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We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as r=hLPlanck, with h=0,1,…,N and N up to 105. The Lagrangian is of the form g(R+αR2) (in units c=ℏ=G=1) and the action is positive-definite, allowing the use of a standard Metropolis algorithm with update probability exp(−βΔS). By minimizing the R+R2 action with respect to conformal modes, Bonanno and Reuter have recently found analytical evidence of a nontrivial “rippled” ground state resembling a kinetic condensate of QCD. Our simulations at low but finite temperature (T=β−1) also display strong localized oscillations of the metric, whose total action S remains ≪ℏ thanks to the indefinite sign of R. The average metric ⟨grr⟩ is significantly different from flat space. The scaling properties of S and ⟨grr⟩ are investigated in dependence on N and β.
G. Modanese. Quantum metrics with very low action in R+R2 gravity. Physical Review D 2021, 103, 106020 .
AMA StyleG. Modanese. Quantum metrics with very low action in R+R2 gravity. Physical Review D. 2021; 103 (10):106020.
Chicago/Turabian StyleG. Modanese. 2021. "Quantum metrics with very low action in R+R2 gravity." Physical Review D 103, no. 10: 106020.
An ongoing debate in the first-principles description of conduction in molecular devices concerns the correct definition of current in the presence of non-local potentials. If the physical current density
F. Minotti; G. Modanese. Are Current Discontinuities in Molecular Devices Experimentally Observable? Symmetry 2021, 13, 691 .
AMA StyleF. Minotti, G. Modanese. Are Current Discontinuities in Molecular Devices Experimentally Observable? Symmetry. 2021; 13 (4):691.
Chicago/Turabian StyleF. Minotti; G. Modanese. 2021. "Are Current Discontinuities in Molecular Devices Experimentally Observable?" Symmetry 13, no. 4: 691.
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 a significant reduction of the pendulum angle amplitude for both the descent and the ascending rides, as predicted from simulations. Furthermore, we measured a smoother progress of the torque at the driving engine during the vehicle support crossings.
Markus Wenin; Siegfried Ladurner; Daniel Reiterer; Maria Bertotti; Giovanni Modanese. Validation of the Velocity Optimization for a Ropeway Passing over a Support. Sustainability 2021, 13, 2986 .
AMA StyleMarkus Wenin, Siegfried Ladurner, Daniel Reiterer, Maria Bertotti, Giovanni Modanese. Validation of the Velocity Optimization for a Ropeway Passing over a Support. Sustainability. 2021; 13 (5):2986.
Chicago/Turabian StyleMarkus Wenin; Siegfried Ladurner; Daniel Reiterer; Maria Bertotti; Giovanni Modanese. 2021. "Validation of the Velocity Optimization for a Ropeway Passing over a Support." Sustainability 13, no. 5: 2986.
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 independent from the simple ansatz knn∝kα but still displays a remarkable scale invariance. A rewiring procedure is then used to explicitely construct synthetic networks using the full correlation P(h|k) from which knn is derived. We consistently find that the knn functions of concrete synthetic networks deviate from ideal assortativity or disassortativity at large k. The consequences of this deviation on a diffusion process (the network Bass diffusion and its peak time) are numerically computed and discussed for some low-dimensional samples. Finally, we check that although the knn functions of the new family have an asymptotic behavior for large networks different from previous estimates, they satisfy the general criterium for the absence of an epidemic threshold.
Maria Letizia Bertotti; Giovanni Modanese. Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks. Symmetry 2021, 13, 141 .
AMA StyleMaria Letizia Bertotti, Giovanni Modanese. Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks. Symmetry. 2021; 13 (1):141.
Chicago/Turabian StyleMaria Letizia Bertotti; Giovanni Modanese. 2021. "Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks." Symmetry 13, no. 1: 141.
The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability e − β | S | . Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action | S | ≪ ħ by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which S = 0 exactly). They exhibit a periodic polarization in the scalar curvature R. In the simulation we fix a length scale L and divide it into N sub-intervals. The continuum limit is investigated by increasing N up to ∼ 10 6 ; the average squared action ⟨ S 2 ⟩ is found to scale as 1 / N 2 and thermalization of the algorithm occurs at a very low temperature (classical limit). This is in qualitative agreement with analytical results previously obtained for theories with stabilized conformal factor in the asymptotic safety scenario.
