ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION

2013 ◽  
Vol 24 (02) ◽  
pp. 405-426 ◽  
Author(s):  
D. KNOPOFF
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Opinion Formation ◽  
Active Particles ◽  
Mathematical Equations

This paper presents a development of the so-called kinetic theory for active particles to the modeling of living, hence complex, systems localized in networks. The overall system is viewed as a network of interacting nodes, mathematical equations are required to describe the dynamics in each node and in the whole network. These interactions, which are nonlinearly additive, are modeled by evolutive stochastic games. The first conceptual part derives a general mathematical structure, to be regarded as a candidate towards the derivation of models, suitable to capture the main features of the said systems. An application on opinion formation follows to show how the theory can generate specific models.

ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES

2013 ◽  
Vol 23 (10) ◽  
pp. 1861-1913 ◽  
Author(s):  
N. BELLOMO ◽  
D. KNOPOFF ◽  
J. SOLER
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Case Studies ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Living Systems ◽  
Active Particles ◽  
Suitable Generalization ◽  
Mathematical Sciences

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

ON THE MATHEMATICAL THEORY OF THE DYNAMICS OF SWARMS VIEWED AS COMPLEX SYSTEMS

2012 ◽  
Vol 22 (supp01) ◽  
pp. 1140006 ◽  
Author(s):  
N. BELLOMO ◽  
J. SOLER
Keyword(s):  
Complex System ◽  
Kinetic Theory ◽  
Statistical Mechanics ◽  
Complex Systems ◽  
Modeling And Simulation ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Microscopic Scale

This paper deals with the modeling and simulation of swarms viewed as a living, hence complex, system. The approach is based on methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlinearly additive and modeled by stochastic games.

LOOKING FOR NEW PARADIGMS TOWARDS A BIOLOGICAL-MATHEMATICAL THEORY OF COMPLEX MULTICELLULAR SYSTEMS

2006 ◽  
Vol 16 (07) ◽  
pp. 1001-1029 ◽  
Author(s):  
NICOLA BELLOMO ◽  
GUIDO FORNI
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Critical Analysis ◽  
Mathematical Theory ◽  
Classical Mechanics ◽  
Biological Functions ◽  
Living Matter ◽  
Research Perspectives ◽  
Active Particles ◽  
Large Systems

This paper deals with the development of new paradigms based on the methods of the mathematical kinetic theory for active particles to model the dynamics of large systems of interacting cells. Interactions are ruled, not only by laws of classical mechanics, but also by a few biological functions which are able to modify the above laws. The paper technically shows, also by reasoning on specific examples, how the theory can be applied to model large complex systems in biology. The last part of the paper deals with a critical analysis and with the indication of research perspectives concerning the challenging target of developing a biological-mathematical theory for the living matter.

On the mathematical theory of post-Darwinian mutations, selection, and evolution

2014 ◽  
Vol 24 (13) ◽  
pp. 2723-2742 ◽  
Author(s):  
E. De Angelis
Keyword(s):  
Kinetic Theory ◽  
Qualitative Analysis ◽  
Mathematical Theory ◽  
Darwinian Selection ◽  
Research Perspectives ◽  
Active Particles ◽  
Selection Processes

This paper is devoted to the modeling, qualitative analysis and simulation of Darwinian selection phenomena and their evolution. The approach takes advantage of the mathematical tools of the kinetic theory of active particles which are applied to describe the selective dynamics of evolution processes. The first part of the paper focuses on a mathematical theory that has been developed to describe mutations and selection processes. The second part deals with different modeling strategies and looks ahead to research perspectives.

scholarly journals From particles to firms: on the kinetic theory of climbing up evolutionary landscapes

2020 ◽  
Vol 30 (07) ◽  
pp. 1441-1460 ◽  
Author(s):  
Nicola Bellomo ◽  
Giovanni Dosi ◽  
Damián A. Knopoff ◽  
Maria Enrica Virgillito
Keyword(s):  
Kinetic Theory ◽  
Mathematical Theory ◽  
Predictive Ability ◽  
Complex Interaction ◽  
Living Systems ◽  
Mathematical Representation ◽  
Future Research ◽  
Industrial Dynamics ◽  
Modeling And Simulations ◽  
Active Particles

This paper constitutes the first attempt to bridge the evolutionary theory in economics and the theory of active particles in mathematics. It seeks to present a kinetic model for an evolutionary formalization of economic dynamics. The new derived mathematical representation intends to formalize the processes of learning and selection as the two fundamental drivers of evolutionary environments [G. Dosi, M.-C. Pereira and M.-E. Virgillito, The footprint of evolutionary processes of learning and selection upon the statistical properties of industrial dynamics, Ind. Corp. Change, 26 (2017) 187–210]. To coherently represent the aforementioned properties, the kinetic theory of active particles [N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems (Birkhäuser-Springer, 2017)] is here further developed, including the complex interaction of two hierarchical functional subsystems. Modeling and simulations enlighten the predictive ability of the approach. Finally, we outline the potential avenues for future research.

Influence of drivers ability in a discrete vehicular traffic model

2017 ◽  
Vol 28 (03) ◽  
pp. 1750030 ◽  
Author(s):  
D. Burini ◽  
S. De Lillo ◽  
G. Fioriti
Keyword(s):  
Kinetic Theory ◽  
Numerical Simulations ◽  
Traffic Flow ◽  
Stochastic Games ◽  
Traffic Model ◽  
Driving Ability ◽  
Experimental Measurements ◽  
Nonlinear Interactions ◽  
Vehicular Traffic ◽  
Active Particles

A vehicular traffic model is presented, based on the so-called Kinetic Theory of Active Particles. Vehicles are characterized by a lattice of discrete speeds and by the driving ability of the drivers. The evolution of the system is modeled through nonlinear interactions, whose output is described by stochastic games. The results of numerical simulations are consistent with experimental measurements of traffic flow.

On the modeling of epidemics under the influence of risk perception

2017 ◽  
Vol 28 (04) ◽  
pp. 1750051 ◽  
Author(s):  
S. De Lillo ◽  
G. Fioriti ◽  
M. L. Prioriello
Keyword(s):  
Risk Perception ◽  
Kinetic Theory ◽  
Numerical Simulations ◽  
Stochastic Games ◽  
Initial Conditions ◽  
Nonlinear Interactions ◽  
Epidemic Spreading ◽  
Spreading Model ◽  
Active Particles

An epidemic spreading model is presented in the framework of the kinetic theory of active particles. The model is characterized by the influence of risk perception which can reduce the diffusion of infection. The evolution of the system is modeled through nonlinear interactions, whose output is described by stochastic games. The results of numerical simulations are discussed for different initial conditions.

HYBRID TWO SCALES MATHEMATICAL TOOLS FOR ACTIVE PARTICLES MODELLING COMPLEX SYSTEMS WITH LEARNING HIDING DYNAMICS

2007 ◽  
Vol 17 (02) ◽  
pp. 171-187 ◽  
Author(s):  
C. CATTANI ◽  
A. CIANCIO
Keyword(s):  
Applied Sciences ◽  
Kinetic Theory ◽  
Differential Equations ◽  
Complex Systems ◽  
Ordinary Differential Equations ◽  
Preliminary Analysis ◽  
Mathematical Structures ◽  
Active Particles ◽  
Model Complex ◽  
Large Systems

This paper deals with the derivation of hybrid mathematical structures to describe the behavior of large systems of active particles by ordinary differential equations with stochastic coefficients whose evolution is modelled by equations of the mathematical kinetic theory. A preliminary analysis shows how the above tools can be used to model complex systems of interest in applied sciences, with special attention to the immune competition.

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