Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponent of Extended Harper's Model

We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic $M_2(\mathbb{C})$-cocycles. This includes rational approximation of the frequency, which so far has not been available

Optical diffraction of focused spots and subwavelength structures

We have developed a numerical diffraction tool for cases in which the incident field is a focused spot and the diffracting structure is a single structure or an aperiodic surface. Our approach uses the integral formulation to solve Maxwell’s equations and is different from previously published methods in its choice of basis function. We compared numerical results with experimental measurements of the far-field intensity for a focused spot incident on an aluminum grating, and the comparison was favorable. Finally, we predict the diffraction behavior of the proposed digital video disk format for the next generation of optical disk. Our analysis shows that the reflected signal for this format has a strong dependence on the polarization of the incident light

Polarization quadrature measurement of subwavelength diffracting structures

The amplitude and the phase of the diffracted far field depends on polarization when the diffracting structure is comparable to or less than the wavelength. When the far-field amplitude and the phase of one polarization with respect to the orthogonal polarization is measured, small changes in the structure can be measured. To make the far-field polarization measurements, we design a detector that measures the relative polarization amplitude and the phase in quadrature. We predict numerically and verify experimentally the polarization amplitude and the phase for an optical disc and a set of gratings with varying depth. Our results show that this measurement technique is sensitive to small variations in the diffracting structure and that it can be useful in applications such as critical dimension and overlay metrology in microelectronics fabrication

Plane wave/pseudopotential implementation of excited state gradients in density functional linear response theory: a new route via implicit differentiation

This work presents the formalism and implementation of excited state nuclear forces within density functional linear response theory (TDDFT) using a plane wave basis set. An implicit differentiation technique is developed for computing nonadiabatic coupling between Kohn-Sham molecular orbital wavefunctions as well as gradients of orbital energies which are then used to calculate excited state nuclear forces. The algorithm has been implemented in a plane wave/pseudopotential code taking into account only a reduced active subspace of molecular orbitals. It is demonstrated for the H$_2$ and N$_2$ molecules that the analytical gradients rapidly converge to the exact forces when the active subspace of molecular orbitals approaches completeness

Ekonomiese en Bedryfswoordeboek: Engels–Afrikaans/Dictionary of Eco-nomics and Business: English–Afrikaans

W.F.J. Steenkamp en/and H.J.J. Reynders (Hoofredakteurs/Chief Edi-tors. Ekonomiese en Bedryfswoordeboek: Engels–Afrikaans/Dictionary of Eco-nomics and Business: English–Afrikaans. Tweede, hersiene en vermeerder-de uitgawe/Second, revised and enlarged edition. 2004, xii + 1199 pp. ISBN 1868900312. Kaapstad/Cape Town: Pharos Woordeboeke/Dic-tionaries. Prys/Price: R400

Paradigms for Parameterized Enumeration

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different notions of efficient enumeration in the context of parameterized complexity. Second we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape

Alternative axiomatics and complexity of deliberative STIT theories

We propose two alternatives to Xu's axiomatization of the Chellas STIT. The first one also provides an alternative axiomatization of the deliberative STIT. The second one starts from the idea that the historic necessity operator can be defined as an abbreviation of operators of agency, and can thus be eliminated from the logic of the Chellas STIT. The second axiomatization also allows us to establish that the problem of deciding the satisfiability of a STIT formula without temporal operators is NP-complete in the single-agent case, and is NEXPTIME-complete in the multiagent case, both for the deliberative and the Chellas' STIT.Comment: Submitted to the Journal of Philosophical Logic; 13 pages excluding anne