The level set method for the two-sided eigenproblem

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomia

An iterative method for extreme optics of two-level systems

We formulate the problem of a two-level system in a linearly polarized laser field in terms of a nonlinear Riccati-type differential equation and solve the equation analytically in time intervals much shorter than half the optical period. The analytical solutions for subsequent intervals are then stuck together in an iterative procedure to cover the scale time of the laser pulse. This approach is applicable to pulses of arbitrary (nonrelativistic) strengths, shapes and durations, thus covering the whole region of light-matter couplings from weak through moderate to strong ones. The method allows quick insight into different problems from the field of light--matter interaction. Very good quality of the method is shown by recovering with it a number of subtle effects met in earlier numerically calculated photon-emission spectra from model molecular ions, double quantum wells, atoms and semiconductors. The method presented is an efficient mathematical tool to describe novel effects in the region of, e.g., extreme nonlinear optics, i.e., when two--level systems are exposed to pulses of only a few cycles in duration and strength ensuring the Rabi frequency to approach and even exceed the laser light frequence

A two-level enriched finite element method for a mixed problem

The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas elemen

Magnus expansion method for two-level atom interacting with few-cycle pulse

Using the Magnus expansion to the fourth order, we obtain analytic expressions for the atomic state of a two-level system driven by a laser pulse of arbitrary shape with small pulse area. We also determine the limitation of our obtained formulas due to limited range of convergence of the Magnus series. We compare our method to the recently developed method of Rostovtsev et al. (PRA 2009, 79, 063833) for several detunings. Our analysis shows that our technique based on the Magnus expansion can be used as a complementary method to the one in PRA 2009.Comment: 12 pages, 4 figure

A simple method to identify significant effects in unreplicated two-level factorial designs

This article proposes a generalization and improvement on the method of Lenth (1989). The problem is solved by fixing outliers in highly contaminated samples. To do this a scale robust estimator is obtained and its performance is analyzed using computer simulations. The method is extremely simple to use and leads to the same results as the more complex one proposed by Box and Meyer (1986)

A two-level four-color SOR method

Bibliography: p. 31.Supported in part by the Army Research Office under grant no. DAAG29-84-K-0005 Supported in part by Advanced Research Projects Agency monitored by ONR under contract N00014-81-K-0742 Supported in part by AFOSR contract F49620-84-0004C.-C. Jay Kuo, Bernard C. Levy