Giovanni Modanese. Quantum-Only Metrics in Spherically Symmetric Gravity. Quantum Reports 2020, 2, 314 -325.
AMA StyleGiovanni Modanese. Quantum-Only Metrics in Spherically Symmetric Gravity. Quantum Reports. 2020; 2 (2):314-325.
Chicago/Turabian StyleGiovanni Modanese. 2020. "Quantum-Only Metrics in Spherically Symmetric Gravity." Quantum Reports 2, no. 2: 314-325.
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 is accepted according to a standard Metropolis probability exp ( ± Δ r / T ) , where T is a variable temperature. We prove a general relation between Δ r and Δ K , thus finding a connection between two variables that have very different definitions and topological meaning. We describe rewiring trajectories in the r-K plane and explore the limits of maximally assortative and disassortative networks, including the case of small minimum degree ( k m i n ≥ 1 ), which has previously not been considered. The size of the giant component and the entropy of the network are monitored in the rewiring. The average number of second neighbors in the branching approximation z ¯ 2 , B is proven to be constant in the rewiring, and independent from the correlations for Markovian networks. As a function of the degree, however, the number of second neighbors gives useful information on the network connectivity and is also monitored.
Maria Letizia Bertotti; Giovanni Modanese. Network Rewiring in the r-K Plane. Entropy 2020, 22, 653 .
AMA StyleMaria Letizia Bertotti, Giovanni Modanese. Network Rewiring in the r-K Plane. Entropy. 2020; 22 (6):653.
Chicago/Turabian StyleMaria Letizia Bertotti; Giovanni Modanese. 2020. "Network Rewiring in the r-K Plane." Entropy 22, no. 6: 653.
We generate numerically on a lattice an ensemble of stationary metrics, with spherical symmetry, which have Einstein action SE « ћ. This is obtained through a Metropolis algorithm with weight exp(−β2S2E) and β » ћ−1. The squared action in the exponential allows to circumvene the problem of the non-positivity of SE. The discretized metrics obtained exhibit a spontaneous polarization in regions of positive and negative scalar curvature. We compare this ensemble with a class of continuous metrics previously found, which satisfy the condition SE = 0 exactly, or in certain cases even the stronger condition R(x) = 0 for any x. All these gravitational field configurations are of considerable interest in quantum gravity, because they represent possible vacuum fluctuations and are markedly different from Wheeler’s “spacetime foam”.
Giovanni Modanese. Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity. Symmetry 2019, 11, 1288 .
AMA StyleGiovanni Modanese. Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity. Symmetry. 2019; 11 (10):1288.
Chicago/Turabian StyleGiovanni Modanese. 2019. "Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity." Symmetry 11, no. 10: 1288.
We generate numerically on a lattice an ensemble of stationary metrics, with spherical symmetry, which have Einstein action $S_E \ll \hbar$. This is obtained through a Metropolis algorithm with weight $\exp(-\beta^2 S^2_E)$ and $\beta \gg \hbar^{-1}$. The squared action in the exponential allows to circumvene the problem of the non-positivity of $S_E$. The discretized metrics obtained exhibit a spontaneous polarization in regions of positive and negative scalar curvature. We compare this ensemble with a class of continuous metrics previously found, which satisfy the condition $S_E=0$ exactly, or in certain cases even the stronger condition $R({\bf x})=0$ for any ${\bf x}$. All these gravitational field configurations are of considerable interest in quantum gravity, because they represent possible vacuum fluctuations and are markedly different from Wheeler's "spacetime foam".
G. Modanese. Metrics with zero and almost-zero Einstein action in quantum gravity. 2019, 1 .
AMA StyleG. Modanese. Metrics with zero and almost-zero Einstein action in quantum gravity. . 2019; ():1.
Chicago/Turabian StyleG. Modanese. 2019. "Metrics with zero and almost-zero Einstein action in quantum gravity." , no. : 1.
