In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaî... more In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaître universes is a special case of generalized Lotka– Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka–Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann–Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka–Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.
We here elaborate on a quantitative argument to support the validity of the
Collatz conjecture, a... more We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct ?fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, in?finite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.
We here elaborate on a quantitative argument to support the validity of the
Collatz conjecture, a... more We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct ?fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, in?finite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.
In this work we propose a new numerical approach to distinguish between regular and chaotic orbit... more In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the H\'enon-Heiles model and the restricted three-body problem.
We propose a metapopulation version of the Schelling model where two kinds of agents relocate the... more We propose a metapopulation version of the Schelling model where two kinds of agents relocate themselves, with unconstrained destination, if their local fitness is lower than a tolerance threshold. We show that, for small values of the latter, the population redistributes highly heterogeneously among the available places. The system thus stabilizes on these heterogeneous skylines after a long quasi-stationary transient period, during which the population remains in a well mixed phase. Varying the tolerance passing from large to small values, we identify three possible global regimes: microscopic clusters with local coexistence of both kinds of agents, macroscopic clusters with local coexistence (soft segregation), macroscopic clusters with local segregation but homogeneous densities (hard segregation). The model is studied numerically and complemented with an analytical study in the limit of extremely large node capacity.
We hereby propose a model of opinion dynamics where individuals update their beliefs because of i... more We hereby propose a model of opinion dynamics where individuals update their beliefs because of interactions in acquaintances' group. The model exhibit a non trivial behavior that we discuss as a function of the main involved parameters. Results are reported on the average number of opinion clusters and the time needed to form such clusters.
We use recent results of [4] on face-to-face contact durations to try to answer the question: why... more We use recent results of [4] on face-to-face contact durations to try to answer the question: why do people engage in face-to-face discussions? In particular we focus on behavior of scientists in academic conferences. We show evidence that macroscopic measured data are compatible with two different micro-founded models of social interaction. We find that the first model, in which discussions are performed with the aim of introducing oneself (networking), explains the data when the group exhibits few well reputed scientists. On the contrary, when the reputation hierarchy is not strong, a model where agents' encounters are aimed at exchanging opinions explains the data better.
The role of competitive markets as efficient aggregators of decentralized information is a fundam... more The role of competitive markets as efficient aggregators of decentralized information is a fundamental problem in economic theory. This paper studies the informational efficiency of a market with a single traded asset, in which agents expectation formation about future price has two kinds of deviations from rationality. First, traders have adaptive expectations, i.e. they give more importance to the past price than a rational agent. Second, the agents are subject to the confirmatory bias, i.e. they tend to discard new information that substantially differs from their priors. Taken separately, each deviation worsens the informational efficiency of the market; however, for some ranges of parameters, when the two biases are combined, they tend to mitigate each others effect (thus increasing the informational efficiency). We also study the robustness of the principal findings to alternative specifications concerning market participation, entry of new agents, and the amount of liquidity ...
Particles accelerators are highly technological instrumental devices allowing to perform studies ... more Particles accelerators are highly technological instrumental devices allowing to perform studies from the "infinitely small scale", e.g. particles responsible for elementary forces, to "extremely large ones", for instance the origin of cosmos. In the simplified version such devices are composed by a sequence of basic cells: focusing magnets, defocus-ing magnets, accelerator electromagnetic fields and trajectory bending in the case of circular accelerators. The result is thus a non–linear, and in good approximation, a conservative system that can be modeled by a symplectic map resulting from the composition of several simpler maps corresponding to each basic cell. One of the main problem in the dynamics of ring accelerator is to study the stability of the nominal orbit, i.e. the circular orbit passing through the center of each element. In fact, each cell can be seen as a non–linear map, that deforms the wanted trajectory. Moreover such maps posses stochastic laye...
ABSTRACT We present a genetic algorithm (GA) we developed for the optimization of light-emitting ... more ABSTRACT We present a genetic algorithm (GA) we developed for the optimization of light-emitting diodes (LED) and solar thermal collectors. The surface of a LED can be covered by periodic structures whose geometrical and material parameters must be adjusted in order to maximize the extraction of light. The optimization of these parameters by the GA enabled us to get a light-extraction efficiency η of 11.0% from a GaN LED (for comparison, the flat material has a light-extraction efficiency η of only 3.7%). The solar thermal collector we considered consists of a waffle-shaped Al substrate with NiCrOx and SnO2 conformal coatings. We must in this case maximize the solar absorption α while minimizing the thermal emissivity ϵ in the infrared. A multi-objective genetic algorithm has to be implemented in this case in order to determine optimal geometrical parameters. The parameters we obtained using the multi-objective GA enable α~97.8% and ϵ~4.8%, which improves results achieved previously when considering a flat substrate. These two applications demonstrate the interest of genetic algorithms for addressing complex problems in physics.
In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaî... more In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaître universes is a special case of generalized Lotka– Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka–Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann–Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka–Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.
