Maria Letizia Bertotti

Free University of Bozen-Bolzano | Unibolzano · Faculty of Science and Technology

Some novel models of opinion dynamics are here formulated in terms of ordinary and of stochastic differential equations. The focus is on the possible emergence of polarization phenomena. Alongside attitudes and mechanisms related to individual-individual interactions, issues such as the influence of persuaders, acting perhaps on different time-scal...

In this paper, a mathematical model is formulated, suitable to explain the evolution of income distribution over a population in the presence of production. The model is conceived from the perspective of complexity. Indeed, the income distribution emerges as the result of a myriad of economic exchanges taking place between individuals. In fact, the...

This study introduces a kinetic model that significantly improves the interpretation of the oxygen radical absorbance capacity (ORAC) assay. Our model accurately simulates and fits the bleaching kinetics of fluorescein in the presence of various antioxidants, achieving high correlation values (R2 > 0.99) with the experimental data. The fit to the e...

We propose and examine a model expressed by stochastic differential equations for the evolution of a complex system. We refer in particular to a market society, in which the state of each individual is identified by the amount of money at his/her disposal. The evolution of such a system over time is described by suitable equations that link the ins...

In this work we present the results of a quasi--static simulation of a ropeway. The mathematical background is sketched and the system of equations is solved numerically for a typical example. The whole computational problem is immediately parallelizable and therefore fast executable. It represents the first approximation of an exact time--dependen...

Reducing inequality is a tremendously important sustainable development goal. Albeit providing stylised frames for modelling, also mathematics can contribute to understanding and explaining the emergence of collective patterns in complex socio-economic systems. It can then effectively help to identify actions and measures to be taken and support po...

In this paper, an elementary mathematical model describing the introduction of a universal basic income in a closed market society is constructed. The model is formulated in terms of a system of nonlinear ordinary differential equations, each of which gives account of how the number of individuals in a certain income class changes in time. Societie...

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale‐free networks. The correlation matrix considered is , where P U is uncorrelated and r (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale expon...

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is $P(h|k)=(1-r)P^U_{hk}+r\delta_{hk}$, where $P^U$ is uncorrelated and $r$ (the Newman assortativity coefficient) can be very small. T...

In this paper, we present a successful experimental validation of the velocity optimization for a cable car passing over a support. We apply the theoretical strategy developed in a previous work, refined by taking into account in a simple manner the hauling cable dynamics. The experiments at the ropeway Postal–Verano (South Tirol, Italy) have shown...

It is well known that dynamical processes on complex networks are influenced by the degree correlations. A common way to take these into account in a mean-field approach is to consider the function knn(k) (average nearest neighbors degree). We re-examine the standard choices of knn for scale-free networks and a new family of functions which is inde...

We generate correlated scale-free networks in the configuration model through a new rewiring algorithm that allows one to tune the Newman assortativity coefficient r and the average degree of the nearest neighbors K (in the range − 1 ≤ r ≤ 1 , K ≥ ⟨ k ⟩ ). At each attempted rewiring step, local variations Δ r and Δ K are computed and then the step...

We generate correlated scale-free networks in the configuration model through a new rewiring algorithm which allows to tune the Newman assortativity coefficient $r$ and the average degree of the nearest neighbors $K$ (in the range $-1\le r \le 1$, $K\ge \langle k \rangle$). At each attempted rewiring step, local variations $\Delta r$ and $\Delta K$...

The dynamics of ropeway vehicles of an aerial ropeway is investigated experimentally by a set of measurements including velocities of the hauling cable at the driving disk and the running gear and time-dependent deviation angles of the hangers. A measurement of the damping characteristics of vehicle oscillations shows that both damper and friction...

In this work we discuss the problem of finding an optimal shape of a cable ropeway support head using optimization techniques. We define a cost function and relevant constraints with the goal to minimize the oscillations of the vehicle when it crosses the support, valid for both driving directions. Our findings reveal potential for practical use by...

This paper deals with a generalization of the Bass model for the description of the diffusion of innovations. The generalization keeps into account heterogeneity of the interactions of the consumers and is expressed by a system of several nonlinear differential equations on complex networks. The following contributions can be singled out: first, ex...

Abstract We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The appl...

We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application o...