We generate numerically on a lattice an ensemble of stationary metrics, with spherical symmetry, which have Einstein action SE « ħ. This is obtained through a Metropolis algorithm with weight exp(-β2SE2) and β » ħ-1. The squared action in the exponential allows to circumvene the problem of the non-positivity of SE. The discretized metrics obtained exhibit a spontaneous polarization in regions of positive and negative scalar curvature. We compare this ensemble with a class of continuous metrics previously found, which satisfy the condition SE=0 exactly, or in certain cases even the stronger condition R(x)=0 for any x. All these gravitational field configurations are of considerable interest in quantum gravity, because they represent possible vacuum fluctuations and are markedly different from Wheeler's ''spacetime foam''.
Giovanni Modanese. Metrics with Zero and Almost-zero Einstein Action in Quantum Gravity. 2019, 1 .
AMA StyleGiovanni Modanese. Metrics with Zero and Almost-zero Einstein Action in Quantum Gravity. . 2019; ():1.
Chicago/Turabian StyleGiovanni Modanese. 2019. "Metrics with Zero and Almost-zero Einstein Action in Quantum Gravity." , no. : 1.
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 of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of “real” networks obtained from preferential attachment. For β≥2 (β is the number of parent nodes in the preferential attachment scheme) the networks obtained with the configuration model are completely connected (giant component equal to 100%). In both generation schemes a clear disassortativity of the small degree nodes is demonstrated from the computation of the function knn. We also develop an efficient rewiring method which produces tunable variations of the assortativity coefficient r, and we use it to obtain maximally disassortative networks having the same degree distribution of BA networks with given β. Possible applications of this method concern assortative social networks.
Maria Letizia Bertotti; Giovanni Modanese. The configuration model for Barabasi-Albert networks. Applied Network Science 2019, 4, 32 .
AMA StyleMaria Letizia Bertotti, Giovanni Modanese. The configuration model for Barabasi-Albert networks. Applied Network Science. 2019; 4 (1):32.
Chicago/Turabian StyleMaria Letizia Bertotti; Giovanni Modanese. 2019. "The configuration model for Barabasi-Albert networks." Applied Network Science 4, no. 1: 32.
In systems with non-local potentials or other kinds of non-locality, the Landauer-Büttiker formula of quantum transport leads to replacing the usual gauge-invariant current density J with a current J e x t which has a non-local part and coincides with the current of the extended Aharonov-Bohm electrodynamics. It follows that the electromagnetic field generated by this current can have some peculiar properties and in particular the electric field of an oscillating dipole can have a long-range longitudinal component. The calculation is complex because it requires the evaluation of double-retarded integrals. We report the outcome of some numerical integrations with specific parameters for the source: dipole length ∼10−7 cm, frequency 10 GHz. The resulting longitudinal field E L turns out to be of the order of 10 2 to 10 3 times larger than the transverse component (only for the non-local part of the current). Possible applications concern the radiation field generated by Josephson tunnelling in thick superconductor-normal-superconductor (SNS) junctions in yttrium barium oxide (YBCO) and by current flow in molecular nanodevices.
Giovanni Modanese. High-Frequency Electromagnetic Emission from Non-Local Wavefunctions. Applied Sciences 2019, 9, 1982 .
AMA StyleGiovanni Modanese. High-Frequency Electromagnetic Emission from Non-Local Wavefunctions. Applied Sciences. 2019; 9 (10):1982.
Chicago/Turabian StyleGiovanni Modanese. 2019. "High-Frequency Electromagnetic Emission from Non-Local Wavefunctions." Applied Sciences 9, no. 10: 1982.
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 compare our results with those obtained for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to a little-known property of these networks: whereas the value of the assortativity coefficient is close to zero, they look disassortative if one considers only a bounded range of degrees, including the smallest ones, and slightly assortative on the range of the higher degrees. We also find that if the trickle-down character of the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model), the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger, reaching, for instance, the 34% in a typical case on a network with 104 nodes.
M. L. Bertotti; G. Modanese. The Bass Diffusion Model on Finite Barabasi-Albert Networks. Complexity 2019, 2019, 1 -12.
AMA StyleM. L. Bertotti, G. Modanese. The Bass Diffusion Model on Finite Barabasi-Albert Networks. Complexity. 2019; 2019 ():1-12.
Chicago/Turabian StyleM. L. Bertotti; G. Modanese. 2019. "The Bass Diffusion Model on Finite Barabasi-Albert Networks." Complexity 2019, no. : 1-12.