We here elaborate on a quantitative argument to support the validity of the
Collatz conjecture, a... more We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct ?fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, in?finite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.
We here elaborate on a quantitative argument to support the validity of the
Collatz conjecture, a... more We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct ?fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8 ), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, in?finite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8^m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.
In this work we propose a new numerical approach to distinguish between regular and chaotic orbit... more In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the H\'enon-Heiles model and the restricted three-body problem.
We propose a metapopulation version of the Schelling model where two kinds of agents relocate the... more We propose a metapopulation version of the Schelling model where two kinds of agents relocate themselves, with unconstrained destination, if their local fitness is lower than a tolerance threshold. We show that, for small values of the latter, the population redistributes highly heterogeneously among the available places. The system thus stabilizes on these heterogeneous skylines after a long quasi-stationary transient period, during which the population remains in a well mixed phase. Varying the tolerance passing from large to small values, we identify three possible global regimes: microscopic clusters with local coexistence of both kinds of agents, macroscopic clusters with local coexistence (soft segregation), macroscopic clusters with local segregation but homogeneous densities (hard segregation). The model is studied numerically and complemented with an analytical study in the limit of extremely large node capacity.
We hereby propose a model of opinion dynamics where individuals update their beliefs because of i... more We hereby propose a model of opinion dynamics where individuals update their beliefs because of interactions in acquaintances' group. The model exhibit a non trivial behavior that we discuss as a function of the main involved parameters. Results are reported on the average number of opinion clusters and the time needed to form such clusters.
We use recent results of [4] on face-to-face contact durations to try to answer the question: why... more We use recent results of [4] on face-to-face contact durations to try to answer the question: why do people engage in face-to-face discussions? In particular we focus on behavior of scientists in academic conferences. We show evidence that macroscopic measured data are compatible with two different micro-founded models of social interaction. We find that the first model, in which discussions are performed with the aim of introducing oneself (networking), explains the data when the group exhibits few well reputed scientists. On the contrary, when the reputation hierarchy is not strong, a model where agents' encounters are aimed at exchanging opinions explains the data better.
The role of competitive markets as efficient aggregators of decentralized information is a fundam... more The role of competitive markets as efficient aggregators of decentralized information is a fundamental problem in economic theory. This paper studies the informational efficiency of a market with a single traded asset, in which agents expectation formation about future price has two kinds of deviations from rationality. First, traders have adaptive expectations, i.e. they give more importance to the past price than a rational agent. Second, the agents are subject to the confirmatory bias, i.e. they tend to discard new information that substantially differs from their priors. Taken separately, each deviation worsens the informational efficiency of the market; however, for some ranges of parameters, when the two biases are combined, they tend to mitigate each others effect (thus increasing the informational efficiency). We also study the robustness of the principal findings to alternative specifications concerning market participation, entry of new agents, and the amount of liquidity ...
Particles accelerators are highly technological instrumental devices allowing to perform studies ... more Particles accelerators are highly technological instrumental devices allowing to perform studies from the "infinitely small scale", e.g. particles responsible for elementary forces, to "extremely large ones", for instance the origin of cosmos. In the simplified version such devices are composed by a sequence of basic cells: focusing magnets, defocus-ing magnets, accelerator electromagnetic fields and trajectory bending in the case of circular accelerators. The result is thus a non–linear, and in good approximation, a conservative system that can be modeled by a symplectic map resulting from the composition of several simpler maps corresponding to each basic cell. One of the main problem in the dynamics of ring accelerator is to study the stability of the nominal orbit, i.e. the circular orbit passing through the center of each element. In fact, each cell can be seen as a non–linear map, that deforms the wanted trajectory. Moreover such maps posses stochastic laye...
ABSTRACT We present a genetic algorithm (GA) we developed for the optimization of light-emitting ... more ABSTRACT We present a genetic algorithm (GA) we developed for the optimization of light-emitting diodes (LED) and solar thermal collectors. The surface of a LED can be covered by periodic structures whose geometrical and material parameters must be adjusted in order to maximize the extraction of light. The optimization of these parameters by the GA enabled us to get a light-extraction efficiency η of 11.0% from a GaN LED (for comparison, the flat material has a light-extraction efficiency η of only 3.7%). The solar thermal collector we considered consists of a waffle-shaped Al substrate with NiCrOx and SnO2 conformal coatings. We must in this case maximize the solar absorption α while minimizing the thermal emissivity ϵ in the infrared. A multi-objective genetic algorithm has to be implemented in this case in order to determine optimal geometrical parameters. The parameters we obtained using the multi-objective GA enable α~97.8% and ϵ~4.8%, which improves results achieved previously when considering a flat substrate. These two applications demonstrate the interest of genetic algorithms for addressing complex problems in physics.
We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, a... more We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod8 m for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion.
