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Volume 1, 2006

Networks & Heterogeneous Media

2006 , Volume 1 , Issue 3

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Asymptotic analysis of an array of closely spaced absolutely conductive inclusions
Leonid Berlyand , Giuseppe Cardone , Yuliya Gorb and  Gregory Panasenko
2006, 1(3): 353-377 doi: 10.3934/nhm.2006.1.353 +[Abstract](37) +[PDF](359.8KB)
We consider the conductivity problem in an array structure with square closely spaced absolutely conductive inclusions of the high concentration, i.e. the concentration of inclusions is assumed to be close to 1. The problem depends on two small parameters: $\varepsilon$, the ratio of the period of the micro-structure to the characteristic macroscopic size, and $\delta$, the ratio of the thickness of the strips of the array structure and the period of the micro-structure. The complete asymptotic expansion of the solution to problem is constructed and justified as both $\varepsilon$ and $\delta$ tend to zero. This asymptotic expansion is uniform with respect to $\varepsilon$ and $\delta$ in the area $\{\varepsilon=O(\delta^{\alpha}),~\delta =O(\varepsilon^{\beta})\}$ for any positive $\alpha, \beta.$
A fluid dynamic model for supply chains
Ciro D'Apice and  Rosanna Manzo
2006, 1(3): 379-398 doi: 10.3934/nhm.2006.1.379 +[Abstract](39) +[PDF](190.3KB)
The paper deals with a fluid dynamic model for supply chains. A mixed continuum-discrete model is proposed and possible choices of solutions at nodes guaranteeing the conservation of fluxes are discussed. Fixing a rule a Riemann solver is defined and existence of solutions to Cauchy problems is proved.
Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity
M.A.J Chaplain and  G. Lolas
2006, 1(3): 399-439 doi: 10.3934/nhm.2006.1.399 +[Abstract](136) +[PDF](993.7KB)
Solid tumours grow through two distinct phases: the avascular and the vascular phase. During the avascular growth phase, the size of the solid tumour is restricted largely by a diffusion-limited nutrient supply and the solid tumour remains localised and grows to a maximum of a few millimetres in diameter. However, during the vascular growth stage the process of cancer invasion of peritumoral tissue can and does take place. A crucial component of tissue invasion is the over-expression by the cancer cells of proteolytic enzyme activity, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs). uPA itself initiates the activation of an enzymatic cascade that primarily involves the activation of plasminogen and subsequently its matrix degrading protein plasmin. Degradation of the matrix then enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body.
    In this paper we consider a relatively simple mathematical model of cancer cell invasion of tissue (extracellular matrix) which focuses on the role of a generic matrix degrading enzyme such as uPA. The model consists of a system of reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the host tissue. The results obtained from numerical computations carried out on the model equations produce dynamic, heterogeneous spatio-temporal solutions and demonstrate the ability of a rather simple model to produce complicated dynamics, all of which are associated with tumour heterogeneity and cancer cell progression and invasion.
Evaluating the effects of parking price and location in multi-modal transportation networks
Massimiliano Caramia and  Giovanni Storchi
2006, 1(3): 441-465 doi: 10.3934/nhm.2006.1.441 +[Abstract](34) +[PDF](417.4KB)
In urban transportation, combined mode trips are increasing in importance due to current urban transportation polices which encourage the use of transit through the creation of apposite parking lots and improvements in the public transportation system. It is widely recognized that parking policy plays an important role in urban management: parking policy measures not only affect the parking system, but also generate impacts to the transport and socioeconomic system of a city. The present paper attempts to expand on previous research concerning the development of models to capture drivers' parking behavior. It introduces in the modeling structure additional variables to the ones usually employed, with which the drivers' behavior to changes in prices and distances (mainly walking) are better captured. We develop a network model that represents trips as a combination of private and transit modes. A graph representing four different modes (car, bus, metro and pedestrian) is defined and a set of free park and ride facilities is introduced to discourage the use of private cars. An algorithm that evaluates the location and the effects of the parking price variation using multi-modal shortest paths is proposed together with an application to the City of Rome. Computational results are shown.
Homogenization of linear spatially periodic electronic circuits
Michel Lenczner
2006, 1(3): 467-494 doi: 10.3934/nhm.2006.1.467 +[Abstract](32) +[PDF](290.1KB)
In this paper we establish a simplified model of general spatially periodic linear electronic analog networks. It has a two-scale structure. At the macro level it is an algebro-differential equation and a circuit equation at the micro level. Its construction is based on the concept of two-scale convergence, introduced by the author in the framework of partial differential equations, adapted to vectors and matrices. Simple illustrative examples are detailed by hand calculation and a numerical simulation is reported.
A Well Posed Riemann Problem for the $p$--System at a Junction
Rinaldo M. Colombo and  Mauro Garavello
2006, 1(3): 495-511 doi: 10.3934/nhm.2006.1.495 +[Abstract](50) +[PDF](212.0KB)
This work is devoted to the solution to Riemann Problems for the $p$-system at a junction, the main goal being the extension to the case of an ideal junction of the classical results that hold in the standard case.
Antoine Gloria Cermics
2006, 1(3): 513-514 doi: 10.3934/nhm.2006.1.513 +[Abstract](32) +[PDF](128.2KB)
In this erratum, we correct a mistake that has propagated in the error analysis of [4].

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