IFSM representation of Brownian motion with applications to simulation

Several methods are currently available to simulate paths of the Brownian motion. In particular, paths of the BM can be simulated using the properties of the increments of the process like in the Euler scheme, or as the limit of a random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve series. In this paper we first propose a IFSM (Iterated Function Systems with Maps) operator whose fixed point is the trajectory of the BM. We then use this representation of the process to simulate its trajectories. The resulting simulated trajectories are self-affine, continuous and fractal by construction. This fact produces more realistic trajectories than other schemes in the sense that their geometry is closer to the one of the true BM's trajectories. The IFSM trajectory of the BM can then be used to generate more realistic solutions of stochastic differential equations

Quantum mechanics in finite dimensional Hilbert space

The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with the infinite dimensional case. The construction of an unbiased basis for state determination is discussed.Comment: 14 pages, no figure

Model-Based Control Using Koopman Operators

This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting model-based control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand.Comment: 8 page

Entanglement for all quantum states

It is shown that a state that is factorizable in the Hilbert space corresponding to some choice of degrees of freedom, becomes entangled for a different choice of degrees of freedom. Therefore, entanglement is not a special case but is ubiquitous in quantum systems. Simple examples are calculated and a general proof is provided. The physical relevance of the change of tensor product structure is mentioned.Comment: 9 page

A Goal Programming Model with Satisfaction Function for Risk Management and Optimal Portfolio Diversification

We extend the classical risk minimization model with scalar risk measures to the general case of set-valued risk measures. The problem we obtain is a set-valued optimization model and we propose a goal programming-based approach with satisfaction function to obtain a solution which represents the best compromise between goals and the achievement levels. Numerical examples are provided to illustrate how the method works in practical situations

Quantum diffusion on a cyclic one dimensional lattice

The quantum diffusion of a particle in an initially localized state on a cyclic lattice with N sites is studied. Diffusion and reconstruction time are calculated. Strong differences are found for even or odd number of sites and the limit N->infinit is studied. The predictions of the model could be tested with micro - and nanotechnology devices.Comment: 17 pages, 5 figure

Geographical Heterogeneities and Externalities in an Epidemiological-Macroeconomic Framework

We analyze the implications of geographical heterogeneities and externalities on health and macroeconomic outcomes by extending a basic epidemiological-macroeconomic model to a spatial dimension. Because of people's migration and commuting across different regions, a disease may spread also in areas far from those in which the outbreak originally occurs and thus the health status (i.e., disease prevalence) in specific regions may depend on the health status in other regions as well. We show that neglecting the existence of cross-regional effects may lead to misleading conclusions about the long run outcome not only in single regions but also in the entire economy, suggesting that single regions need to coordinate their efforts to achieve disease eradication. We analyze such a coordination by focusing on a control problem in which the social planner determines globally the level of intervention showing that to achieve eradication it is essential to accompany traditional disease control policies (i.e., prevention and treatment) with regulations limiting people's movements. Focusing on COVID-19 we present a calibration based on Italian data showing that, because of the infections generated by cross-regional commuting, even vaccination may not be enough to achieve disease eradication, and limitations on people's movements need to accompany vaccination to preclude COVID-19 from reaching an endemic state

In-Flight Cost Index Optimisation Upon Weather Forecast Updates

This paper presents an optimisation framework to compute the altitude and speed profiles of a trajectory in the execution phase of the flight, such that the expected total cost (ETC) of the operation is minimised (i.e., modelling the expected cost of delay and fuel – including arrival uncertainties – at the arrival gate). This is achieved with a two-stage optimisation strategy: a trajectory optimiser that minimises a generalised direct operating cost function, for a given cost index; and an upper-level optimiser, which obtains the best cost index that minimises the ETC. Several case studies are presented for different departure delays, while considering the impact of two different weather forecast updates too: a region with relative high head-winds appearing half way across the flight; and a cold atmosphere scenario, with a tropopause altitude lower than standard conditions