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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, February 2017
1 
Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods
Volume 6, Number 3, Pages: 521  544, 2011
Daniel Roggen,
Martin Wirz,
Gerhard Tröster
and Dirk Helbing
Abstract
References
Full Text
Related Articles
Mobile onbody sensing has distinct advantages for the analysis and understanding of crowd dynamics: sensing is not geographically restricted to a specific instrumented area, mobile phones offer onbody sensing and they are already deployed on a large scale, and the rich sets of sensors they contain allows one to characterize the behavior of users through pattern recognition techniques.
In this paper we present a methodological framework for the machine recognition of crowd behavior from onbody sensors, such as those in mobile phones.
The recognition of crowd behaviors opens the way to the acquisition of largescale datasets for the analysis and understanding of crowd dynamics.
It has also practical safety applications by providing improved crowd situational awareness in cases of emergency.
The framework comprises: behavioral recognition with the user's mobile device, pairwise analyses of the activity relatedness of two users, and graph clustering in order to uncover globally, which users participate in a given crowd behavior.
We illustrate this framework for the identification of groups of persons walking, using empirically collected data.
We discuss the challenges and research avenues for theoretical and applied mathematics arising from the mobile sensing of crowd behaviors.

2 
Numerical network models and entropy principles for isothermal junction flow
Volume 9, Number 1, Pages: 65  95, 2014
Gunhild A. Reigstad
Abstract
References
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We numerically explore network models which are derived for the isothermal Euler equations.
Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal crosssectional areas for all connected pipe sections.
In the present paper we extend this proof to the case of pipe sections of different crosssectional areas.
We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical highresolution Roe scheme.
We propose a numerical treatment of the boundary conditions at the pipejunction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
Numerical results are provided for two different network configurations and for three different network models.
Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness.
Analytical predictions for junctions connecting three pipe sections are verified for both network configurations.
Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different crosssectional area is shown.

3 
Modeling opinion dynamics: How the network enhances consensus
Volume 10, Number 4, Pages: 877  896, 2015
Marina Dolfin
and Mirosław Lachowicz
Abstract
References
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In this paper we
analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensusdissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered.
We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing
a consensusdissent dynamics of interactions and a qualitative analysis is given.

4 
Constructing setvalued fundamental diagrams from Jamiton solutions in second order traffic models
Volume 8, Number 3, Pages: 745  772, 2013
Benjamin Seibold,
Morris R. Flynn,
Aslan R. Kasimov
and Rodolfo R. Rosales
Abstract
References
Full Text
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Fundamental diagrams of vehicular traffic flow are generally multivalued in the congested flow regime. We show that such setvalued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical PayneWhitham model or the inhomogeneous AwRascleZhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multivalued parts in the fundamental diagram correspond precisely to jamitondominated solutions. This study shows that transitions from functionvalued to setvalued parts in a fundamental diagram arise naturally in wellknown second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

5 
Liquidity generated by heterogeneous beliefs and costly estimations
Volume 7, Number 2, Pages: 349  361, 2012
Min Shen
and Gabriel Turinici
Abstract
References
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We study the liquidity, defined as the size of the trading volume, in a situation where an infinite number of agents
with heterogeneous beliefs reach a tradeoff between the cost of a precise estimation (variable depending on the agent) and the expected wealth from trading. The "true" asset price is not known and the market price is set at a level that clears the market. We show that, under some technical assumptions, the model has natural properties such as monotony of supply and demand functions with respect to the price, existence of an equilibrium and monotony with respect to the
marginal cost of information. We also situate our approach within the Mean Field Games (MFG) framework of Lions and Lasry which allows to obtain an interpretation as a limit of Nash equilibrium for an infinite number of agents.

6 
Analysis of a system of nonlocal conservation laws for multicommodity flow on networks
Volume 10, Number 4, Pages: 749  785, 2015
Martin Gugat,
Alexander Keimer,
Günter Leugering
and Zhiqiang Wang
Abstract
References
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We consider a system of scalar nonlocal conservation laws on networks that model a highly reentrant multicommodity manufacturing system as encountered in semiconductor production.
Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress.
We illustrate the dynamics for two commodities.
In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered.
On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs.
For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass.
We provide results of regularity, existence and wellposedness of the multicommodity network model for $L^{p}$, $BV$ and $W^{1,p}$data.
Moreover, we define an $L^{2}$tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem.

7 
Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics
Volume 10, Number 4, Pages: 787  807, 2015
SeungYeal Ha,
Se Eun Noh
and Jinyeong Park
Abstract
References
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We study the practical synchronization of the Kuramoto dynamics of units distributed over networks. The unit dynamics on the nodes of the network are governed by the interplay between their own intrinsic dynamics and Kuramoto coupling dynamics. We present two sufficient conditions for practical synchronization under homogeneous and heterogeneous forcing. For practical synchronization estimates, we employ the configuration diameter as a Lyapunov functional, and derive a Gronwalltype differential inequality for this value.

8 
(Almost) Everything you always wanted to know about deterministic control problems in stratified domains
Volume 10, Number 4, Pages: 809  836, 2015
Guy Barles
and Emmanuel Chasseigne
Abstract
References
Full Text
Related Articles
We revisit the pioneering work of Bressan & Hong on
deterministic control problems in stratified domains, i.e. control problems
for which the dynamic and the cost may have discontinuities on submanifolds
of $\mathbb{R}^N$. By using slightly different methods, involving more partial
differential equations arguments, we $(i)$ slightly improve the assumptions
on the dynamic and the cost; $(ii)$ obtain a comparison result for general
semicontinuous sub and supersolutions (without any continuity assumptions
on the value function nor on the sub/supersolutions); $(iii)$ provide a general
framework in which a stability result holds.

9 
A destinationpreserving model for simulating Wardrop equilibria in traffic flow on networks
Volume 10, Number 4, Pages: 857  876, 2015
Emiliano Cristiani
and Fabio S. Priuli
Abstract
References
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Related Articles
In this paper we propose a LWRlike model for traffic flow on networks which allows to track several groups of drivers, each of them being characterized only by their destination in the network.
The path actually followed to reach the destination is not assigned a priori, and can be chosen by the drivers during the journey, taking decisions at junctions.
The model is then used to describe three possible behaviors of drivers, associated to three different ways to solve the route choice problem: 1. Drivers ignore the presence of the other vehicles; 2. Drivers react to the current distribution of traffic, but they do not forecast what will happen at later times; 3. Drivers take into account the current and future distribution of vehicles. Notice that, in the latter case, we enter the field of differential games, and, if a solution exists, it likely represents a global equilibrium among drivers.
Numerical simulations highlight the differences between the three behaviors and offer insights into the existence of equilibria.

10 
Optima and
equilibria for traffic flow on networks
with backward propagating queues
Volume 10, Number 4, Pages: 717  748, 2015
Alberto Bressan
and Khai T. Nguyen
Abstract
References
Full Text
Related Articles
This paper studies an optimal decision problem for several
groups of drivers on a network of roads. Drivers have different
origins and destinations,
and different costs, related to their departure and arrival time.
On each road the flow is governed by a conservation law, while intersections
are modeled using buffers of limited capacity, so that queues can spill backward
along roads leading to a crowded intersection.
Two main results are proved: (i)
the existence of a globally optimal solution, minimizing the sum of the costs
to all drivers, and (ii) the existence of
a Nash equilibrium solution, where no driver can lower his own cost by changing
his departure time or the route taken to reach destination.

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