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The complex interactions between human behaviors and social economic sciences is critically analyzed in this paper in view of possible applications of mathematical modeling as an attainable interdisciplinary approach to understand and simulate the aforementioned dynamics. The quest is developed along three steps: Firstly an overall analysis of social and economic sciences indicates the main requirements that a contribution of mathematical modeling should bring to these sciences; subsequently the focus moves to an overview of mathematical tools and to the selection of those which appear, according to the authors bias, appropriate to the modeling; finally, a survey of applications is presented looking ahead to research perspectives.

In a previous paper a mathematical model was developed for the dynamics of activation and clonal expansion of T cells during the immune response to a single type of antigen challenge, constructed phenomenologically in the macroscopic framework of a thermodynamic theory of continuum mechanics for reacting and proliferating fluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.

In this paper we investigate the critical conditions under which a small perturbation in an homogeneous continuum can possibly grows into a shear band instability. In particular, we analyze from a thermodynamical viewpoint the phenomenon of shear bands in viscoanelastic media with memory. It is emphasized, in the scientific literature, that the specific adopted rheology strongly affects the results so that a special attention has to be paid, also for engineering purposes, to the accuracy of the rheological model. Several well-known rheological model (for instance the so called Maxwell or Jeffreys media) are particular cases of the general model we adopt in the paper to analyze shear bands. Instability conditions, giving rise to shear bands formation, are obtained by introducing small perturbations around an homogeneous deformation into the system of differential equations governing the problem of homogeneous deformations in the considered continuous medium; as a result a non-homogeneous linear dynamical system is obtained whose stability is analyzed. A research perspective in view of a possible comparison with experimental results is proposed; in particular the simple methodology proposed in the paper should be applied in view of using the phenomenon of the initiation of shear bands to calculate the thermomechanical coefficients of real materials.

In a previous paper we outlined a geometric model for the thermodynamical description of ferrimagnetic crystals with a non-Euclidean structure. Applying a geometrization technique based on a model for magnetizable deformable media earlier introduced by Maugin, starting from an appropriate dynamical system on the fiber bundle of processes for simple material elements of these media, the expressions of the entropy function and the entropy 1-form were obtained. In this contribution we deepen the study of this geometrical model. We give a detailed description of the media under consideration, we introduce the transformation induced by the process and, applying the closure conditions for the entropy 1-form, we work out the necessary conditions for the existence of the entropy function. The derivation of the entropy 1-form is the starting point to introduce an extended thermodynamical phase space. Finally, from the exploitation of the Clausius–Duhem inequality, we derive, using Maugin's techniques, the residual dissipation inequality and the heat equation in the first and second form.

This paper proposes a two scale modeling approach to vehicular traffic, where macroscopic conservation equations are closed by models at the microscopic scale obtained by a mathematical interpretation of driver behaviors to local flow conditions. The paper focuses on the closure of the mass conservation equations by phenomenological models derived by a detailed analysis at the scale of individual vehicles.

This paper deals with the modeling of the dynamics of clonal expansion and immune competition of T cells. The modeling is based on the approach of continuum mechanics, namely by conservation equations closed by phenomenological constitutive equations concerning the material behaviors. The qualitative properties of the model are analyzed in detail. A critical analysis concludes this paper with indication of research perspectives.

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) [Formula: see text] and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model. In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.