufficiently nice test functions f. We study the fluctuations of Pn-K i=1 f(λi), where K randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when K is fixed as well as for the case when K tends to infinity with n. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes, [Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 93-117]. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of Xn to the circular law in Wasserstein distance, which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this article, we evaluate integrals which are connected to the complex multivariate beta integral. These results are useful in computing expected values of trace functions of complex random matrices. [ABSTRACT FROM AUTHOR]

Given multiple point clouds, how to find the rigid transform (rotation, reflection, and shifting) such that these point clouds are well aligned? This problem, known as the generalized orthogonal Procrustes problem (GOPP), has found numerous applications in statistics, computer vision, and imaging science. While one commonly-used method is finding the least squares estimator, it is generally an NP-hard problem to obtain the least squares estimator exactly due to the notorious nonconvexity. In this work, we apply the semidefinite programming (SDP) relaxation and the generalized power method to solve this generalized orthogonal Procrustes problem. In particular, we assume the data are generated from a signal-plus-noise model: each observed point cloud is a noisy copy of the same unknown point cloud transformed by an unknown orthogonal matrix and also corrupted by additive Gaussian noise. We show that the generalized power method (equivalently alternating minimization algorithm) with spectral initialization converges to the unique global optimum to the SDP relaxation, provided that the signal-to-noise ratio is high. Moreover, this limiting point is exactly the least squares estimator and also the maximum likelihood estimator. Our theoretical bound is near-optimal in terms of the information-theoretic limit (only loose by a factor of the dimension and a log factor). Our results significantly improve the state-of-the-art results on the tightness of the SDP relaxation for the generalized orthogonal Procrustes problem, an open problem posed by Bandeira et al. (2014) [8]. [ABSTRACT FROM AUTHOR]

Let x 1 , ... , x n be a random sample of size n from a p-dimensional population distribution, where p = p (n) → ∞ . Consider a symmetric matrix W = X ⊤ X with parameters n and p, where X = (x 1 , ... , x n) ⊤ . In this paper, motivated by model selection theory in high-dimensional statistics, we mainly investigate the asymptotic behavior of the eigenvalues of the principal minors of the random matrix W. For the Gaussian case, under a simple condition that m = o (n / log p) , we obtain the asymptotic results on maxima and minima of the eigenvalues of all m × m principal minors of W. We also extend our results to general distributions with some moment conditions. Moreover, we gain the asymptotic results of the extreme eigenvalues of the principal minors in the case of the real Wigner matrix. Finally, similar results for the maxima and minima of the eigenvalues of all the principal minors with a size smaller than or equal to m are also given. [ABSTRACT FROM AUTHOR]

We bound the second eigenvalue of random d$$ d $$‐regular graphs, for a wide range of degrees d$$ d $$, using a novel approach based on Fourier analysis. Let Gn,d$$ {G}_{n,d} $$ be a uniform random d$$ d $$‐regular graph on n$$ n $$ vertices, and λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ be its second largest eigenvalue by absolute value. For some constant c>0$$ c>0 $$ and any degree d$$ d $$ with log10n≪d≤cn$$ {\log}^{10}n\ll d\le cn $$, we show that λ(Gn,d)=(2+o(1))d(n−d)/n$$ \lambda \left({G}_{n,d}\right)=\left(2+o(1)\right)\sqrt{d\left(n-d\right)/n} $$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ for all d≤cn$$ d\le cn $$. To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d$$ d $$‐regular random graphs—especially those of Liebenau and Wormald. [ABSTRACT FROM AUTHOR]

This paper examines the distribution of eigenvalues for a 2×2 random confusion matrix used in machine learning evaluation. We also analyze the distributions of the matrix’s trace and the difference between the traces of random confusion matrices. Furthermore, we demonstrate how these distributions can be applied to calculate the superiority probability of machine learning models. By way of example, we use the superiority probability to compare the accuracy of four disease outcomes machine learning prediction tasks.

In this paper, we investigate the invertibility of the fourth-order cubic tensors and present several necessary and sufficient conditions for such tensors to be invertible. We also introduce tensors in statistics and use the fourth-order tensors to simplify the expressions of the higher order derivatives of a multivariate function. Finally, we define the covariance tensor of a random matrix X as a fourth-order tensor D [ X ] and show that D [ X ] is positive definite if X is square symmetric. [ABSTRACT FROM AUTHOR]

