1,446 research outputs found
Spectra of large time-lagged correlation matrices from Random Matrix Theory
We analyze the spectral properties of large, time-lagged correlation matrices
using the tools of random matrix theory. We compare predictions of the
one-dimensional spectra, based on approaches already proposed in the
literature. Employing the methods of free random variables and diagrammatic
techniques, we solve a general random matrix problem, namely the spectrum of a
matrix , where is an Gaussian random
matrix and is \textit{any} , not necessarily symmetric
(Hermitian) matrix. As a particular application, we present the spectral
features of the large lagged correlation matrices as a function of the depth of
the time-lag. We also analyze the properties of left and right eigenvector
correlations for the time-lagged matrices. We positively verify our results by
the numerical simulations.Comment: 44 pages, 11 figures; v2 typos corrected, final versio
Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
Using large arguments, we propose a scheme for calculating the two-point
eigenvector correlation function for non-normal random matrices in the large
limit. The setting generalizes the quaternionic extension of free
probability to two-point functions. In the particular case of biunitarily
invariant random matrices, we obtain a simple, general expression for the
two-point eigenvector correlation function, which can be viewed as a further
generalization of the single ring theorem. This construction has some striking
similarities to the freeness of the second kind known for the Hermitian
ensembles in large . On the basis of several solved examples, we conjecture
two kinds of microscopic universality of the eigenvectors - one in the bulk,
and one at the rim. The form of the conjectured bulk universality agrees with
the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL,
\textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.Comment: 20 pages + 4 pages of references, 12 figs; v2: typos corrected, refs
added; v3: more explanator
Universal transient behavior in large dynamical systems on networks
We analyze how the transient dynamics of large dynamical systems in the
vicinity of a stationary point, modeled by a set of randomly coupled linear
differential equations, depends on the network topology. We characterize the
transient response of a system through the evolution in time of the squared
norm of the state vector, which is averaged over different realizations of the
initial perturbation. We develop a mathematical formalism that computes this
quantity for graphs that are locally tree-like. We show that for unidirectional
networks the theory simplifies and general analytical results can be derived.
For example, we derive analytical expressions for the average squared norm for
random directed graphs with a prescribed degree distribution. These analytical
results reveal that unidirectional systems exhibit a high degree of
universality in the sense that the average squared norm only depends on a
single parameter encoding the average interaction strength between the
individual constituents. In addition, we derive analytical expressions for the
average squared norm for unidirectional systems with fixed diagonal disorder
and with bimodal diagonal disorder. We illustrate these results with numerical
experiments on large random graphs and on real-world networks.Comment: 19 pages, 7 figures. Substantially enlarged version. Submitted to
Physical Review Researc
Measuring crack initiation and growth in the presence of large strains using the potential drop technique
Accurate laboratory measurements of crack initiation and growth are of vital importance for characterising material behaviour for use in the residual life assessment of structural components. The Potential Drop (PD) technique is one of the most common methods of performing these measurements, but such measurements are also sensitive to large inelastic strains which are often erroneously interpreted as crack growth. Despite the maturity of the PD technique, the extent of these errors is not fully understood and the most appropriate method of suppressing them is unknown.
In this thesis typical errors in the measurement of crack extension due to large inelastic strains have been quantified experimentally. These errors depend on the PD configuration and in some cases the configurations recommended in the standards are susceptible to particularly large errors. Optimum configurations for common fracture specimens have been identified but despite these mitigating measures, when testing ductile materials, the errors due to strain remain large compared to other sources of error common to the PD technique.
A sequentially coupled structural-electrical FE modelling approach has been developed which is capable of predicting the influence of strain on PD. This provides a powerful tool for decoupling the effects of strain from crack extension. It has been used in conjunction with experimental measurements, performed using a novel low frequency ACPD system (which behaves in a quasi-DC manner), to develop procedures for accurately measuring crack initiation and growth during fracture toughness and creep crack growth testing. It is demonstrated that some of the common methods of interpreting PD measurements during these tests are not fit for purpose. The proposed method of interpreting creep crack growth data has been used to re-validate creep crack initiation prediction models provided in the R5 assessment procedure.Open Acces
Christine de Pizan, Le Livre des Faits et Bonnes Moeurs du roi Charles V le Sage. Eric Hicks and Thérèse Moreau, trans., Stock/Moyen Age, 1997
Universal spectra of random Lindblad operators
To understand typical dynamics of an open quantum system in continuous time,
we introduce an ensemble of random Lindblad operators, which generate Markovian
completely positive evolution in the space of density matrices. Spectral
properties of these operators, including the shape of the spectrum in the
complex plane, are evaluated by using methods of free probabilities and
explained with non-Hermitian random matrix models. We also demonstrate
universality of the spectral features. The notion of ensemble of random
generators of Markovian qauntum evolution constitutes a step towards
categorization of dissipative quantum chaos.Comment: 6 pages, 4 figures + supplemental materia
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