Topological invariance and spatial scaling of surface roughness in two highly eroded zones of Mexico: a comparative study

The Fractal Image Informatics toolbox (Oleschko et al., 2008 a; Torres-Argüelles et al., 2010) was applied to extract, classify and model the topological structure and dynamics of surface roughness in two highly eroded catchments of Mexico. Both areas are affected by gully erosion (Sidorchuk, 2005) and characterized by avalanche-like matter transport. Five contrasting morphological patterns were distinguished across the slope of the bare eroded surface of Faeozem (Queretaro State) while only one (apparently independent on the slope) roughness pattern was documented for Andosol (Michoacan State). We called these patterns ?the roughness clusters? and compared them in terms of metrizability, continuity, compactness, topological connectedness (global and local) and invariance, separability, and degree of ramification (Weyl, 1937). All mentioned topological measurands were correlated with the variance, skewness and kurtosis of the gray-level distribution of digital images. The morphology0 spatial dynamics of roughness clusters was measured and mapped with high precision in terms of fractal descriptors. The Hurst exponent was especially suitable to distinguish between the structure of ?turtle shell? and ?ramification? patterns (sediment producing zone A of the slope); as well as ?honeycomb? (sediment transport zone B) and ?dinosaur steps? and ?corals? (sediment deposition zone C) roughness clusters. Some other structural attributes of studied patterns were also statistically different and correlated with the variance, skewness and kurtosis of gray distribution of multiscale digital images. The scale invariance of classified roughness patterns was documented inside the range of five image resolutions. We conjectured that the geometrization of erosion patterns in terms of roughness clustering might benefit the most semi-quantitative models developed for erosion and sediment yield assessments (de Vente and Poesen, 2005).

Hybrid topological derivative and gradient-based methods for electrical impedance tomography

We present a technique to reconstruct the electromagnetic properties of a medium or a set of objects buried inside it from boundary measurements when applying electric currents through a set of electrodes. The electromagnetic parameters may be recovered by means of a gradient method without a priori information on the background. The shape, location and size of objects, when present, are determined by a topological derivative-based iterative procedure. The combination of both strategies allows improved reconstructions of the objects and their properties, assuming a known background.

Topological measure locating the effective crossover between segregation and integration in a modular network

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hypergraph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intracluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed topological measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity and total integration, and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.

Modeling the evolution of item rating networks using time-domain preferential attachment

The understanding of the structure and dynamics of the intricate network of connections among people that consumes products through Internet appears as an extremely useful asset in order to study emergent properties related to social behavior. This knowledge could be useful, for example, to improve the performance of personal recommendation algorithms. In this contribution, we analyzed five-year records of movie-rating transactions provided by Netflix, a movie rental platform where users rate movies from an online catalog. This dataset can be studied as a bipartite user-item network whose structure evolves in time. Even though several topological properties from subsets of this bipartite network have been reported with a model that combines random and preferential attachment mechanisms [Beguerisse Díaz et al., 2010], there are still many aspects worth to be explored, as they are connected to relevant phenomena underlying the evolution of the network. In this work, we test the hypothesis that bursty human behavior is essential in order to describe how a bipartite user-item network evolves in time. To that end, we propose a novel model that combines, for user nodes, a network growth prescription based on a preferential attachment mechanism acting not only in the topological domain (i.e. based on node degrees) but also in time domain. In the case of items, the model mixes degree preferential attachment and random selection. With these ingredients, the model is not only able to reproduce the asymptotic degree distribution, but also shows an excellent agreement with the Netflix data in several time-dependent topological properties.

Bifurcation diagrams for polymer blends with diffuse interfaces in confined and adaptive geometries

Dynamics of binary mixtures such as polymer blends, and fluids near the critical point, is described by the model-H, which couples momentum transport and diffusion of the components [1]. We present an extended version of the model-H that allows to study the combined effect of phase separation in a polymer blend and surface structuring of the film itself [2]. We apply it to analyze the stability of vertically stratified base states on extended films of polymer blends and show that convective transport leads to new mechanisms of instability as compared to the simpler diffusive case described by the Cahn- Hilliard model [3, 4]. We carry out this analysis for realistic parameters of polymer blends used in experimental setups such as PS/PVME. However, geometrically more complicated states involving lateral structuring, strong deflections of the free surface, oblique diffuse interfaces, checkerboard modes, or droplets of a component above of the other are possible at critical composition solving the Cahn Hilliard equation in the static limit for rectangular domains [5, 6] or with deformable free surfaces [6]. We extend these results for off-critical compositions, since balanced overall composition in experiments are unusual. In particular, we study steady nonlinear solutions of the Cahn-Hilliard equation for bidimensional layers with fixed geometry and deformable free surface. Furthermore we distinguished the cases with and without energetic bias at the free surface. We present bifurcation diagrams for off-critical films of polymer blends with free surfaces, showing their free energy, and the L2-norms of surface deflection and the concentration field, as a function of lateral domain size and mean composition. Simultaneously, we look at spatial dependent profiles of the height and concentration. To treat the problem of films with arbitrary surface deflections our calculations are based on minimizing the free energy functional at given composition and geometric constraints using a variational approach based on the Cahn-Hilliard equation. The problem is solved numerically using the finite element method (FEM).

Numerical issues in Lagrangian tracking and topological evolution of fluid particles in wallbounded turbulent flows

The determination of the local Lagrangian evolution of the flow topology in wall-bounded turbulence, and of the Lagrangian evolution associated with entrainment across the turbulent / non-turbulent interface into a turbulent boundary layer, require accurate tracking of a fluid particle and its local velocity gradients. This paper addresses the implementation of fluid-particle tracking in both a turbulent boundary layer direct numerical simulation and in a fully developed channel flow simulation. Determination of the sub-grid particle velocity is performed using both cubic B-spline, four-point Hermite spline and higher-order Hermite spline interpolation. Both wall-bounded flows show similar oscillations in the Lagrangian tracers of both velocity and velocity gradients, corresponding to the movement of particles across the boundaries of computational cells. While these oscillation in the particle velocity are relatively small and have negligible effect on the particle trajectories for time-steps of the order of CFL = 0.1, they appear to be the cause of significant oscillations in the evolution of the invariants of the velocity gradient tensor.