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Quantum Chaotic Systems and Random Matrix Theory
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2019
Akhilesh Pandey, Avanish Kumar, Sanjay Puri
It is relevant to ask why the GOE works so well for nuclear spectra. We emphasize that the system's physical symmetries are important in determining the relevant level statistics. Thus, e.g., the GUE is not encountered in TRI systems, e.g., nuclear spectra. Moreover, the quantum numbers should not be mixed as this leads to the superposition of independent spectra leading to Poisson statistics (see Section 12.3). Apart from nuclear spectra, random matrix statistics is also found in complex atomic and molecular spectra. Rosenzweig and Porter [31] have shown that levels having the same quantum numbers show level repulsion, and the spacing distribution follows Wigner's prediction. More generally, it has emerged that quantum chaotic systems, viz., quantum systems whose classical analogs are chaotic, follow random matrix statistics. In this section, we will provide a brief overview of some important developments in quantum chaos.
Random Matrix Theory
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Romain Couillet, Merouane Debbah
Definition 8.2.1. An N × n matrix X is said to be a random matrix if it is a matrix-valued random variable on some probability space (Ω, F, P) with entries in some measurable space (R, G), where F is a σ-field on Ω with probability measure P and G is a σ-field on R. As per conventional notations, we denote X(ω) the realization of the variable X at point ω ∈ Ω.
Rotational Motion
Published in P. F. Bortignon, A. Bracco, R. A. Broglia, Giant Resonances, 2019
P. F. Bortignon, A. Bracco, R. A. Broglia
Rotational damping, as well as the mean energy of rotational transitions, depends on the details of the Hamiltonian. On the other hand, compound nucleus damping is expected to be insensitive to the particular details of the Hamiltonian. It will lead, once diagonalized, to universal fluctuations of intensities and energies. In principle, this part of the Hamiltonian can be represented by random matrix Hamiltonian, in particular that associated with the Gaussian Orthogonal Ensemble (GOE). Quite independently from Γrot, the quantity Γμ controls, together with the average level spacing D = ϱ−1, ϱ, being the density of levels, the range of validity of concepts like chaos and the associated presence of GOE level statistics and Porter-Thomas intensity fluctuations. It is expected that most of the experimental information on the damping of rotational motion arises from regions of the spectrum where (cf. Figs. 1.13 and 11.4): Γrot < ϱ−1 and Γμ < ϱ−1 that is, in the case of discrete rotational transitions giving rise to ridge-like structures in the γ–γ coincidence spectra.Γrot > Γμϱ−1 (Γμ > Γrot ϱ−, that is, rotational transitions displaying damping without (with) motional narrowing, and leading to valleylike regions in the γ–γ coincidence spectra.
Blind false data injection attacks in smart grids subject to measurement outliers
Published in Journal of Control and Decision, 2022
However, as smart grids control system information more strictly, it limits the attackers' ability to obtain matrix H. In view of this, Yu and Chin (2015) proposed blind false data injection attacks (BFDIAs). Deng and Liang (2019) proved the feasibility of BFDIAs in case that attackers grasp partial susceptance on branches, i.e. partial information of matrix H. Chen et al. (2016) dynamically obtained the elements of matrix H in real time by means of calculation based on linear Total Least Squares estimation (TLS). From another perspective, BFDIAs are constructed by subspace methods with all or part of measurement data (Kim et al., 2015). Esmalifalak et al. (2011) and Yu and Chin (2015) separately used principal component analysis (PCA) and independent component analysis (ICA) to approximate matrix H and obtain corresponding reasonable matrix , furthermore, Li and Wang (2019) proposed kernel ICA-based BFDIAs by mapping partial measurement data into matrix . Based on the linear combination of measurement data, Chin et al. (2018) proposed new BFDIAs called ad-hoc attacks. Conditionally, with the limitation on the length of measurement data observation window, Lakshminarayana et al. (2021) used a random matrix theory spiked model as a criterion to estimate the optimal eigenvalues and eigenvectors of measurement data covariance matrix, so that attack vector can be constructed.
Personal reflections on 50 years of scientific computing: 1967–2017
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
At Michigan, I was privileged to have learned from Wilkinson and Moler, took the core numerical analysis courses from Carl de Boor, studied under David Kahaner, and officially finished the Ph.D. under George Fix (my thesis topic and early supervision actually came from Katta G. Murty, Department of Industrial and Operations Engineering, before Fix arrived). From my notes during those years, I have some quotes that I find especially trenchant. Cleve Moler, on the inverse of a matrix: ‘there is nothing you can do with the inverse that you can't do better without it’. Carl de Boor, on splines: ‘the only good thing about the natural cubic spline is its name’. George Fix, on theorems concerning random matrix properties, in the context of PDEs: ‘there is no such thing as a random matrix’. Larry Shampine, on comparing ODE algorithms: ‘the implementation of an algorithm is more important than the algorithm itself’. Richard Hamming, on everything: ‘the purpose of computing is insight, not numbers’.
Distributed filtering for delayed nonlinear system with random sensor saturation: a dynamic event-triggered approach
Published in Systems Science & Control Engineering, 2021
Let be a real-valued matrix and be a diagonal random matrix. Then, one has where ° is the Hadamard product and is defined as