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Networks & Heterogeneous Media

2012 , Volume 7 , Issue 1

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From the Newton equation to the wave equation in some simple cases
Xavier Blanc , Claude Le Bris and  Pierre-Louis Lions
2012, 7(1): 1-41 doi: 10.3934/nhm.2012.7.1 +[Abstract](31) +[PDF](551.8KB)
Abstract:
We prove that, in some simple situations at least, the one-dimensional wave equation is the limit as the microscopic scale goes to zero of some time-dependent Newton type equation of motion for atomistic systems. We address both some linear and some nonlinear cases.
Differential equation approximations of stochastic network processes: An operator semigroup approach
András Bátkai , Istvan Z. Kiss , Eszter Sikolya and  Péter L. Simon
2012, 7(1): 43-58 doi: 10.3934/nhm.2012.7.43 +[Abstract](70) +[PDF](394.2KB)
Abstract:
The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size ($N$). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.
A sufficient condition for classified networks to possess complex network features
Xianmin Geng , Shengli Zhou , Jiashan Tang and  Cong Yang
2012, 7(1): 59-69 doi: 10.3934/nhm.2012.7.59 +[Abstract](52) +[PDF](320.4KB)
Abstract:
We investigate network features for complex networks. A sufficient condition for the limiting random variable to possess the scale free property and the high clustering property is given. The uniqueness and existence of the limit of a sequence of degree distributions for the process is proved. The limiting degree distribution and a lower bound of the limiting clustering coefficient of the graph-valued Markov process are obtained as well.
Robot's finger and expansions in non-integer bases
Anna Chiara Lai and  Paola Loreti
2012, 7(1): 71-111 doi: 10.3934/nhm.2012.7.71 +[Abstract](45) +[PDF](1864.2KB)
Abstract:
We study a robot finger model in the framework of the theory of expansions in non-integer bases. We investigate the reachable set and its closure. A control policy to get approximate reachability is also proposed.
Robustness of finite element simulations in densely packed random particle composites
Daniel Peterseim
2012, 7(1): 113-126 doi: 10.3934/nhm.2012.7.113 +[Abstract](63) +[PDF](483.4KB)
Abstract:
This paper presents some weighted $H^2$-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., perfectly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of thermal conductivity of the material as the volume fraction of the particles is close to the jammed state. We prove that the characteristic percolation behavior of the material is well captured by standard conforming finite element models.
Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length
Paolo Tilli
2012, 7(1): 127-136 doi: 10.3934/nhm.2012.7.127 +[Abstract](38) +[PDF](305.7KB)
Abstract:
In this paper we prove some lower bounds for the compliance functional, in terms of the $1$-dimensional Hausdorff measure of the Dirichlet region and the number of its connected components. When the measure of the Dirichlet region is large, these estimates are asymptotically optimal and yield a proof of a conjecture by Buttazzo and Santambrogio.
Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients
Patrick W. Dondl and  Michael Scheutzow
2012, 7(1): 137-150 doi: 10.3934/nhm.2012.7.137 +[Abstract](43) +[PDF](382.7KB)
Abstract:
We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.
Steklov problems in perforated domains with a coefficient of indefinite sign
Valeria Chiado Piat , Sergey S. Nazarov and  Andrey Piatnitski
2012, 7(1): 151-178 doi: 10.3934/nhm.2012.7.151 +[Abstract](29) +[PDF](486.7KB)
Abstract:
We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in periodically perforated domain under the assumption that the spectral weight function changes sign. We show that the limit behaviour of the spectrum depends essentially on wether the average of the weight function over the boundary of holes is positive, or negative or equal to zero. In all these cases we construct the asymptotics of the eigenpairs.
Renormalized Ginzburg-Landau energy and location of near boundary vortices
Leonid Berlyand , Volodymyr Rybalko and  Nung Kwan Yip
2012, 7(1): 179-196 doi: 10.3934/nhm.2012.7.179 +[Abstract](35) +[PDF](418.5KB)
Abstract:
We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.

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