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Editor in Chief, Benedetto Piccoli, at nhmaims@camden.rutgers.edu.
NHM offers a strong combination of three features: Interdisciplinary character, specific focus,
and deep mathematical content. Also, the journal aims to create a link between the discrete
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,
physical, engineering, socioeconomical and biomedical problems.
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TOP 10 Most Read Articles in NHM, May 2015
1 
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.

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 
Conservation laws with discontinuous flux
Volume 2, Number 1, Pages: 159  179, 2006
Mauro Garavello,
Roberto Natalini,
Benedetto Piccoli
and Andrea Terracina
Abstract
Full Text
Related Articles
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.

4 
Nonlinear transmission problems for quasilinear diffusion systems
Volume 2, Number 2, Pages: 359  381, 2007
F. R. Guarguaglini
and R. Natalini
Abstract
Full Text
Related Articles
We study degenerate quasilinear parabolic systems in two different domains, which are connected by a nonlinear transmission condition at their interface. For a large class of models, including those modeling pollution aggression on stones and chemotactic movements of bacteria, we prove global existence, uniqueness and stability of the solutions.

5 
A review of conservation laws on networks
Volume 5, Number 3, Pages: 565  581, 2010
Mauro Garavello
Abstract
Full Text
Related Articles
This paper deals with various applications of conservation
laws on networks. In particular we consider the car traffic,
described by the LighthillWhithamRichards model and by the
AwRascleZhang model, the telecommunication case, by using the
model introduced by D'ApiceManzoPiccoli and, finally, the
case of a gas pipeline, modeled by the classical $p$system.
For each of these models we present a review of some results about Riemann
and Cauchy problems in the case of a network, formed by a single vertex
with $n$ incoming and $m$ outgoing arcs.

6 
Globally stable quasistatic evolution in plasticity with softening
Volume 3, Number 3, Pages: 567  614, 2008
G. Dal Maso,
Antonio DeSimone,
M. G. Mora
and M. Morini
Abstract
Full Text
Related Articles
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stressstrain response.

7 
Discretetocontinuum limits for strainalignmentcoupled systems:
Magnetostrictive solids, ferroelectric crystals and nematic
elastomers
Volume 4, Number 4, Pages: 667  708, 2009
Marco Cicalese,
Antonio DeSimone
and Caterina Ida Zeppieri
Abstract
Full Text
Related Articles
In the framework of linear elasticity, we study the limit of a class
of discrete free energies modeling strainalignmentcoupled systems
by a rigorous coarsegraining procedure, as the number of molecules
diverges. We focus on three paradigmatic examples: magnetostrictive
solids, ferroelectric crystals and nematic elastomers, obtaining in
the limit three continuum models consistent with those commonly
employed in the current literature. We also derive the correspondent
macroscopic energies in the presence of displacement boundary
conditions and of various kinds of applied external fields.

8 
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.

9 
Homogenization approach to filtration through a fibrous medium
Volume 2, Number 3, Pages: 529  550, 2007
Mohamed Belhadj,
Eric Cancès,
JeanFrédéric Gerbeau
and Andro Mikelić
Abstract
Full Text
Related Articles
We study the flow through fibrous media using homogenization techniques. The fibre network under study is the one already used by M. Briane in the context of heat conduction of biological tissues. We derive and justify the effective Darcy equation and the permeability tensor for such fibrous media. The theoretical results on the permeability are illustrated by some numerical simulations. Finally, the low solid fraction limit is considered. Applying results by G. Allaire to our setting, we justify rigorously the leading order term in the empirical formulas for the effective permeability used in engineering. The results are also confirmed by a direct numerical calculation of the permeability, in which the small diameter of the fibres requires high accuracy approximations.

10 
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.

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