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D. Brunetto, C. Andrà, N. Parolini and M. Verani

Abstract

This paper aims at bridging Mathematical Modelling and Mathematics Education by studying the opinion dynamics of students who work in small groups during mathematics classrooms. In particular, we propose a model which hinges upon the pioneering work of Hegselmann and Krause on opinion dynamics and integrates recent results of interactionist research in Mathematical Education. More precisely, the proposed model incorporates the following features: 1) the feelings of each student towards the classmates (building upon the so-called \I can" -\you can" framework); 2) the different levels of preparation of the students; 3) the presence of the teacher, who may or may not intervene to drive the students towards the correct solution of the problem. Several numerical experiments are presented to assess the capability of the model in reproducing typical realistic scenarios.

Open access

Giacomo Albi and Lorenzo Pareschi

Abstract

In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of a controller, acting in order to enhance consensus. Two types of selective controls have been presented: an homogeneous control filtered by a selective function and a distributed control active only on a selective set. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we derive the mean-field limit of the feedback selective constrained dynamics, which eventually will be solved numerically by means of a stochastic algorithm, able to simulate efficiently the selective constrained dynamics. Finally, several numerical simulations are reported to show the efficiency of the proposed techniques.

Open access

M. Dolfin

Abstract

The political replacement effect is an interesting socio-political hypothesis introduced by Acemoglu and Robinson and statistically tested. It may determine, under some conditions, the phenomenon of innovation blocking, possibly leading to economic backwardness in a society. In a previous paper, we have introduced a kinetic model with stochastic evolutive game-type interactions, analyzing the relationship between the level of political competition in a society and the degree of economic liberalization. In the present paper we model we model the possibility of having a sort of phase transition occurring in the system when the phenomenon of blocking of the introduction of technological innovation, intended in a broad sense, appears. Crossing a critical point, the rules of interactions change by means of slightly different transition probabilities nevertheless determining very significant differences in the resulting long-term solutions.

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Nathan Muyinda, Bernard De Baets and Shodhan Rao

Abstract

We identify sufficient conditions for the stability of some well-known finite difference schemes for the solution of the multivariable reaction-diffusion equations that model chemical reaction networks. Since the equations are mainly nonlinear, these conditions are obtained through local linearization. A recurrent condition is that the Jacobian matrix of the reaction part evaluated at some positive unknown solution is either D-semi-stable or semi-stable. We demonstrate that for a single reversible chemical reaction whose kinetics are monotone, the Jacobian matrix is D-semi-stable and therefore such schemes are guaranteed to work well.

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Damián A. Knopoff and Germán A. Torres

Abstract

Kinetic models have so far been used to model wealth distribution in a society. In particular, within the framework of the kinetic theory for active particles, some important models have been developed and proposed. They involve nonlinear interactions among individuals that are modeled according to game theoretical tools by introducing parameters governing the temporal dynamics of the system. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile.

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M. Penati, E. Miglio, N. Parolini and R. Porcù

Abstract

The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to the simulation of brownout phenomena occurring in helicopter take-of and landing.

Open access

Lidia Saluto, David Jou and Maria Stella Mongiovì

Abstract

We consider heat rectification in radial flows of turbulent helium II, where heat flux is not described by Fourier's law, but by a more general law. This is different from previous analyses of heat rectification, based on such law. In our simplified analysis we show that the coupling between heat flux and the gradient of vortex line density plays a decisive role in such rectification. Such rectification will be low at low and high values of the heat rate, but it may exhibit a very high value at an intermediate value of the heat rate. In particular, for a given range of values for the incoming heat ow, the outgoing heat flow corresponding to the exchange of internal and external temperatures would be very small. This would imply difficulties in heat removal in a given range of temperature gradients.

Open access

Najat M. Omar Dabnoun and Maria Stella Mongiovì

Abstract

This paper deals with the modeling of interactions between the immune system and cancer cells, in the framework of the mathematical kinetic theory for active particles. The work deepens a previous paper of Belloquid et al. that assumes spatial homogeneity and discrete values of the activity of cancer and immune cells. A number of simulations are made with the aim to investigate how the state of the various cell populations evolves in time depending on the choice of the free parameters.

Open access

Jacques Tagoudjeu and Abdelghani Bellouquid

Abstract

In this paper, we propose a numerical approach to solve a kinetic model of chemotaxis phenomena. This scheme is shown to be uniformly stable with respect to the small parameter, consistent with the fluid-diffusion limit (Keller-Segel model). Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model that couples a kinetic equation with macroscopic ones. This method is validated by various test cases and compared to other standard methods.