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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, December 2016
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
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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 
Computational models for fluid exchange between microcirculation and tissue interstitium
Volume 9, Number 1, Pages: 135  159, 2014
Laura Cattaneo
and Paolo Zunino
Abstract
References
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The aim of this work is to develop a computational model able to capture the interplay between microcirculation and interstitial flow. Such phenomena are at the basis of the exchange of nutrients, wastes and pharmacological agents between the cardiovascular system and the organs. They are particularly interesting for the study of effective therapies to treat vascularized tumors with drugs. We develop a model applicable at the microscopic scale, where the capillaries and the interstitial volume can be described as independent structures capable to propagate flow. We facilitate the analysis of complex capillary bed configurations, by representing the capillaries as a onedimensional network, ending up with a heterogeneous system characterized by channels embedded into a porous medium. We use the immersed boundary method to couple the onedimensional with the threedimensional flow through the network and the interstitial volume, respectively. The main idea consists in replacing the immersed network with an equivalent concentrated source term. After discussing the details for the implementation of a computational solver, we apply it to compare flow within healthy and tumor tissue samples.

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
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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 
The derivation of continuum limits of neuronal networks with gapjunction couplings
Volume 9, Number 1, Pages: 111  133, 2014
Claudio Canuto
and Anna Cattani
Abstract
References
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We consider an idealized network, formed by $N$ neurons individually described by the FitzHughNagumo equations
and connected by electrical synapses. The limit for $N \to \infty$ of the resulting discrete model is thoroughly
investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies
for passing to the limit are analysed: i) a more conventional approach, based on a fixed nearestneighbour
connection topology accompanied by a suitable scaling of the diffusion coefficients; ii) a new approach, in which the number of connections to any given neuron varies with $N$ according to a precise law, which simultaneously
guarantees the nontriviality of the limit and the locality of neuronal interactions. Both approaches yield in the limit
a pdebased model, in which the distribution of action potential obeys a nonlinear reactionconvectiondiffusion equation;
convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are
discussed, both theoretically and numerically.

6 
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.

7 
A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function
Volume 9, Number 1, Pages: 191  196, 2014
Boris Muha
Abstract
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The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous
function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph
of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous
mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluidstructure interaction problems.

8 
Asymptotic synchronous behavior of Kuramoto type models with frustrations
Volume 9, Number 1, Pages: 33  64, 2014
SeungYeal Ha,
Yongduck Kim
and Zhuchun Li
Abstract
References
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We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization.
We consider three Kuramototype models with frustrations.
First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed.
Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phaselocked states via the segregation stage. Finally, we consider a Kuramototype model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Lück and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained.
Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.

9 
Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Volume 9, Number 1, Pages: 1  31, 2014
Mattia Bongini
and Massimo Fornasier
Abstract
References
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Conditional selforganization and patternformation are relevant phenomena arising in biological, social, and economical contexts, and received a growing attention in recent years in mathematical modeling. An important issue related to optimal government strategies is how to design external parsimonious interventions, aiming at enforcing systems to converge to specific patterns. This is in contrast to other models where the players of the systems are allowed to interact freely and are supposed autonomously, either by game rules or by embedded decentralized feedback control rules, to converge to patterns.
In this paper we tackle the problem of designing optimal centralized feedback controls for systems of moving particles, subject to mutual attraction and repulsion forces, and friction.
Under certain conditions on the attraction and repulsion forces, if the total energy of the system, composed of the sum of its kinetic and potential parts, is below a certain critical threshold, then such systems are known to
converge autonomously to the stable configuration of keeping confined and collision avoiding in space, uniformly in time. If the energy is above such a critical level, then the space coherence can be lost.
We show that in the latter situation of lost selforganization, one can nevertheless steer the system to return to stable energy levels by feedback controls defined as the minimizers of a certain functional with $l_1$norm penalty and constraints. Additionally we show that the optimal strategy in this class of controls is necessarily sparse, i.e., the control acts on at most one agent at each time. This is another remarkable example of how homophilious systems, i.e., systems where agents tend to be strongly more influenced by near agents than far ones, are naturally prone to sparse stabilization, explaining the effectiveness of parsimonious interventions of governments in societies.

10 
Motion of discrete interfaces in lowcontrast periodic media
Volume 9, Number 1, Pages: 169  189, 2014
Giovanni Scilla
Abstract
References
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Related Articles
We study the motion of discrete interfaces driven by ferromagnetic interactions in a twodimensional lowcontrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discretetocontinuum analysis. As in a recent paper by Braides and Scilla dealing with highcontrast periodic media, we give an example showing that in general the effective motion does not depend only on the $\Gamma$limit, but also on geometrical features that are not detected in the static description. We show that there exists a critical value $\widetilde{\delta}$ of the contrast parameter $\delta$ above which the discrete motion is constrained and coincides with the highcontrast case. If $\delta<\widetilde{\delta}$ we have a new pinning threshold and a new effective velocity both depending on $\delta$. We also consider the case of nonuniform inclusions distributed into periodic uniform layers.

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