# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

Open Access Articles

2021, 16(1): 1-29 doi: 10.3934/nhm.2020031 +[Abstract](895) +[HTML](290) +[PDF](681.95KB)
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The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

2020, 15(3): i-i doi: 10.3934/nhm.2020020 +[Abstract](686) +[HTML](248) +[PDF](90.18KB)
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2019, 14(1): i-ii doi: 10.3934/nhm.20191i +[Abstract](4785) +[HTML](534) +[PDF](97.78KB)
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2018, 13(1): 69-94 doi: 10.3934/nhm.2018004 +[Abstract](5182) +[HTML](2047) +[PDF](6157.76KB)
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We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

2017, 12(3): i-ii doi: 10.3934/nhm.201703i +[Abstract](2715) +[HTML](203) +[PDF](130.1KB)
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2017, 12(2): 217-243 doi: 10.3934/nhm.2017009 +[Abstract](4285) +[HTML](1660) +[PDF](819.2KB)
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We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

2017, 12(2): i-ii doi: 10.3934/nhm.201702i +[Abstract](2469) +[HTML](218) +[PDF](130.1KB)
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2016, 11(2): i-ii doi: 10.3934/nhm.2016.11.2i +[Abstract](2681) +[PDF](113.0KB)
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During last 20 years the theory of Conservation Laws underwent a dramatic development. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

2016, 11(1): i-ii doi: 10.3934/nhm.2016.11.1i +[Abstract](2503) +[PDF](105.3KB)
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During last 20 years the theory of Conservation Laws underwent a dramatic developmen. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

2015, 10(3): i-ii doi: 10.3934/nhm.2015.10.3i +[Abstract](3446) +[PDF](105.5KB)
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This Special Issue is based on research presented at the Workshop Modeling and Control of Social Dynamics", hosted by the Center of Computational and Integrative Biology and the Department of Mathematical Sciences at Rutgers University - Camden. The Workshop is part of the activities of the NSF Research Network in Mathematical Sciences: Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # 1107444, which is also acknowledged for funding the workshop.

2015, 10(1): i-iii doi: 10.3934/nhm.2015.10.1i +[Abstract](3425) +[PDF](156.8KB)
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The real world surrounding us is full of complex systems from various types and categories. Internet, the World Wide Web, biological and biochemical networks (brain, metabolic, protein and genomic networks), transport networks (underground, train, airline networks, road networks), communication networks (computer servers, Internet, online social networks), and many others (social community networks, electric power grids and water supply networks,...) are a few examples of the many existing kinds and types of networks [1,2,3,4,6,8,9,10,11]. In the recent past years, the study of structure and dynamics of complex networks has been the subject of intense interest. Recent advances in the study of complex networked systems has put the spotlight on the existence of more than one type of links whose interplay can affect the structure and function of those systems [5,7]. In these networks, relevant information may not be captured if the single layers are analyzed separately, since these different components and units interact with others through different channels of connectivity and dependencies. The global characteristics and behavior of these systems depend on multiple dimensions of integration, relationship or cleavage of its units.

2014, 9(4): i-ii doi: 10.3934/nhm.2014.9.4i +[Abstract](2209) +[PDF](112.9KB)
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Although the concrete is a simple man-made material with initially-controlled composition (for instance, all ingredients are known beforehand, the involved chemical mechanisms are well studied, the mechanical strength of test samples is measured accurately), forecasting its behaviour for large times under variable external (boundary) conditions is not properly understood. The main reason is that the simplicity of the material is only apparent. The combination of the heterogeneity of the material together with the occurrence of a number of multiscale phase transitions either driven by aggressive chemicals (typically ions, like in corrosion situations), or by extreme heating, or by freezing/thawing of the ice lenses within the microstructure, and the inherent non-locality of the mechanical damage leads to mathematically challenging nonlinear coupled systems of partial differential equations (PDEs).

2014, 9(1): 65-95 doi: 10.3934/nhm.2014.9.65 +[Abstract](3264) +[PDF](1803.6KB)
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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 cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction 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 cross-sectional area is shown.
2013, 8(3): 745-772 doi: 10.3934/nhm.2013.8.745 +[Abstract](3968) +[PDF](1439.2KB)
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Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
2013, 8(3): i-ii doi: 10.3934/nhm.2013.8.3i +[Abstract](2698) +[PDF](90.8KB)
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This Special Issue gathers contributions, most of which were presented at the Workshop Mathematics of Traffic Flow Modeling, Estimation and Control", organized at the Institute for Pure and Applied Mathematics of the University of California Los Angeles on December 7--9 2011.

2013, 8(2): 529-540 doi: 10.3934/nhm.2013.8.529 +[Abstract](2082) +[PDF](208.1KB)
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Assume that a stochastic process can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.
2013, 8(1): i-iii doi: 10.3934/nhm.2013.8.1i +[Abstract](2015) +[PDF](93.9KB)
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Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He is a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

2012, 7(4): i-iii doi: 10.3934/nhm.2012.7.4i +[Abstract](2405) +[PDF](109.3KB)
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Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He has been a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

2012, 7(3): i-iii doi: 10.3934/nhm.2012.7.3i +[Abstract](2533) +[PDF](177.2KB)
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The study of networks has become one of the paradigms of the science of complexity as well as a fascinating branch of research in applied mathematics, physics, engineering, sociology, biology and science in general. Different systems such as transport networks (underground, train, airline networks, road networks), communication networks (computer servers, Internet, online social networks), neural networks (neural interaction networks and brain networks), biochemical networks (metabolic, protein and genomic networks), trophic networks, social community networks, marketing and recommendation networks, other infrastructure networks (electric power grids, water supply networks) and many others (including the World Wide Web)([1],[3],[4],[7],[8],[9],[10]) are known to have behavioral and structural characteristics in common, and they can be studied by using non-linear mathematical techniques and computer modeling approaches. The interest on complex networks has certainly been promoted by the optimized rating of computing facilities, and by the availability of data on large real networks (including the World Wide Web, cortical networks, citation networks from Scientific Citation Index and online social networks). This focused section is characterized for emphasizing the latest applications of complex networks rather than the theoretical aspects, but covering several aspects as topological properties, algorithms and computation tools, models of interactions between complex systems, synchronization, control and some other related topics.

2012, 7(2): i-ii doi: 10.3934/nhm.2012.7.2i +[Abstract](2340) +[PDF](101.9KB)
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The theory of Mean Field Games (MFG, in short) is a branch of the theory of Differential Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the mass of the other agents. The name comes from particle physics where it is common to consider interactions among particles as an external mean field which influences the particles. In spite of the optimization made by rational agents, playing the role of particles in such models, appropriate mean field equations can be derived to replace the many particles interactions by a single problem with an appropriately chosen external mean field which takes into account the global behavior of the individuals.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9