Using a heterogeneous mean-field network formulation of the Bass innovation diffusion model and recent exact results on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We...

In this work we discuss the problem of finding an optimal shape of a cable ropeway support head using optimization techniques. We define a cost function and relevant constraints with the goal to minimize the oscillations of the vehicle when it crosses the support, valid for both driving directions. Our findings reveal potential for practical use by...

In the dynamic behaviour of a cable railway oscillations of cables and cars play an important role. We present a simple model to describe and investigate oscillations of a cable, spanned over a support and charged with an arbitrary number of point loads with arbitrary masses. We construct a time-dependent propagator, which contains the full intrins...

In this paper, we present a theoretical model that solves the problem of minimization of aerial ropeway vehicle oscillations that are induced as the vehicle passes over a support. The task is formulated as an inverse problem, where the vehicle oscillations are minimized by an appropriate choice of the velocity profile of the hauling cable. We study...

Microscopic models describing a whole of economic interactions in a closed society are considered. The presence of a tax system combined with a redistribution process is taken into account, as well as the occurrence of tax evasion. In particular, the existence is postulated, in relation to the level of evasion, of different individual taxpayer beha...

Using a mean-field network formulation of the Bass innovation diffusion model and exact results by Fotohui and Rabbat on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We...

In our recently proposed stochastic version of discretized kinetic theory, the exchange of wealth in a society is modelled through a large system of Langevin equations. The deterministic part of the equations is based on non-linear transition probabilities between income classes. The noise terms can be additive, multiplicative or mixed, both with w...

In this article, we consider a stylized dynamic model to describe the economics of a population, expressed by a Langevin-type kinetic equation. The dynamics is defined by a combination of terms, one of which represents monetary exchanges between individuals mutually engaged in trade, while the uncertainty in barter (trade exchange) is modeled throu...

In the dynamic behaviour of a cable railway oscillations of cables and cars play an important role. We present a simple model to describe and investigate oscillations of a cable, spanned over a support and charged with an arbitrary number of point loads with arbitrary masses. We construct a time-dependent propagator, which contains the full intrins...

Why does the Maxwell-Boltzmann energy distribution for an ideal classical gas have an exponentially thin tail at high energies, while the Kaniadakis distribution for a relativistic gas has a power-law fat tail? We argue that a crucial role is played by the kinematics of the binary collisions. In the classical case the probability of an energy excha...

Starting from a class of stochastically driven kinetic models of economic exchange, here we present results highlighting the correlation of the Gini inequality index with the social mobility rate, close to dynamical equilibrium. Except for the ”canonical-additive case”, our numerical results consistently indicate negative values of the correlation...

In this article, we discuss a dynamical stochastic model that represents the time evolution of income distribution of a population, where the dynamics develop from an interplay of multiple economic exchanges in the presence of multiplicative noise. The model remit stretches beyond the conventional framework of a Langevin-type kinetic equation in th...

Planning a cable railway is a complex task. One has to take into account many aspects and an optimal solution is not well defined. The calculation of the cable configurations for given support positions, cable pretension and cable types however is rather formalizable and presents a direct problem. In this research work we study the first steps to s...

Microscopic models describing a whole of economic interactions in a closed society are considered. The presence of a tax system combined with a redistribution process is taken into account, as well as the occurrence of tax evasion. In particular, the existence is postulated, in relation to the level of evasion, of different individual taxpayer beha...

Also motivated by the topical problem of growing economic inequality, we propose and investigate models describing the formation process of income distribution in a closed society. Our approach fits in with a complex system perspective. We look at society as a system composed by a large number of individuals, divided into income classes. The inform...

Statistical evaluations of the economic mobility of a society are more difficult than measurements of the income distribution, because they require to follow the evolution of the individuals’ income for at least one or two generations. In micro-to-macro theoretical models of economic exchanges based on kinetic equations, the income distribution dep...

We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times...

We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times...

Linear stochastic models and discretized kinetic theory are two complementary analytical techniques used for the investigation of complex systems of economic interactions. The former employ Langevin equations, with an emphasis on trade; the latter is based on systems of ordinary differential equations and is better suited for the description of bin...

The Bass model, which is an effective forecasting tool for innovation diffusion based on large collections of empirical data, assumes an homogeneous diffusion process. We introduce a network structure into this model and we investigate numerically the dynamics in the case of networks with link density where . The resulting curve of the total adopti...

