Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schr\"odinger equation $i\partial \Psi/\partial t = {H}\Psi$, where ${H}$ is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector $\Psi$ composed of the electromagnetic fields and auxiliary matter fields describing the medium response. In this representation, the initial value problem is solved by applying the fundamental solution $\exp(-itH)$ to the initial field configuration. The Faber polynomial approximation of the fundamental solution is used to develop a numerical algorithm for propagation of broad band wave packets in passive media. The action of the Hamiltonian on the wave function $\Psi$ is approximated by the Fourier grid pseudospectral method. The algorithm is global in time, meaning that the entire propagation can be carried out in just a few time steps. A typical time step is much larger than that in finite differencing schemes, $\Delta t_F \gg \|H\|^{-1}$. The accuracy and stability of the algorithm is analyzed. The Faber propagation method is compared with the Lanczos-Arnoldi propagation method with an example of scattering of broad band laser pulses on a periodic grating made of a dielectric whose dispersive properties are described by the Rocard-Powels-Debye model. The Faber algorithm is shown to be more efficient. The Courant limit for time stepping, $\Delta t_C \sim \|H\|^{-1}$, is exceeded at least in 3000 times in the Faber propagation scheme.Comment: Latex, 17 pages, 4 figures (separate png files); to appear in J. Comput. Phy

Demand scenario analysis and planned capacity expansion: A system dynamics framework

This paper establishes an approach to develop models for forecasting demand and evaluating policy scenarios related to planned capacity expansion for meeting optimistic and pessimistic future demand projections. A system dynamics framework is used to model and to generate scenarios because of their capability of representing physical and information flows, which will enable us to understand the nonlinear dynamics behavior in uncertain conditions. These models can provide important inputs such as construction growth, GDP growth, and investment growth to specific business decisions such as planned capacity expansion policies that will improve the system performance

Capacity expansion under a service-level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service-level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up-and-out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes

Firms as Bundles of Discrete Resources - Towards an Explanation of the Exponential Distribution of Firm Growth Rates

A robust feature of the corporate growth process is the Laplace, or symmetric exponential, distribution of firm growth rates. In this paper, we sketch out a class of simple theoretical models capable of explaining this empirical regularity. We do not attempt to generalize on where growth opportunities comme from, but rather we focus on how firms build upon growth opportunites. We borrow ideas from the self-organizing criticality literature to explain how the interdependent nature of discrete resources may lead to the triggering off of a series of additions to a firm's resources. In a first formal model we consider the case of employment growth in a hierarchy, and observe that growth rates follow an exponential distribution. In a second model we include plant and capital as resources and we are able to reproduce a number of stylized facts about firm growth.Firm growth rates, exponential distribution, hierarchy, growth autocorrelation.

Virus Replication as a Phenotypic Version of Polynucleotide Evolution

In this paper we revisit and adapt to viral evolution an approach based on the theory of branching process advanced by Demetrius, Schuster and Sigmund ("Polynucleotide evolution and branching processes", Bull. Math. Biol. 46 (1985) 239-262), in their study of polynucleotide evolution. By taking into account beneficial effects we obtain a non-trivial multivariate generalization of their single-type branching process model. Perturbative techniques allows us to obtain analytical asymptotic expressions for the main global parameters of the model which lead to the following rigorous results: (i) a new criterion for "no sure extinction", (ii) a generalization and proof, for this particular class of models, of the lethal mutagenesis criterion proposed by Bull, Sanju\'an and Wilke ("Theory of lethal mutagenesis for viruses", J. Virology 18 (2007) 2930-2939), (iii) a new proposal for the notion of relaxation time with a quantitative prescription for its evaluation, (iv) the quantitative description of the evolution of the expected values in in four distinct "stages": extinction threshold, lethal mutagenesis, stationary "equilibrium" and transient. Finally, based on these quantitative results we are able to draw some qualitative conclusions.Comment: 23 pages, 1 figure, 2 tables. arXiv admin note: substantial text overlap with arXiv:1110.336