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NHM offers a strong combination of three features: Interdisciplinary character, specific focus,
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and the continuous communities, which distinguishes it from other journals with strong PDE orientation.
NHM publishes original contributions of high quality in networks,
heterogeneous media and related fields.
NHM is thus devoted to research work on complex media arising in mathematical,
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TOP 10 Most Read Articles in NHM, January 2015
1 
Spectral stiff problems in domains surrounded
by thin stiff and heavy bands: Local effects for eigenfunctions
Volume 6, Number 1, Pages: 1  35, 2011
Delfina Gómez,
Sergey A. Nazarov
and Eugenia Pérez
Abstract
References
Full Text
Related Articles
We consider the Neumann spectral problem for a second order differential operator,
with piecewise constants coefficients, in a domain
$\Omega_\varepsilon$ of $R^2$. Here $\Omega_\varepsilon$ is
$\Omega \cup \omega_\varepsilon \cup \Gamma$, where $\Omega$ is
a fixed bounded domain with boundary $\Gamma$,
$\omega_\varepsilon$ is a curvilinear band of variable width
$O(\varepsilon)$, and $\Gamma=\overline{\Omega}\cap \overline
{\omega_\varepsilon}$. The density and stiffness
constants are of order $O(\varepsilon^{m1})$ and $O(\varepsilon^{1})$
respectively in this band, while they are of order $O(1)$ in
$\Omega$; $m$ is a positive parameter and $\varepsilon \in
(0,1)$, $\varepsilon\to 0$. Considering the range of the low, middle
and high frequencies, we provide asymptotics for the eigenvalues
and the corresponding eigenfunctions. For $m>2$, we highlight the middle
frequencies for which the corresponding eigenfunctions may be
localized asymptotically in small neighborhoods of certain points
of the boundary.

2 
Selforganized network flows
Volume 2, Number 2, Pages: 193  210, 2007
Dirk Helbing,
Jan Siegmeier
and Stefan Lämmer
Abstract
Full Text
Related Articles
A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other significant features of traffic flows such as jam formation, spillovers, and loaddependent transportation times. Furthermore, conflicts or coordination problems of intersecting or merging flows are considered as well. Making assumptions regarding the permeability of the intersection as a function of the conflicting flows and the queue lengths, we find selforganized oscillations in the flows similar to the operation of traffic lights.

3 
Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks
Volume 3, Number 2, Pages: 185  200, 2008
Jan Lorenz
and Stefano Battiston
Abstract
Full Text
Related Articles
We study the mean field approximation of a recent model of cascades on networks relevant to the investigation of systemic risk control in financial networks. In the model, the hypothesis of a trend reinforcement in the stochastic process describing the fragility of the nodes, induces a tradeoff in the systemic risk with respect to the density of the network. Increasing the average link density, the network is first less exposed to systemic risk, while above an intermediate value the systemic risk increases. This result offers a simple explanation for the emergence of instabilities in financial systems that get increasingly interwoven. In this paper, we study the dynamics of the probability density function of the average fragility. This converges to a unique stationary distribution which can be computed numerically and can be used to estimate the systemic risk as a function of the parameters of the model.

4 
A distributed model of traffic flows on extended regions
Volume 5, Number 3, Pages: 525  544, 2010
Fabio Della Rossa,
Carlo D’Angelo
and Alfio Quarteroni
Abstract
Full Text
Related Articles
This work deals with the modelling of traffic flows in complex
networks, spanning twodimensional regions whose size
( macroscale ) is much greater than the characteristic size of
the network arcs ( microscale). A typical example is the
modelling of traffic flow in large urbanized areas with diameter of
hundreds of kilometers, where standard models of traffic flows on
networks resolving all the streets are computationally too
expensive. Starting from a stochastic lattice gas model with simple
constitutive laws, we derive a distributed twodimensional model of
traffic flow, in the form of a nonlinear diffusionadvection
equation for the particle density. The equation is formally
equivalent to a (nonlinear) Darcy's filtration law. In particular,
it contains two parameters that can be seen as the porosity and the
permeability tensor of the network. We provide suitable algorithms
to extract these parameters starting from the geometry of the
network and a given microscale model of traffic flow (for instance
based on cellular automata). Finally, we compare the fully
microscopic simulation with the finite element solution of
our upscaled model in realistic cases, showing that our model is
able to capture the largescale feature of the flow.