It has recently been proven that certain effective wavefunctions in fractional quantum mechanics and condensed matter do not have a locally conserved current; as a consequence, their coupling to the electromagnetic field leads to extended Maxwell equations, featuring non-local, formally simple additional source terms. Solving these equations in general form or finding analytical approximations is a formidable task, but numerical solutions can be obtained by performing some bulky double-retarded integrals. We focus on concrete experimental situations which may allow to detect an anomalous quasi-static magnetic field generated by these (collective) wavefunctions in cuprate superconductors. We compute the spatial dependence of the field and its amplitude as a function of microscopic parameters including the fraction η of supercurrent that is not locally conserved in Josephson junctions between grains, the thickness a of the junctions and the size ε of their current sinks and sources. The results show that the anomalous field is actually detectable at the macroscopic level with sensitive experiments, and can be important at the microscopic level because of virtual charge effects typical of the extended Maxwell equations.
G. Modanese. Design of a test for the electromagnetic coupling of non-local wavefunctions. Results in Physics 2018, 12, 1056 -1061.
AMA StyleG. Modanese. Design of a test for the electromagnetic coupling of non-local wavefunctions. Results in Physics. 2018; 12 ():1056-1061.
Chicago/Turabian StyleG. Modanese. 2018. "Design of a test for the electromagnetic coupling of non-local wavefunctions." Results in Physics 12, no. : 1056-1061.
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 two general cases numerically, a single vehicle system (FUNIFOR), as well as a classical aerial ropeway with two vehicles. In both cases we find optimal velocity profiles that show a considerable improvement of the oscillatory behavior of the vehicles as compared to constant velocity profiles and optimal profiles that have been obtained analytically by loosening some of the constraints for the system. In addition to a minimization of the vehicle oscillations, we also optimize the time that elapses as the vehicle is hauled through the system. We believe that this exploratory study lays a sound basis for various possible future studies and practical applications (Computer Aided Engineering).
M. Wenin; A. Windisch; S. Ladurner; M.L. Bertotti; G. Modanese. Optimal velocity profile for a cable car passing over a support. European Journal of Mechanics - A/Solids 2018, 73, 366 -372.
AMA StyleM. Wenin, A. Windisch, S. Ladurner, M.L. Bertotti, G. Modanese. Optimal velocity profile for a cable car passing over a support. European Journal of Mechanics - A/Solids. 2018; 73 ():366-372.
Chicago/Turabian StyleM. Wenin; A. Windisch; S. Ladurner; M.L. Bertotti; G. Modanese. 2018. "Optimal velocity profile for a cable car passing over a support." European Journal of Mechanics - A/Solids 73, no. : 366-372.
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.
Giovanni Modanese. Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current. Mathematics 2018, 6, 155 .
AMA StyleGiovanni Modanese. Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current. Mathematics. 2018; 6 (9):155.
Chicago/Turabian StyleGiovanni Modanese. 2018. "Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current." Mathematics 6, no. 9: 155.
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 intrinsic information of the mechanical system and represents a linear map between the initial state, t = 0 (initial condition of a set of linear differential equations) and the state at a time t. We consider undamped and damped oscillations, where damping is introduced by a phenomenological way (Onsager’s lineare ansätze). A numerical example is given.
Markus Wenin; Michael Irschara; Stephan Obexer; Maria Letizia Bertotti; Giovanni Modanese. Cable Railway Simulation: A Two-Span Oscillator Model. Advanced Structured Materials 2018, 65 -79.
AMA StyleMarkus Wenin, Michael Irschara, Stephan Obexer, Maria Letizia Bertotti, Giovanni Modanese. Cable Railway Simulation: A Two-Span Oscillator Model. Advanced Structured Materials. 2018; ():65-79.
Chicago/Turabian StyleMarkus Wenin; Michael Irschara; Stephan Obexer; Maria Letizia Bertotti; Giovanni Modanese. 2018. "Cable Railway Simulation: A Two-Span Oscillator Model." Advanced Structured Materials , no. : 65-79.
