We define a mathematical notion of complex adaptive system by following the original intuition of... more We define a mathematical notion of complex adaptive system by following the original intuition of G.K. Zipf about the principle of least effort, an intuitive idea which is nowadays informally widespread in complex systems modeling. We call generalized evolution principle this mathematical notion of interaction spaces theory. Formalizing and generalizing Mandelbrot's ideas, we also prove that a large class of these systems satisfy a power law. We finally illustrate the notion of complex adaptive system with theorems describing a Von Thünen-like model. The latter can be easily generalized to other complex systems and describes the appearance of emergent patterns. Every notion is introduced both using an intuitive description with lots of examples, and using a modern mathematical language.
We present the first steps of interaction spaces theory, a universal mathematical theory of compl... more We present the first steps of interaction spaces theory, a universal mathematical theory of complex systems which is able to embed cellular automata, agent based models, master equation based models, stochastic or deterministic, continuous or discrete dynamical systems, networked dynamical models, artificial neural networks and genetic algorithms in a single notion. Therefore, interaction spaces represent a common mathematical language that can be used to describe several complex systems modeling frameworks. This is the first step to start a mathematical theory of complex systems. Every notion is introduced both using an intuitive description by listing lots of examples, and using a modern mathematical language.
Journal of Mathematical Analysis and Applications, 2013
Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalize... more Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized points that induce the standard Euclidean topology on the reals. We also give a new description of the sharp topology in terms of the natural extension of the absolute value (or of the defining semi-norms in the case of locally convex spaces) that allows to preserve a number of classical notions. Building on a new point value characterization of Colombeau generalized functions we prove a Fermat-Reyes theorem that forms the basis of an approach to differentiation on spaces of generalized functions close to the classical one.
The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, a... more The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.
The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, a... more The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.
NILPOTENT INFINITESIMALS AND SYNTHETIC DIFFERENTIAL GEOMETRY IN CLASSICAL LOGIC Paolo Giordano Ac... more NILPOTENT INFINITESIMALS AND SYNTHETIC DIFFERENTIAL GEOMETRY IN CLASSICAL LOGIC Paolo Giordano Accademia di architettura, Universitd della Svizzera italiana Largo Bernasconi, CH-6658 Mendrisio, Switzerland pgiordano6arch. unisi. ch Keywords: ...
Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals ... more Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals proceeds by imposing a suitable equivalence relation. More generally, the completion of a metric space starts from an analogous equivalence relation among sequences of points of the space. Can Cantor's relation among Cauchy sequences of reals be refined so as to produce a Cauchy complete and infinitesimal-enriched continuum? We present two possibilities: one leads to invertible infinitesimals and the hyperreals; the other to nilpotent infinitesimals (e.g. h nonzero infinitesimal such that h^2=0) and Fermat reals. One of our themes is the trade-off between formal power and intuition.
Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing... more Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study spaces of mappings. We define a full embedding of the category Man${}^n$ of finite dimensional $\mathcal{C}^n$ manifolds in a cartesian closed category. In it we have a functor ${}^\bullet (-)$ which extends these spaces adding new infinitesimal points and with values in another full cartesian closed embedding of Man${}^n$. We present a first development of Differential Geometry using these infinitesimals.
We present a mathematical model for urban systems based on a continuous valued cellular automaton... more We present a mathematical model for urban systems based on a continuous valued cellular automaton. In the modelling we have an urban system, described through a specification cell by cell of built volumes and surfaces for different land uses and a system of agents interacting with the urban system and governed by fuzzy decision processes depending on the configuration of the urban system. For developers e.g. a point in the decision space specifies the cell and a set of continuous parameters describing the building quantitatively (e.g. surface and volume). The decision process of an agent is described as a random motion in the decision space where the agent can follow four possibilities: begin a decision process; make the decision modifying correspondingly the configuration of the urban system; jump towards another point in the space; abandon the decision space. Computer simulations on an artificial case with detailed real characteristics are presented.