5 
Random homogenization of fractional obstacle problems
Volume 3, Number 3, Pages: 523  554, 2008
Luis Caffarelli
and Antoine Mellet
Abstract
Full Text
Related Articles
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

6 
Quasistatic evolution for CamClay plasticity: The spatially
homogeneous case
Volume 5, Number 1, Pages: 97  132, 2010
Gianni Dal Maso
and Francesco Solombrino
Abstract
Full Text
Related Articles
We study the spatially uniform case of the quasistatic evolution in
CamClay plasticity, a relevant example of small strain
nonassociative elastoplasticity. Introducing a viscous
approximation, the problem reduces to determine the limit behavior
of the solutions of a singularly perturbed system of ODE's in a
finite dimensional Banach space. Depending on the sign of two
explicit scalar indicators, we see that the limit dynamics presents,
under quite generic assumptions, the alternation of three possible
regimes: the elastic regime, when the limit equation is just the
equation of linearized elasticity; the slow dynamics, when the
stress evolves smoothly on the yield surface and plastic flow is
produced; the fast dynamics, which may happen only in the softening
regime, when viscous solutions exhibit a jump determined by the
heteroclinic orbit of an auxiliary system. We give an iterative
procedure to construct a viscous solution.

7 
Improving on computation of homogenized coefficients in the
periodic and quasiperiodic settings
Volume 5, Number 1, Pages: 1  29, 2010
Xavier Blanc
and Claude Le Bris
Abstract
Full Text
Related Articles
In quasiperiodic homogenization of elliptic equations or nonlinear
periodic homogenization of systems, the cell problem must be in
general set on the whole space. Numerically computing the
homogenization coefficient therefore implies a truncation error, due
to the fact that the problem is approximated on a bounded, large
domain. We present here an approach that improves the rate of
convergence of this approximation.

8 
Rateindependent phase transitions in elastic materials: A Youngmeasure approach
Volume 5, Number 2, Pages: 257  298, 2010
Alice Fiaschi
Abstract
Full Text
Related Articles
A quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the usage of suitable regularity arguments in order to prove an existence result for a notion of evolution presenting some improvements with respect to the one defined in [13], for infinitely many phases.

9 
Gas flow in pipeline networks
Volume 1, Number 1, Pages: 41  56, 2006
Mapundi K. Banda,
Michael Herty
and Axel Klar
Abstract
Full Text
Related Articles
We introduce a model for gas flow in pipeline networks based on
the isothermal Euler equations. We model the intersection of multiple pipes
by posing an additional assumption on the pressure at the interface. We give a
method to obtain solutions to the gas network problem and present numerical
results for sample networks.

10 
A mathematical model for spaghetti cooking with free boundaries
Volume 6, Number 1, Pages: 37  60, 2011
Antonio Fasano,
Mario Primicerio
and Andrea Tesi
Abstract
References
Full Text
Related Articles
We propose a mathematical model for the process of dry pasta cooking with specific reference to spaghetti.
Pasta cooking is a twostage process: water penetration followed by starch gelatinization. Differently from the approach adopted so far in the
technical literature, our model includes free boundaries: the water penetration front and the gelatinization onset front representing a fast stage
of the corresponding process. Behind the respective fronts water sorption and gelatinization proceed according to some kinetics.
The outer boundary is also moving and unknown as a consequence of swelling. Existence and uniqueness are proved and numerical simulations are presented.

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