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 white or Ornstein–Uhlenbeck spectrum. The most important measured correlations are those between Gini inequality index G and social mobility M, between total income and G, and between M and total income. We describe numerical results concerning these correlations and a quantity which gives average stochastic deviations from the equilibrium solutions in dependence on the noise amplitude.
Maria Letizia Bertotti; Amit K. Chattopadhyay; Giovanni Modanese. Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. Entropy 2018, 20, 166 .
AMA StyleMaria Letizia Bertotti, Amit K. Chattopadhyay, Giovanni Modanese. Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange. Entropy. 2018; 20 (3):166.
Chicago/Turabian StyleMaria Letizia Bertotti; Amit K. Chattopadhyay; Giovanni Modanese. 2018. "Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange." Entropy 20, no. 3: 166.
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 exchange far from the average (i.e., close to 0% or 100%) is quite large, while in the extreme relativistic case it is small. We compare these properties with the concept of “saving propensity”, employed in econophysics to define the fraction of their money that individuals put at stake in economic interactions.
Maria Letizia Bertotti; Giovanni Modanese. Statistics of Binary Exchange of Energy or Money. Entropy 2017, 19, 465 .
AMA StyleMaria Letizia Bertotti, Giovanni Modanese. Statistics of Binary Exchange of Energy or Money. Entropy. 2017; 19 (9):465.
Chicago/Turabian StyleMaria Letizia Bertotti; Giovanni Modanese. 2017. "Statistics of Binary Exchange of Energy or Money." Entropy 19, no. 9: 465.
The Aharonov–Bohm electrodynamics is a generalization of Maxwell theory with reduced gauge invariance. It allows to couple the electromagnetic field to a charge which is not locally conserved, and has an additional degree of freedom, the scalar field [Formula: see text], usually interpreted as a longitudinal wave component. By reformulating the theory in a compact Lagrangian formalism, we are able to eliminate S explicitly from the dynamics and we obtain generalized Maxwell equation with interesting properties: they give [Formula: see text] as the (conserved) sum of the (possibly non-conserved) physical current density [Formula: see text], and a “secondary” current density [Formula: see text] which is a nonlocal function of [Formula: see text]. This implies that any non-conservation of [Formula: see text] is effectively “censored” by the observable field [Formula: see text], and yet it may have real physical consequences. We give examples of stationary solutions which display these properties. Possible applications are to systems where local charge conservation is violated due to anomalies of the Adler–Bell–Jackiw (ABJ) kind or to macroscopic quantum tunnelling with currents which do not satisfy a local continuity equation.
Giovanni Modanese. Generalized Maxwell equations and charge conservation censorship. Modern Physics Letters B 2017, 31, 1750052 .
AMA StyleGiovanni Modanese. Generalized Maxwell equations and charge conservation censorship. Modern Physics Letters B. 2017; 31 (6):1750052.
Chicago/Turabian StyleGiovanni Modanese. 2017. "Generalized Maxwell equations and charge conservation censorship." Modern Physics Letters B 31, no. 6: 1750052.
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 that our model dynamics is self-consistently constrained by dynamical conservation laws emerging from population and wealth conservation. This model is numerically solved and analyzed to interpret the inequality of income as a function of relevant dynamical parameters like the {\it mobility} $M$ and the {\it total income} $\mu$. In our model, inequality is quantified by the {\it Gini index} $G$. In particular, correlations between any two of the mobility index $M$ and/or the total income $\mu$ with the Gini index $G$ are investigated and compared with the analogous correlations resulting from an equivalent additive noise model. Our findings highlight the importance of a multiplicative noise based economic modeling structure in the analysis of inequality. The model also depicts the nature of correlation between mobility and total income of a population from the perspective of inequality measure.
Maria Letizia Bertotti; Amit K Chattopadhyay; Giovanni Modanese. Economic inequality and mobility for stochastic models with multiplicative noise. 2017, 1 .
AMA StyleMaria Letizia Bertotti, Amit K Chattopadhyay, Giovanni Modanese. Economic inequality and mobility for stochastic models with multiplicative noise. . 2017; ():1.
Chicago/Turabian StyleMaria Letizia Bertotti; Amit K Chattopadhyay; Giovanni Modanese. 2017. "Economic inequality and mobility for stochastic models with multiplicative noise." , no. : 1.