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Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers.

This paper concerns the study of matrix polynomials of arbitrary degree. In terms of L ( λ ) = λ r I - ∑ j = 1 r λ r - j C j with or without commuting coefficients ( C i C j = C j C i , o r C i C j ≠ C j C i f o r C i ∈ C t × t , i , j = 1 , … , r ) by determinant of block Toeplitz-Hessenberg matrices.

Image watermarking is a key technology for copyright protection, and how to better balance the invisibility and robustness of algorithms is a challenge. To tackle this challenge, a watermarking optimization method based on matrix decomposition and discrete wavelet transform (DWT) for multi-size images is proposed. The DWT, Hessenberg matrix decomposition (HMD), singular value decomposition (SVD), particle swarm optimization (PSO), Arnold transform and logistic mapping are combined for the first time to achieve an image watermarking optimization algorithm. The multi-level decomposition of DWT is used to be adapted to multi-size host images, the Arnold transform, logistic mapping, HMD and SVD are used to enhance the security and robustness, and the PSO optimized scaling factor to balance invisibility and robustness. The simulation results of the proposed method show that the PSNRs are higher than 44.9 dB without attacks and the NCs are higher than 0.98 under various attacks. Compared with the existing works, the proposed method shows high robustness against various attacks, such as noise, filtering and JPEG compression and in particular, the NC values are at least 0.44% higher than that in noise attacks.

We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile, we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis, along with several numerical experiments, shows that the proposed variants generally outperform the classical iterations.

Let = ) denote the number defined recursively by = + + for ≥ 2, where = = 1. Terms of the sequence (1) are referred to simply as numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ ≤ 7, where several connections are made to entries in the . By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

Advancements in digital medical imaging technologies have significantly impacted the healthcare system. It enables the diagnosis of various diseases through the interpretation of medical images. In addition, telemedicine, including teleradiology, has been a crucial impact on remote medical consultation, especially during the COVID-19 pandemic. However, with the increasing reliance on digital medical images comes the risk of digital media attacks that can compromise the authenticity and ownership of these images. Therefore, it is crucial to develop reliable and secure methods to authenticate these images that are in NIfTI image format. The proposed method in this research involves meticulously integrating a watermark into the slice of the NIfTI image. The Slantlet transform allows modification during insertion, while the Hessenberg matrix decomposition is applied to the LL subband, which retains the most energy of the image. The Affine transform scrambles the watermark before embedding it in the slice. The hybrid combination of these functions has outperformed previous methods, with good trade-offs between security, imperceptibility, and robustness. The performance measures used, such as NC, PSNR, SNR, and SSIM, indicate good results, with PSNR ranging from 60 to 61 dB, image quality index, and NC all close to one. Furthermore, the simulation results have been tested against image processing threats, demonstrating the effectiveness of this method in ensuring the authenticity and ownership of NIfTI images. Thus, the proposed method in this research provides a reliable and secure solution for the authentication of NIfTI images, which can have significant implications in the healthcare industry.

In this paper, we disprove a remaining conjecture about Bohemian matrices, in which the numbers of distinct determinants of a normalized Bohemian upper-Hessenberg matrix were conjectured.

The inverses of Toeplitz-Hessenberg matrices are investigated. It is known that each inverse of such a matrix is a sum of a lower triangular matrix 𝐿 and a matrix 𝑅 of rank 1. The formulas of 𝐿 and 𝑥, 𝑦 such that 𝑥𝑦𝑇 = 𝑅 are derived. Using this result we propose an algorithm for inverting Toeplitz-Hessenberg matrices. Moreover, from the expression of the inverse a formula for the determinant is deduced.