DNA has become a popular choice for next-generation storage media due to its high storage density and stability. As the storage medium of life's information, DNA has significant storage capacity and low-cost, low-power replication and transcription capabilities. However, utilizing long double-stranded DNA for storage can introduce unstable factors that make it difficult to meet the constraints of biological systems. To address this challenge, we have designed a highly robust coding scheme called the "random code system," inspired by the idea of fountain codes. The random code system includes the establishment of a random matrix, Gaussian preprocessing, and random equilibrium. Compared to Luby transform codes (LT codes), random code (RC) has better robustness and recovery ability of lost information. In biological experiments, we successfully stored 29,390 bits of data in 25,700 bp chains, achieving a storage density of 1.78 bits per nucleotide. These results demonstrate the potential for using long double-stranded DNA and the random code system for robust DNA-based data storage. [ABSTRACT FROM AUTHOR]

Targeting the problem of the power grid facing greater risks with the connection of large-scale wind power, a method for power grid state analysis using big data is proposed. First, based on the big data, the wind power matrix and the branch power matrix are each constructed. Second, for the wind energy matrix, the eigenvalue index in the complex domain and the spectral density index in the real domain are constructed based on the circular law and the M-P law, respectively, to describe the variation of wind energy. Then, based on the concept of entropy and the M-P law, the index for describing the variation of the branch power is constructed. Finally, in order to analyze the real-time status of the grid connected to large-scale wind power, the proposed index is combined with the sliding time window. The simulation results based on the enhanced IEEE-33 bus system show that the proposed method can perform real-time analysis on the grid state of large-scale wind power connection from different perspectives, and its sensitivity is good. [ABSTRACT FROM AUTHOR]

Let X= \{X(t), t \in \mathbb {R}^N\} be a centered Gaussian random field with values in \mathbb {R}^d satisfying certain conditions and let F \subset \mathbb {R}^d be a Borel set. In our main theorem, we provide a sufficient condition for F to be polar for X, i.e. \mathbb P\big (X(t) \in F \text { for some } t \in \mathbb {R}^N\big) = 0, which improves significantly the main result in Dalang et al. [Ann. Probab. 45 (2017), pp. 4700–4751], where the case of F being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [Random Matrices Theory Appl. 9 (2020), p. 26] and Song et al. [J. Math. Anal. Appl. 502 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

We study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system. [ABSTRACT FROM AUTHOR]

The over-current capacity of half-bridge modular multi-level converter (MMC) is quite weak, which requests protections to detect faults accurately and reliably in several milliseconds after DC faults. The sensitivity and reliability of the existing schemes are vulnerable to high resistance and data errors. To improve the insufficiencies, this paper proposes a pilot protection scheme by using the random matrix for DC lines in the symmetrical bipolar MMC high-voltage direct current (HVDC) grid. Firstly, the 1-mode voltage time-domain characteristics of the line end, DC bus, and adjacent line end are analyzed by the inverse Laplace transform to find indicators of fault direction. To combine the actual model with the data-driven method, the methods to construct the data expansion matrix and to calculate additional noise are proposed. Then, the mean spectral radiuses of two random matrices are used to detect fault directions, and a novel pilot protection criterion is proposed. The protection scheme only needs to transmit logic signals, decreasing the communication burden. It performs well in high-resistance faults, abnormal data errors, measurement errors, parameters errors, and different topology conditions. Numerous simulations in PSCAD/EMTDC confirm the effectiveness and reliability of the proposed protection scheme.

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E (tr (W h)) g (tr (W j)) i and E (tr (W − h)) g (tr (W − j)) i , respectively, for particular values of g, h, i, j, g + h + i + j ≤ 5 , where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr (W h)) g (tr (W j)) i and (tr (W − h)) g (tr (W − j)) i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived. [ABSTRACT FROM AUTHOR]

Heterogeneity plays an important role in cancer genesis and progression. In this paper, we theoretically and computationally show that the increase of heterogeneity signals predeterioration during tumor progression by unified random ordinary differential equations (RODEs). Tumor formation results from a systematic change in organisms, and its causality is complex. Heterogeneity may be one of the key factors under some conditions. Specifically, the interactions between cell populations are modeled by random community matrices in normal tissues, benign tumors, and malignant tumors, and then we prove that the RODEs system describing the cell density of normal tissues or benign tumors becomes unstable as heterogeneity of cell populations or species increases. The increase of heterogeneity is an important signal. Heterogeneity can be viewed as a feature to distinguish benign tumors from malignant tumors under certain circumstances. Furthermore, we show that with the increase of heterogeneity, the divergence speed of the RODEs system becomes faster, which implies that malignant tumors with higher heterogeneity may develop faster and have poorer prognoses. Our theoretical findings can explain some noteworthy phenomena in various datasets in tumor patients and our biological experiments in mice from a mathematical viewpoint. Particularly, clinical data and mutation information from the cancer genome atlas and our experiments in mice revealed that tumors with higher heterogeneity usually show shorter survival time, whereas tumors with lower heterogeneity tend to have better prognosis, which also indicates that heterogeneity can be used as a potential biomarker in future clinical diagnosis. Targeting the heterogeneity may be a potential strategy for the cancer treatment. [ABSTRACT FROM AUTHOR]

