Collective behavior in the North Rhine-Westphalia motorway network

To understand the dynamics on complex networks, measurement of correlations is indispensable. In a motorway network, it is not sufficient to collect information on fluxes and velocities on all individual links, i.e. parts of the freeways between ramps and highway crosses. The interdependencies and mutual connections are also of considerable interest. We analyze correlations in the complete motorway network in North Rhine-Westphalia, the most populous state in Germany. We view the motorway network as a complex system consisting of road sections which interact via the motion of vehicles, implying structures in the corresponding correlation matrices. In particular, we focus on collective behavior, i.e. coherent motion in the whole network or in large parts of it. To this end, we study the eigenvalue and eigenvector statistics and identify significant sections in the motorway network. We find collective behavior in these significant sections and further explore its causes. We show that collectivity throughout the network cannot directly be related to the traffic states (free, synchronous and congested) in Kerner’s three-phase theory. Hence, the degree of collectivity provides a new, complementary observable to characterize the motorway network.

An optimised one-way wave migration method for complex media with arbitrarily varying velocity based on eigen-decomposition

The classical one-way generalised screen propagator (GSP) and Fourier finite-difference (FFD) schemes have limitations in imaging large angles in complex media with substantial lateral variations in wave velocity. Some improvements to the classical one-way wave scheme have been proposed with optimised methods. However, the performance of these methods in imaging complex media remains unsatisfying. To overcome this issue, a new strategy for wavefield extrapolation based on the eigenvalue and eigenvector decomposition of the Helmholtz operator is presented herein. In this method, the square root operator is calculated after the decomposition of the Helmholtz operator at the product of the eigenvalues and eigenvectors. Then, Euler transformation is applied using the best polynomial approximation of the trigonometric function based on the infinite norm, and the propagator for one-way wave migration is calculated. According to this scheme, a one-way operator can be computed more accurately with a lower-order expansion. The imaging performance of this scheme was compared with that of the classical GSP, FFD and the recently developed full-wave-equation depth migration (FWDM) schemes. The impulse responses in media with arbitrary velocity inhomogeneity demonstrate that the proposed migration scheme performs better at large angles than the classical GSP scheme. The wavefronts calculated in the dipping and salt dome models illustrate that this scheme can provide a precise wavefield calculation. The applications of the Marmousi model further demonstrate that the proposed approach can achieve better-migrated results in imaging small-scale and complex structures, especially in media with steep-dipping faults.

An instrumental orchestration for online course at the university level

In this article, we present a proposal for instrumental orchestration that organizes the use of technological environments in online mathematics education, in the synchronous mode for the concepts of eigenvalue and eigenvector of a first linear algebra course with engineering students. We used the instrumental orchestration approach as a theoretical framework to plan and organize the artefacts involved in the environment (didactic configuration) and the ways in which they are implemented (exploitation modes). The activities were designed using interactive virtual didactic scenarios, in a dynamic geometry environment, guided exploration worksheets with video and audio recordings of the work of the students, individually or in pairs. The results obtained are presented and the orchestrations of a pedagogical sequence to introduce the concepts of eigenvalue and eigenvector are briefly discussed. This work allowed us to identify new instrumental orchestrations for online mathematics education.

PolarGUI: A MATLAB-Based Tool for Polarization Analysis of the Three-Component Seismic Data Using Different Algorithms

We present an open-source and MATLAB-based tool with an easy-to-use graphical user interface (GUI) consisting of four polarization analysis approaches: the particle-motion trajectory (a hodogram in a 3D plane), eigenvalue decomposition (EVD) based on the covariance matrix (including two calculation methods), singular value decomposition using principal component analysis, and EVD based on a constructed analytic signal matrix (EVD-ASM). We review the calculation processes and features of the four cited methods. The eigenvalue and eigenvector are applied to obtain the polarization attributes of the three-component (3C) seismic data. Using rose graphs and histograms, the corresponding azimuth and incidence angle are calculated to determine the propagation direction of the seismic wave. Statistical distribution curves of the corresponding rectilinearity and planarity of the waves are also plotted. The polarization analysis GUI can simultaneously analyze two selected data sections in a seismic recording corresponding to P and S waves. We evaluate the performance of these algorithms using real 3C earthquake datasets. Comparison tests indicate that the aforementioned four methods have different time consumption, and the differences between the results of the EVD-ASM and those of the other methods are very small.

Newton's method for the eigenvalue problem of a symmetric matrix

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

Decomposition-Based Soil Moisture Estimation Using UAVSAR Fully Polarimetric Images

Polarimetric decomposition extracts scattering features that are indicative of the physical characteristics of the target. In this study, three polarimetric decomposition methods were tested for soil moisture estimation over agricultural fields using machine learning algorithms. Features extracted from model-based Freeman–Durden, Eigenvalue and Eigenvector based H/A/α, and Van Zyl decompositions were used as inputs in random forest and neural network regression algorithms. These algorithms were applied to retrieve soil moisture over soybean, wheat, and corn fields. A time series of polarimetric Uninhabited Aerial Vehicle Synthetic Aperture Radar (UAVSAR) data acquired during the Soil Moisture Active Passive Experiment 2012 (SMAPVEX12) field campaign was used for the training and validation of the algorithms. Three feature selection methods were tested to determine the best input features for the machine learning algorithms. The most accurate soil moisture estimates were derived from the random forest regression algorithm for soybeans, with a correlation of determination (R2) of 0.86, root mean square error (RMSE) of 0.041 m3 m−3 and mean absolute error (MAE) of 0.030 m3 m−3. Feature selection also impacted results. Some features like anisotropy, Horizontal transmit and Horizontal receive (HH), and surface roughness parameters (correlation length and RMS-H) had a direct effect on all algorithm performance enhancement as these parameters have a direct impact on the backscattered signal.

On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain

ABSTRACT In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) and discrete system (boundary element method, BEM) for a circular case. The infinite-dimensional degree of freedom for the continuous system can be reduced to finite-dimensional space using the generalized Fourier coordinates. The property of the second-order tensor for the influence matrix under different observers is also examined. On the other hand, the discrete system in the BEM can be analytically studied, thanks to the spectral property of circulant matrix. We adopt the circulant matrix of odd dimension, (2N + 1) by (2N + 1), instead of the previous even one of 2N by 2N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.

Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides

The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwells equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwells equations.

The characteristics analysis of strain variation associated with Wenchuan earthquake using principal component analysis

Borehole strainmeters that are installed deeply into bedrock are capable of recording both continuous stress and strain measurements, and have consequently become an important tool for monitoring crustal deformation. A YRY-4 borehole strainmeter installed at the Guza Station recorded anomalous changes in borehole strain data preceding the Wenchuan earthquake on May 12, 2008 (UTC) (=8.0). We apply principal component analysis (PCA) to analyze borehole strain data from the Guza Station. The first principal component eigenvalues and eigenvectors are calculated. The fitted results of the cumulative number of anomalous eigenvalues demonstrate that an acceleration occurred approximately 4 months before the earthquake (from January 2008). The results of the combined eigenvalue and eigenvector analyses show that the spatial distribution of eigenvectors and accelerated occurrence of eigenvalue anomalies represents the stress evolution characteristics of the fault from a steady state to a sub-instability state in rock experiments. We tentatively infer that this process may also be linked to the preparation phase of a large earthquake.