Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets.

We report a search for new gravitational physics phenomena based on Riemann-Cartan theory of general relativity including spacetime torsion. Starting from the parametrized torsion framework of Mao, Tegmark, Guth, and Cabi, we analyze the motion of test bodies in the presence of torsion, and, in particular, we compute the corrections to the perihelion advance and to the orbital geodetic precession of a satellite. We consider the motion of a test body in a spherically symmetric field, and the motion of a satellite in the gravitational field of the Sun and the Earth. We describe the torsion field by means of three parameters, and we make use of the autoparallel trajectories, which in general differ from geodesics when torsion is present. We derive the specific approximate expression of the corresponding system of ordinary differential equations, which are then solved with methods of celestial mechanics. We calculate the secular variations of the longitudes of the node and of the pericenter of the satellite. The computed secular variations show how the corrections to the perihelion advance and to the orbital de Sitter effect depend on the torsion parameters. All computations are performed under the assumptions of weak field and slow motion. To test our predictions, we use the measurements of the Moon's geodetic precession from lunar laser ranging data, and the measurements of Mercury's perihelion advance from planetary radar ranging data. These measurements are then used to constrain suitable linear combinations of the torsion parameters.

We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical properties of the model and global estimates. The asymptotic behavior and the stability of solutions in the case of two strategies is analyzed in details. Numerical schemes for two and three strategies which are able to capture the correct equilibrium states are also proposed together with several numerical examples.

The flow in the stern region of a fully appended hull is analyzed by both computational and experimental fluid dynamics. The study is focused on the velocity field induced by the rotating propellers. Measurements have been performed by laser Doppler velocimetry (LDV) on the vertical midplane of the rudder and in two transversal planes behind the propeller and behind the rudder. In the numerical approach, the real geometry of the propeller has been considered. To this purpose, a dynamic overlapping grids method has been used, which is implemented in the unsteady Reynolds averaged Navier-Stokes equations (URANSE) solver developed at INSEAN. Uncertainty analysis has been performed on both data sets and the results from the two approaches are compared. The agreement between the two data sets is found to be good, the deviation in the velocity and vorticity fields lying within the evaluated uncertainties. Numerical data allowed the analysis of further details of the flow that could not be measured, like load conditions of the single blades, interaction of the propeller wake with the rudder, and pressure oscillations induced by the propeller on the vault of the stern.

This paper presents the results of a large experimental and numerical campaign aimed to the analysis of the interference effect for a fast catamaran. Several separation distances are considered; data for resistance, sinkage and trim are collected by towing tank experiments for Froude number ranging from 0.2 to 0.8. Monohull tests are also carried out, the analysis of the interference and its dependency on the separation length being the main objective of the paper. Resistance coefficient curves reveal the presence of two humps, the second one strongly depending on the separation length; high interference is observed in correspondence of the second hump. It is found that the narrower is the configuration, the higher is the interference and the speed at which this maximum occurs. To gain a deeper insight into these behaviors, a complementary analysis, in terms of wave field, surface pressure and velocity field is carried out by an in-house unsteady RANS solver. Verification of numerical results is provided, together with validation, which is made by the comparison with both present and other experimental data. Agreement in terms of resistance coefficient is rather good, comparison error being always smaller than 2.2%.

The simulations of the flow around a high-speed vessel in both catamaran and monohull configurations are carried out by the numerical solution of the Reynold averaged Navier–Stokes (RANS) equations. The goal of the analysis is the investigation of the interference phenomena between the two hulls, with focus on its dependence on the Reynolds number (Re). To this aim, numerical simulations are carried out for values of Re ranging from 106 to 108 for two different values of the Froude number (Fr = 0.30, 0.45). Wave patterns, wave profiles, limiting treamlines, surface pressure and velocity fields are analyzed; comparison is made between the catamaran and the monohull configurations. Dependence of the pressure and viscous resistance coefficients, as well as of the interference factor, on the Reynolds number is investigated. Verification and validation for both resistance coefficients and wave cuts is also performed.

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows one to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This way it is possible to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper. © 2011 Society for Industrial and Applied Mathematics.

In this work a physical modelling framework is presented, describing the intelligent, non-local, and anisotropic behaviour of pedestrians. Its phenomenological basics and constitutive elements are detailed, and a qualitative analysis is provided. Within this common framework, two first-order mathematical models, along with related numerical solution techniques, are derived. The models are oriented to specific real world applications: a one-dimensional model of crowd-structure interaction in footbridges and a two-dimensional model of pedestrian flow in an underground station with several obstacles and exits. The noticeable heterogeneity of the applications demonstrates the significance of the physical framework and its versatility in addressing different engineering problems. The results of the simulations point out the key role played by the physiological and psychological features of human perception on the overall crowd dynamics. © 2010 Elsevier Inc.

This paper introduces a new model of pedestrian flow, formulated within a measure-theoretic framework. It consists of a macroscopic representation of the system via a family of measures which, pushed forward by some flow maps, provide an estimate of the space occupancy by pedestrians at successive times. From the modeling point of view, this setting is particularly suitable for treating nonlocal interactions among pedestrians, obstacles, and wall boundary conditions. In addition, the analysis and numerical approximation of the resulting mathematical structures, which are the principal objectives of this work, follow more easily than for models based on standard hyperbolic conservation laws. © 2010 Springer-Verlag.

In this paper we consider first order differential models of collective behaviors of groups of agents, based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.