A microscopic dynamic model is here constructed and analyzed, describing the evolution of the income distribution in the presence of taxation and redistribution in a society in which also tax evasion and auditing processes occur. The focus is on effects of enforcement regimes, characterized by different choices of the audited taxpayer fraction and...

Planning a cable railway is a complex task. One has to take into account many aspects and an optimal solution is not well defined. The calculation of the cable configurations for given support positions, cable pretension and cable types however is rather formalizable and presents a direct problem. In this research work we study the first steps to s...

The network of interpersonal connections is one of the possible heterogeneous factors which affect the income distribution emerging from micro-to-macro economic models. In this paper we equip our model discussed in [1,2] with a network structure. The model is based on a system of $n$ differential equations of the kinetic discretized-Boltzmann kind....

Statistical evaluations of the economic mobility of a society are more difficult than measurements of the income distribution, because they require to follow the evolution of the individuals' income for at least one or two generations. In micro-to-macro theoretical models of economic exchanges based on kinetic equations, the income distribution dep...

We formulate a flexible micro-to-macro kinetic model which is able to explain the emergence of income profiles out of a whole of individual economic interactions. The model is expressed by a system of several nonlinear differential equations which involve parameters defined by probabilities. Society is described as an ensemble of individuals divide...

We investigate the effect of tax evasion on the income distribution and the inequality index of a society through a kinetic model described by a set of nonlinear ordinary differential equations. The model allows to compute the global outcome of binary and multiple microscopic interactions between individuals. When evasion occurs, both individuals i...

We discuss a framework for the microscopic modelling of taxation and redistribution processes in a closed trading market society. For a prototype model and some variants of it, we examine the emergence of income distribution curves which exhibit “fat” power-law tails as the real world ones. We also incorporate tax evasion into the models and we inv...

We discuss a family of models expressed by nonlinear differential equation systems describing closed market societies in the presence of taxation and redistribution. We focus in particular on three example models obtained in correspondence to different parameter choices. We analyse the influence of the various choices on the long time shape of the...

This paper deals with some methodological aspects related to the discretization of a class of integro-differential equations modelling the evolution of the probability distribution over the microscopic state of a large system of interacting individuals. The microscopic state includes both mechanical and socio-biological variables. The discretizatio...

We present here a general framework, expressed by a system of nonlinear differential equations, suitable for the modelling of taxation and redistribution in a closed (trading market) society. This framework allows to describe the evolution of the income distribution over the population and to explain the emergence of collective features based on th...

A mathematical approach is here discussed, addressed to try and explain the emergence of collective properties of socio-economic systems with a large number of individuals, based on the knowledge of pairwise individual interactions. In particular, two different issues, namely the opinion formation and the taxation and redistribution process, are co...

In this paper we discuss and analyze a two-parameter family of systems of quadratic ordinary differential equations of interest in applied sciences, whose dynamics exhibits an emerging cluster structure.

In this paper a general framework is proposed, suitable for the modelling of the taxation and redistribution process in a closed society. This framework arises within a discrete kinetic approach for active particle systems, and is expressed by a system of nonlinear ordinary differential equations. It is intended to describe the evolution of the wea...

In this paper we formulate a discrete version of the bounded confidence model (Deffuant etal. in Adv Complex Syst 3:87–98, 2000; Weisbuch etal. in Complexity 7:55–63, 2002), which is representable as a family of ordinary differential equation systems. Then, we analytically study these systems. We establish the existence of equilibria which correspo...

This paper deals with the definition of a general framework, inspired by the discrete generalized kinetic theory, suitable for the description of the evolution of opinions within a population in the presence of some external actions. As a conceivable application, a specific model of opinion formation is formulated, relying on the interactions of si...

This paper concerns a model of opinion formation in a population of interacting individuals under the influence of external leaders or persuaders, which act in a time periodic fashion. The model is formulated within a general framework inspired to a discrete generalized kinetic approach, which has been developed in Ref. 6. It is expressed by a syst...

A conservative social dynamics model is developed within a discrete kinetic framework for active particles, which has been proposed in [M.L. Bertotti, L. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Mod. Meth. Appl. Sci. 14 (2004) 1061–1084]. The model concerns a society in which...