1. Abstract We present a mathematical model based decision support system for the placement of bi... more 1. Abstract We present a mathematical model based decision support system for the placement of big traffic generators (BTG) like shopping centres. The model aims to forecast the traffic flows induced by BTG on the road network of Canton Ticino in Switzerland, and is beginning to be used by the public administration as a support to decide master plan rules
ABSTRACT The present work aimed at defining an urban efficiency index taking into account the rel... more ABSTRACT The present work aimed at defining an urban efficiency index taking into account the relationships among urban structure, energy consumption, emissions of pollutants and depletion of resources. Core of the work is the analysis of the most significant energy and mass flows characterizing urban metabolism and their representation through hierarchically organized system of variables and their fuzzy evaluations using a cascade of Takagi–Sugeno models. The performance of the urban system is evaluated splitting it in defined subsystems about transportation, built environment and the socioeconomic sector. A chain of Takagi–Sugeno models starts from a fuzzy evaluation of urban data, evaluates the efficiency of these subsystems and eventually recursively obtains the final evaluation of the whole system. This structure and the use of fuzzy methods permit the definition of an index which is very clear from the point of view of interdisciplinary approach. Therefore, this kind of index can support policy makers, public utilities, urban designers and other stake holders in defining strategies for improving the performance of the different urban areas looking at their features, weaknesses and potentialities.
We present a mathematical model for urban systems based on a continuous valued cellular automaton... more We present a mathematical model for urban systems based on a continuous valued cellular automaton. In the modelling we have an urban system, described through a specification cell by cell of built volumes and surfaces for different land uses and a system of agents ...
A new kind of cellular automaton (CA) for the study of the dynamics of urban systems is proposed.... more A new kind of cellular automaton (CA) for the study of the dynamics of urban systems is proposed. The state of a cell is not described using a finite set, but by means of continuum variables. A population sector is included, taking into account migration processes from ...
Abstract Housing market dynamics and modeling of housing market evolution in Lugano (southern par... more Abstract Housing market dynamics and modeling of housing market evolution in Lugano (southern part of Switzerland) is the main goal of this work. An agent-based modeling has been used to model the individual behavior and decision-making process of people in the ...
We introduce the notion of functionally compact sets into the theory of nonlinear generalized fun... more We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties
We develop the integral calculus for quasi-standard smooth functions defined on the ring of Ferma... more We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.
We define a mathematical notion of complex adaptive system by following the original intuition of... more We define a mathematical notion of complex adaptive system by following the original intuition of G.K. Zipf about the principle of least effort, an intuitive idea which is nowadays informally widespread in complex systems modeling. We call generalized evolution principle this mathematical notion of interaction spaces theory. Formalizing and generalizing Mandelbrot's ideas, we also prove that a large class of these systems satisfy a power law. We finally illustrate the notion of complex adaptive system with theorems describing a Von Thünen-like model. The latter can be easily generalized to other complex systems and describes the appearance of emergent patterns. Every notion is introduced both using an intuitive description with lots of examples, and using a modern mathematical language.
We present the first steps of interaction spaces theory, a universal mathematical theory of compl... more We present the first steps of interaction spaces theory, a universal mathematical theory of complex systems which is able to embed cellular automata, agent based models, master equation based models, stochastic or deterministic, continuous or discrete dynamical systems, networked dynamical models, artificial neural networks and genetic algorithms in a single notion. Therefore, interaction spaces represent a common mathematical language that can be used to describe several complex systems modeling frameworks. This is the first step to start a mathematical theory of complex systems. Every notion is introduced both using an intuitive description by listing lots of examples, and using a modern mathematical language.
Journal of Mathematical Analysis and Applications, 2013
Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalize... more Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized points that induce the standard Euclidean topology on the reals. We also give a new description of the sharp topology in terms of the natural extension of the absolute value (or of the defining semi-norms in the case of locally convex spaces) that allows to preserve a number of classical notions. Building on a new point value characterization of Colombeau generalized functions we prove a Fermat-Reyes theorem that forms the basis of an approach to differentiation on spaces of generalized functions close to the classical one.
The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, a... more The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.
The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, a... more The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.
NILPOTENT INFINITESIMALS AND SYNTHETIC DIFFERENTIAL GEOMETRY IN CLASSICAL LOGIC Paolo Giordano Ac... more NILPOTENT INFINITESIMALS AND SYNTHETIC DIFFERENTIAL GEOMETRY IN CLASSICAL LOGIC Paolo Giordano Accademia di architettura, Universitd della Svizzera italiana Largo Bernasconi, CH-6658 Mendrisio, Switzerland pgiordano6arch. unisi. ch Keywords: ...
Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals ... more Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals proceeds by imposing a suitable equivalence relation. More generally, the completion of a metric space starts from an analogous equivalence relation among sequences of points of the space. Can Cantor's relation among Cauchy sequences of reals be refined so as to produce a Cauchy complete and infinitesimal-enriched continuum? We present two possibilities: one leads to invertible infinitesimals and the hyperreals; the other to nilpotent infinitesimals (e.g. h nonzero infinitesimal such that h^2=0) and Fermat reals. One of our themes is the trade-off between formal power and intuition.
Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing... more Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study spaces of mappings. We define a full embedding of the category Man${}^n$ of finite dimensional $\mathcal{C}^n$ manifolds in a cartesian closed category. In it we have a functor ${}^\bullet (-)$ which extends these spaces adding new infinitesimal points and with values in another full cartesian closed embedding of Man${}^n$. We present a first development of Differential Geometry using these infinitesimals.
We present a mathematical model for urban systems based on a continuous valued cellular automaton... more We present a mathematical model for urban systems based on a continuous valued cellular automaton. In the modelling we have an urban system, described through a specification cell by cell of built volumes and surfaces for different land uses and a system of agents interacting with the urban system and governed by fuzzy decision processes depending on the configuration of the urban system. For developers e.g. a point in the decision space specifies the cell and a set of continuous parameters describing the building quantitatively (e.g. surface and volume). The decision process of an agent is described as a random motion in the decision space where the agent can follow four possibilities: begin a decision process; make the decision modifying correspondingly the configuration of the urban system; jump towards another point in the space; abandon the decision space. Computer simulations on an artificial case with detailed real characteristics are presented.
1. Abstract We present a mathematical model based decision support system for the placement of bi... more 1. Abstract We present a mathematical model based decision support system for the placement of big traffic generators (BTG) like shopping centres. The model aims to forecast the traffic flows induced by BTG on the road network of Canton Ticino in Switzerland, and is beginning to be used by the public administration as a support to decide master plan rules
ABSTRACT The present work aimed at defining an urban efficiency index taking into account the rel... more ABSTRACT The present work aimed at defining an urban efficiency index taking into account the relationships among urban structure, energy consumption, emissions of pollutants and depletion of resources. Core of the work is the analysis of the most significant energy and mass flows characterizing urban metabolism and their representation through hierarchically organized system of variables and their fuzzy evaluations using a cascade of Takagi–Sugeno models. The performance of the urban system is evaluated splitting it in defined subsystems about transportation, built environment and the socioeconomic sector. A chain of Takagi–Sugeno models starts from a fuzzy evaluation of urban data, evaluates the efficiency of these subsystems and eventually recursively obtains the final evaluation of the whole system. This structure and the use of fuzzy methods permit the definition of an index which is very clear from the point of view of interdisciplinary approach. Therefore, this kind of index can support policy makers, public utilities, urban designers and other stake holders in defining strategies for improving the performance of the different urban areas looking at their features, weaknesses and potentialities.
We present a mathematical model for urban systems based on a continuous valued cellular automaton... more We present a mathematical model for urban systems based on a continuous valued cellular automaton. In the modelling we have an urban system, described through a specification cell by cell of built volumes and surfaces for different land uses and a system of agents ...
A new kind of cellular automaton (CA) for the study of the dynamics of urban systems is proposed.... more A new kind of cellular automaton (CA) for the study of the dynamics of urban systems is proposed. The state of a cell is not described using a finite set, but by means of continuum variables. A population sector is included, taking into account migration processes from ...
Abstract Housing market dynamics and modeling of housing market evolution in Lugano (southern par... more Abstract Housing market dynamics and modeling of housing market evolution in Lugano (southern part of Switzerland) is the main goal of this work. An agent-based modeling has been used to model the individual behavior and decision-making process of people in the ...
We introduce the notion of functionally compact sets into the theory of nonlinear generalized fun... more We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties
We develop the integral calculus for quasi-standard smooth functions defined on the ring of Ferma... more We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.
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Papers by Paolo Giordano