Narrow Results

This paper, that deals with the modelling of crowd dynamics, is the first one of a project finalized to develop a mathematical theory refereing to the modelling of the complex systems constituted by several interacting individuals in bounded and unbounded domains. The first part of the paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions. The second part of the paper deals with a qualitative analysis of the models with the aim of analyzing their properties. Finally, a critical analysis is proposed in view of further development of the modelling approach. Additional reasonings are devoted to understanding the conceptual differences between crowd and swarm modelling.

The cells in a tissue occupying a region Ω_t are divided according to their cycling phase. The density p_i of cells in phase i depends on the spatial variable x, the time t, and the time s_i since the cells entered in phase i. The p_i(x, t, s_i) and the oxygen concentration w(x, t) satisfy a system of PDEs in Ω_t, and the boundary of Ω_t is a free boundary. We denote by the oxygen concentration on the free boundary and consider the radially symmetric case, so that Ω_t = {r < R(t)}. We prove that R(t) is always bounded; furthermore, if is small, then R(t) → 0 as t → ∞, and if is large, then R(t) ≥ c > 0 for all t. Finally, we prove the existence and uniqueness of a stationary solution in a special case.

A wide range of applications requires the modeling of wave propagation phenomena in media with variable physical properties in the domain of interest, while highly accurate algorithms are needed to avoid unphysical effects. Spectral element methods (SEM), based on either a Chebyshev or a Legendre polynomial basis, have excellent properties of accuracy and flexibility in describing complex models, outperforming other techniques. In the standard SEM approach the computational domain is discretized by using very coarse meshes and constant-property elements, but in some cases the accuracy and the computational efficiency may be seriously reduced. For instance, a finely heterogeneous medium requires grid resolution down to the finest scales, leading to an extremely large problem dimension. In such problems the wavelength scale of interest is much larger but cannot be exploited in order to reduce the problem size. A poly-grid Chebyshev spectral element method (PG-CSEM) can overcome this limitation. In order to accurately deal with continuous variation in the properties, or even with small scale fluctuations, temporary auxiliary grids are introduced which avoid the need of using any finer global grid, and at the macroscopic level the wave field propagation is solved maintaining the SEM accuracy and computational efficiency.

Based on the observation that the elasticity of variance of risky assets is randomly varying around a constant, we take an underlying asset model in which the averaged constant elasticity of variance is perturbed by a small fast fluctuating process and study the Merton type portfolio optimization problem using dynamic programming as well as asymptotic expansions. The Hamilton–Jacobi–Bellman equation for each of the power and exponential utility functions leads to an optimal trading strategy as a perturbation around the well known one. We reveal the impact of both the constant elasticity of variance upon the Merton investment optimal control under the Black–Scholes model and the stochastic elasticity of variance upon the investment optimal control under the constant elasticity of variance model. The concavity of the investment policy with respect to the excess return is characteristic of a market economy with the constant or stochastic elasticity of variance.

In this paper, the multiscale stochastic 3D generalized Navier–Stokes equations are studied. By using Khasminkii’s time discretization approach and the technique of stopping time, the strong averaging principle for stochastic 3D generalized Navier–Stokes equations is proved in the space ${ℍ}^1({𝕋}^3)$.

This paper constructs a class of semi-orthogonal and bi-orthogonal wavelet transforms on possibly irregular point sets with the property that the scaling coefficients are independent from the order of refinement. That means that scaling coefficients at a given scale can be constructed with the configuration at that scale only. This property is of particular interest when the refinement operation is data dependent, leading to adaptive multiresolution analyses. Moreover, the proposed class of wavelet transforms are constructed using a sequence of just two lifting steps, one of which contains a linear interpolating prediction operator. This operator easily allows extensions towards directional offsets from predictions, leading to an edge-adaptive nonlinear multiscale decomposition.