Given a Riemannian submersion \phi : M \to N, we construct a stochastic process X on M such that the image Y≔\phi (X) is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping \pi : GL(n) \to GL(n)/O(n), whose image is equivalent to the space of n-by-n positive definite matrices, \mathcal {S}_+(n,n), and the said flow has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincaré's upper half plane and the Bures-Wasserstein geometry on \mathcal {S}_+(n,n), on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian motion. By choosing the background metric via natural GL(n) action, we arrive at an invariant control system on the GL(n)-homogenous space GL(n)/O(n). We investigate the feasibility of developing stochastic algorithms using the mean curvature flow. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of joint detection, tracking, and classification (JDTC) for multiple extended objects (EOs) within a Poisson multi-Bernoulli (MB) mixture (PMBM) filter, where an EO is described as an ellipse, and the ellipse is modeled by a random matrix. The EOs are classified according to the size information of the ellipse. Usually, detection, tracking, and classification are processed step-by-step. However, step-by-step processing ignores the coupling relationship between detection, tracking, and classification, resulting in information loss. In fact, detection, tracking, and classification affect each other, and JDTC is expected to be beneficial for achieving better overall performance. In the multi-target tracking problem based on RFS, the overall performance of the PMBM filter satisfying the conjugate priors has been verified to be superior to other filters. Specifically, the PMBM filter propagates multiple MB simultaneously during iterative updates and model the distribution of hitherto undetected EOs. At present, the PMBM filter is only applied to multiple extended objects tracking problem. Therefore, we consider using the PMBM filter to solve the JDTC problem of multiple EOs and further improve JDTC performance. Furthermore, the closed-form implementation based on the product of a gamma Gaussian inverse Wishart (GGIW) and class probability mass function (PMF) is proposed. The details of parameters calculation in the implementation process and the derivation of class PMF are presented in this paper. Simulation experiments verify that the proposed algorithm, named the JDTC-PMBM-GGIW filter, performs well in comparison to the existing JDTC strategies for multiple extended objects. [ABSTRACT FROM AUTHOR]

Fisher matrix is one of the most important statistics in multivariate statistical analysis. Its eigenvalues are of primary importance for many applications, such as testing the equality of mean vectors, testing the equality of covariance matrices and signal detection problems. In this paper, we establish the limiting spectral distribution of high-dimensional noncentral Fisher matrices and investigate its analytic behavior. In particular, we show the determination criterion for the support of the limiting spectral distribution of the noncentral Fisher matrices, which is the base of investigating the high-dimensional problems concerned with noncentral Fisher matrices. [ABSTRACT FROM AUTHOR]

Image coding technology has become an indispensable technology in the field of modern information. With the vigorous development of the big data era, information security has received more attention. Image steganography is an important method of image encoding and hiding, and how to protect information security with this technology is worth studying. Using a basis of mathematical modeling, this paper makes innovations not only in improving the theoretical system of kernel function but also in constructing a random matrix to establish an information-hiding scheme. By using the random matrix as the reference matrix for secret-information steganography, due to the characteristics of the random matrix, the secret information set to be retrieved is very small, reducing the modification range of the steganography image and improving the steganography image quality and efficiency. This scheme can maintain the steganography image quality with a PSNR of 49.95 dB and steganography of 1.5 bits per pixel and can ensure that the steganography efficiency is improved by reducing the steganography set. In order to adapt to different steganography requirements and improve the steganography ability of the steganography schemes, this paper also proposes an adaptive large-capacity information-hiding scheme based on the random matrix. In this scheme, a method of expanding the random matrix is proposed, which can generate a corresponding random matrix according to different steganography capacity requirements to achieve the corresponding secret-information steganography. Two schemes are demonstrated through simulation experiments as well as an analysis of the steganography efficiency, steganography image quality, and steganography capacity and security. The experimental results show that the latter two schemes are better than the first two in terms of steganography capacity and steganography image quality. [ABSTRACT FROM AUTHOR]

The theorem developed by John Bell constituted the starting point of a revolution that translated a philosophical question about the nature of reality into the broad and intense field of research of the quantum information technologies. We focus on a system of two qubits prepared in a random, mixed state, and we study the typical behavior of their nonlocality via the CHSH–Bell inequality. Afterward, motivated by the necessity of accounting for inefficiency in the state preparation, we address to what extent states close enough to one with a high degree of nonclassicality can violate local realism with a previously chosen experimental setup. [ABSTRACT FROM AUTHOR]

The limit theorems for the estimators G 9 and G 10 for the solutions of the Kolmogorov–Wiener filter are proved. [ABSTRACT FROM AUTHOR]