We here provide a proof of the Collatz conjecture, also known as the (3x+1) or Syracuse conjectur... more We here provide a proof of the Collatz conjecture, also known as the (3x+1) or Syracuse conjecture. The proof is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, in the second part of the proof, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system is contracting, when mapped on the original space of the positive integers. Building on these premises, and invoking ergodicity of the introduced Markov model, we prove that each deterministic trajectory of the third iterate of the Collatz map is bound to converge to the fixed points, as identified above. Stated differently, all positive integers converge to the period–3 orbit formed by the numbers {1, 2, 4}, under repeated application of the Collatz map, a conclusion which proves the conjecture. A modified version of the Markov process is also studied which explicitly accounts for the presence of an absorbing sink, the Collatz cycle {1, 2, 4}.
We here provide a proof of the Collatz conjecture, also known as the (3x+1) or Syracuse conjectur... more We here provide a proof of the Collatz conjecture, also known as the (3x+1) or Syracuse conjecture. The proof is organised as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8, obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates. Then, in the second part of the proof, we consider a probabilistic analogue of the deterministic map, where the period-3 orbit {1, 2, 4} acts as an absorbing sink. Working in this setting, we demonstrate that the quasi-stationary state sampled by the system while seeking for its deputed equilibrium is contracting and that each trajectory is bound to converge to the fixed points, as identified above. This implies in turn that all positive integer numbers converge to the period–3 orbit formed by the numbers {1, 2, 4}, under repeated application of the Collatz map. This proves in turn the conjecture.
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Papers by Timoteo Carletti
Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The
analysis is structured as follows. First, three distinct ?fixed points are
found for the third iterate of the Collatz map, which hence organise in a
period 3 orbit of the original map. These are 1, 2 and 4, the elements which
defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out
the calculation we write the positive integers in modulo 8 (mod8 ), obtain a
closed analytical form for the associated map and determine the transitions
that yield contracting or expanding iterates in the original,
in?finite-dimensional, space of positive integers. Then, we consider a Markov
chain which runs on the reduced space of mod8 congruence classes of integers.
The transition probabilities of the Markov chain are computed from the
deterministic map, by employing a measure that is invariant for the map itself.
Working in this setting, we demonstrate that the stationary distribution
sampled by the stochastic system induces a contracting behaviour for the orbits
of the deterministic map on the original space of the positive integers.
Sampling the equilibrium distribution on the congruence classes mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.
Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The
analysis is structured as follows. First, three distinct ?fixed points are
found for the third iterate of the Collatz map, which hence organise in a
period 3 orbit of the original map. These are 1, 2 and 4, the elements which
defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out
the calculation we write the positive integers in modulo 8 (mod8 ), obtain a
closed analytical form for the associated map and determine the transitions
that yield contracting or expanding iterates in the original,
in?finite-dimensional, space of positive integers. Then, we consider a Markov
chain which runs on the reduced space of mod8 congruence classes of integers.
The transition probabilities of the Markov chain are computed from the
deterministic map, by employing a measure that is invariant for the map itself.
Working in this setting, we demonstrate that the stationary distribution
sampled by the stochastic system induces a contracting behaviour for the orbits
of the deterministic map on the original space of the positive integers.
Sampling the equilibrium distribution on the congruence classes mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.
Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The
analysis is structured as follows. First, three distinct ?fixed points are
found for the third iterate of the Collatz map, which hence organise in a
period 3 orbit of the original map. These are 1, 2 and 4, the elements which
defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out
the calculation we write the positive integers in modulo 8 (mod8 ), obtain a
closed analytical form for the associated map and determine the transitions
that yield contracting or expanding iterates in the original,
in?finite-dimensional, space of positive integers. Then, we consider a Markov
chain which runs on the reduced space of mod8 congruence classes of integers.
The transition probabilities of the Markov chain are computed from the
deterministic map, by employing a measure that is invariant for the map itself.
Working in this setting, we demonstrate that the stationary distribution
sampled by the stochastic system induces a contracting behaviour for the orbits
of the deterministic map on the original space of the positive integers.
Sampling the equilibrium distribution on the congruence classes mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.
Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The
analysis is structured as follows. First, three distinct ?fixed points are
found for the third iterate of the Collatz map, which hence organise in a
period 3 orbit of the original map. These are 1, 2 and 4, the elements which
defi?ne the unique attracting cycle, as hypothesised by Collatz. To carry out
the calculation we write the positive integers in modulo 8 (mod8 ), obtain a
closed analytical form for the associated map and determine the transitions
that yield contracting or expanding iterates in the original,
in?finite-dimensional, space of positive integers. Then, we consider a Markov
chain which runs on the reduced space of mod8 congruence classes of integers.
The transition probabilities of the Markov chain are computed from the
deterministic map, by employing a measure that is invariant for the map itself.
Working in this setting, we demonstrate that the stationary distribution
sampled by the stochastic system induces a contracting behaviour for the orbits
of the deterministic map on the original space of the positive integers.
Sampling the equilibrium distribution on the congruence classes mod8^m for any
m, which amounts to arbitrarily reducing the degree of imposed coarse graining,
returns an identical conclusion.