Article
Mathematics, Applied
Manuel Bogoya, Sergei M. M. Grudsky
Summary: This work aims to construct an asymptotic expansion for the eigenvalues of a Toeplitz matrix T-n(a) as n goes to infinity. The matrix has a continuous and real-valued symbol a with a power singularity of degree gamma at one point. The authors apply the simple-loop (SL) method to obtain and justify a uniform asymptotic expansion for all the eigenvalues.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2023)
Article
Mathematics
Mauricio Barrera, Sergei M. Grudsky
Summary: This study examines the asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices using a different method and explores a more general case. The construction of uniform asymptotic expansions allows for the refinement of classical results.
COMPLEX VARIABLES AND ELLIPTIC EQUATIONS
(2022)
Article
Engineering, Electrical & Electronic
Zhichao Zhang, Yangfan He
Summary: The article introduces a new type of time-frequency analysis tool called the Matrix Wigner Distribution (MWD), which can overcome the time-frequency resolution limit of the traditional Wigner Distribution. By using a generalized symplectic matrix to transform coordinates, the researchers have discovered optimal conditions for enhancing the time-frequency resolution of this specific MWD. A simulation example with linear frequency-modulated signals is provided to verify the feasibility of the established superresolution theory.
Article
Physics, Mathematical
Giorgio Cipolloni, Laszlo Erdos, Dominik Schroder
Summary: In the physics literature, the spectral form factor (SFF) is commonly used to test universality for disordered quantum systems. Previous mathematical results were limited to two solvable models. However, using the multi-resolvent local laws, we rigorously prove the physics prediction on SFF for a large class of random matrices, including the monoparametric ensemble. Our findings supplement the recently proven Wigner-Dyson universality and extensive numerics support the accuracy of our formulas in predicting the SFF.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Materials Science, Multidisciplinary
Andrea Cepellotti, Boris Kozinsky
Summary: This study introduces a new first principles electronic transport model that includes contributions from interband coupling and off-diagonal components, aiming to explain electronic transport behavior in narrow gap semiconductors. Experimental results show that interband tunneling dominates the electron transport dynamics at low doping concentrations.
MATERIALS TODAY PHYSICS
(2021)
Article
Mathematics, Applied
Manuel Bogoya, Sven-Erik Ekstrom, Stefano Serra-Capizzano
Summary: This paper discusses the asymptotic expansions of eigenvalues of Toeplitz matrices based on the simple-loop theory and presents a matrix-less algorithm for efficient eigenvalue computation. Numerical experiments demonstrate higher precision and comparable computational cost compared to existing procedures.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Sergei M. Grudsky, Egor A. Maximenko, Alejandro Soto-Gonzalez
Summary: In this paper, we investigate the eigenvalues of the laplacian matrices of cyclic graphs with one edge of weight alpha and the others of weight 1. We find that the characteristic polynomial and eigenvalues depend only on Re(alpha). We also study the individual behavior of the eigenvalues, including their localization, numerical solution of the characteristic equation, and asymptotic formulas.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics, Applied
Shulin Lyu, Yang Chen
Summary: The study focuses on the probability of eigenvalues of Hermitian matrices from the Jacobi unitary ensemble with a specific weight lying in a certain interval, and provides the asymptotic constant in the determinant of the Bessel kernel. A specialization of the results yields the constant for the probability of eigenvalues within a specific interval in the Jacobi unitary ensemble with a symmetric weight.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Statistics & Probability
Laszlo Erdos, Yuanyuan Xu
Summary: This article establishes precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.
Article
Statistics & Probability
Jianqing Fan, Yingying Fan, Xiao Han, Jinchi Lv
Summary: This article introduces a general framework for the asymptotic theory of eigenvectors in large random matrices and establishes the asymptotic properties for spiked eigenvectors and eigenvalues in the scenario of generalized Wigner matrix noise. Simulation studies validate the theoretical results.
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
(2022)
Article
Mathematics, Applied
De Yu Zhang, Wen Guang Zhai
Summary: This paper investigates the mean value of Fourier coefficients of holomorphic eigenforms for the full modular group under certain conditions and their properties under Lebesgue measure.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2021)
Article
Mathematics
Maryam Baghipur, Modjtaba Ghorbani, Shariefuddin Pirzada, Najaf Amraei
Summary: This paper introduces the concepts of the generalized adjacency matrix, A(a)-spread of a graph, and the smallest S(A(a)) of the path graph. It answers the question raised in a previous paper and establishes a relationship between S(A(a)) and S(A). It also obtains several bounds for S(A(a)).
Article
Statistics & Probability
Yiting Li, Yuanyuan Xu
Summary: We consider an N by N real or complex generalized Wigner matrix with independent centered random variables. Gaussian fluctuations for linear eigenvalue statistics are established on global scales and mesoscopic scales, up to the spectral edges, with universal mesoscopic central limit theorems obtained for statistics inside the bulk and at the edges.
Article
Mathematics, Applied
Bin He, Min Wang, Guangsheng Wei
Summary: This paper discusses the inverse eigenvalue problem of constructing a Jacobi matrix under the given eigenvalues and partial matrix data. The necessary and sufficient conditions for the solvability of the problem are derived, and a numerical algorithm and example are provided.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics
Shin-ichi Tsukada
Summary: This study investigates the asymptotic distribution of the estimator for the correlation matrix and proposes a hypothesis testing method in a three-step monotone incomplete sample, which is validated by numerical simulation.
COMMUNICATIONS IN MATHEMATICS AND STATISTICS
(2023)