This paper deals with the modeling of complex social systems by methods of the mathematical kinetic theory for active particles. Specifically, a recent model by the last two authors is analyzed from the social sciences point of view. The model shows, despite its simplicity, some interesting features. In particular, this paper investigates the abili...

This work deals with a family of dynamical systems which were introduced in [M.L. Bertotti, M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Methods Appl. Sci. 7 (2004) 1061–1084], modelling the evolution of a population of interacting individuals, distinguished by their so...

Without Abstract

This paper deals with the solution of a boundary value problem related to a steady nonuniform description of a class of traffic flow models. The models are obtained by the closure of the mass conservation equation with a phenomenological relation linking the local mass velocity to the local density. The analysis is addressed to define the proper fr...

We consider a natural Lagrangian system on a torus and give sufficient conditions for the existence of chaotic trajectories for energy values slightly below the maximum of the potential energy. It turns out that chaotic trajectories always exist except when the system is "variationally separable", i.e. minimizers of the action functional behave lik...

We consider natural systems (T,Π) on ℝ 2 described by the Lagrange equation d dt∂T ∂q ˙-∂T ∂q=-∂Π ∂q, where T(q,q ˙) is a positive definite quadratic form in q ˙ and Π(q) has a critical point at 0. It is proved that there exist a C ∞ potential energy Π and two C ∞ kinetic energies T and T ˜ such that the equilibrium q(t)≡0 is stable for the system...

Natural Lagrangian systems (T,Π) on R 2 described by the equation \(\frac{d}{{dt}}\frac{{\partial T}}{{\partial \dot q}} - \frac{{\partial T}}{{\partial q}} = - \frac{{\partial \Pi }}{{\partial q}}\) are considered, where T(q, q̇) is a positive definite quadratic form in q̇ and Π(q) has a critical point at 0. It is constructively proved that there...

We study via variational methods some chaotic features of a class of almost periodic Lagrangian systems on a torus. In particular we show that slowly oscillating perturbations of such systems admit a multibump dynamics relative to possibly degenerate equilibria.

We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t . As an example, we study doubly asymptoti...

The existence is proved, by means of variational arguments, of infinitely many heteroclinic solutions connecting possibly degenerate equilibria for a class of almost periodic Lagrangian system. An analogous multiplicity result is then established for homoclinic solutions of systems with an almost periodic singular potential.

We consider a class of second-order systems , with q(t) ∊ℝn, for which the potential energy V: ℝn\S→ℝ admits a (possibly unbounded) singular set S ⊂ℝn and has a unique absolute maximum at 0 ∈ℝn. Under some conditions on S and V, we prove the existence of several solutions homoclinic to 0.

We consider quasiperiodic positive definite Lagrangian systems and establish the existence of one or more solutions homoclinic (namely asymptotic as $t\to\pm\infty$) to a quasiperiodic solution. These results are obtained by means of a variational approach. An application is carried out to quasiperiodically perturbed Lagrangian systems.

We consider Hamiltonian systems whose Hamiltonian function is quadratic in the momenta and almost periodic in time. We assume that the configuration space M is not simply connected and there is an equilibrium, non-necessarily hyperbolic, corresponding to a minimum of the Lagrangian. We establish by variational arguments, the existence of several so...

We consider forced dynamical systems with two degrees of freedom having singular potentials and we prove existence of infinitely many classical (noncollision) periodic solutions. These solutions have a prescribed rotation behavior with respect to the singularities and a prescribed period (the same of the systems). They are obtained variationally as...

We outline a variational approach, developed to find periodic orbits of the satellite in the elliptic restricted problem with any value of the masses of the primaries. This approach leads to a multiplicity of generalized periodic solutions (namely solutions x which satisfy a boundary condition: x(T) = x(0) and which possibly experience collisions)....

We prove existence and multiplicity of long time periodic solutions for a class of nonlinear nonautonomous Hamiltonian systems locally near an equilibrium solution. The result relies on a variational principle and on the spectral analysis of an associated linear operator.

We consider Hamiltonian systems whose Hamiltonian function is quadratic in momenta and almost periodic in time. We assume that the configuration space M is not simply connected and there is an equilibrium, non necessarily hyperbolic, corresponding to a minimum of the Lagrangian. We establish, by variational arguments, the existence of several solut...