Projected equation methods for approximate solution of large linear systems

We consider linear systems of equations and solution approximations derived by projection on a low-dimensional subspace. We propose stochastic iterative algorithms, based on simulation, which converge to the approximate solution and are suitable for very large-dimensional problems. The algorithms are extensions of recent approximate dynamic programming methods, known as temporal difference methods, which solve a projected form of Bellman’s equation by using simulation-based approximations to this equation, or by using a projected value iteration method.     

Fourier time spectral method for subsonic and transonic flows

The time accuracy of the exponentially accu-rate Fourier time spectral method (TSM) is examined and compared with a conventional 2nd-order backward differ-ence formula (BDF) method for periodic unsteady flows. In particular, detailed error analysis based on numerical com-putations is performed on the accuracy of resolving the local pressure coefficient and global integrated force coefficients for smooth subsonic and non-smooth transonic flows with moving shock waves on a pitching airfoil. For smooth sub-sonic flows, the Fourier TSM method offers a significant accuracy advantage over the BDF method for the predic-tion of both the local pressure coefficient and integrated force coefficients. For transonic flows where the motion of the discontinuous shock wave contributes significant higher-order harmonic contents to the local pressure fluctuations, a sufficient number of modes must be included before the Fourier TSM provides an advantage over the BDF method. The Fourier TSM, however, still offers better accuracy than the BDF method for integrated force coefficients even for transonic flows. A problem of non-symmetric solutions for symmetric periodic flows due to the use of odd numbers of intervals is uncovered and analyzed. A frequency-searching method is proposed for problems where the frequency is not known a priori. The method is tested on the vortex shedding problem of the flow over a circular cylinder.     

Pulsed laser deposition of fluoride glass thin films

The development of integrated waveguide lasers for different applications such as marking, illumination or medical technology has become highly desirable. Diode pumped planar waveguide lasers emitting in the green visible spectral range, e.g. thin films from praseodymium doped fluorozirconate glass matrix (called ZBLAN, owing to the main components ZrF₄, BaF₂, LaF₃, AlF₃ and NaF) as the active material pumped by a blue laser diode, have aroused great interest. In this work we have investigated the deposition of Pr:ZBLAN thin films using pulsed laser radiation of λ = 193 and λ = 248 nm. The deposition has been carried out on MgF₂ single crystal substrates in a vacuum chamber by varying both processing gas pressure and energy fluence. The existence of an absorption line at 210 nm in Pr:ZBLAN leads to absorption and radiative relaxation of the absorbed laser energy of λ = 193 nm preventing the evaporation of target material. The deposited thin films consist of solidified and molten droplets and irregular particulates only. Furthermore, X-ray radiation has been applied to fluoride glass targets to enhance the absorption in the UV spectral region and to investigate the deposition of X-ray treated targets applying laser radiation of λ = 248 nm. It has been shown that induced F-centres near the target surface are not thermally stable and can be easily ablated. Therefore, λ = 248 nm is not suitable for evaporation of Pr:ZBLAN.     

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).     

A realistic world from intersecting D6-branes

We describe a three-family Pati–Salam model from intersecting D6-branes in type IIA string theory on the T⁶/(Z₂×Z₂)${\mathbf{T}}^6/({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$ orientifold which is of strong phenomenological interest. In the model, the gauge coupling unification is achieved naturally at the string scale, and the gauge symmetry can be broken down to the Standard Model (SM) close to the string scale. Moreover, we find that it is possible to obtain the correct SM quark masses and mixings, and the tau lepton mass. Additionally, neutrino masses and mixings may be generated via the seesaw mechanism. Furthermore, we calculate the supersymmetry breaking soft terms, and the corresponding low-energy supersymmetric particle spectra which may potentially be tested at the Large Hadron Collider (LHC), and provide the observed dark matter density.     

Intrinsic nonlinearity from LAOStrain—experiments on various strain- and stress-controlled rheometers: a quantitative comparison

Strain-controlled large amplitude oscillatory shear (LAOStrain) experiments on a polyisoprene melt and a polyisobutylene solution were conducted on four different rheometers. The results are compared using nonlinear quantities such as the normalized intensity of the third harmonic (I _3/1) and the intrinsic nonlinearity in order to assess the reproducibility of the experiments. Two of the investigated instruments were strain-controlled rheometers, another two, were advanced stress-controlled rheometers. Since the stress-controlled rheometers are able to conduct strain-controlled tests when employing an active deformation control loop, the two different rheometer types could be compared. Experimental details like the gain of the deformation control loop, and the method of temperature control have been shown to play crucial roles in achieving reasonable reproducibility across the different instruments. Furthermore, deviations from the quadratic scaling of I _3/1 with the strain amplitude and the influence of instrument inertia on nonlinear quantities were observed for one of the stress-controlled instruments. The standard deviation of the intrinsic nonlinearity Q ₀(ω ₀) at a specific angular frequency as determined by measurements on the same instrument was found to be 8 % or lower. The relative deviations of Q ₀ across different instruments were instead up to 12 % in the investigated frequency range with an exception for a specific instrument and one of the samples, where the deviation was considerably larger.     

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

The reduced density matrix is variationally optimized for the two-dimensional Hubbard model. Exploiting all symmetries present in the system, we have been able to study 6 × 6 lattices at various fillings and different values for the on-site repulsion, using the highly accurate but computationally expensive three-index conditions. To reduce the computational cost we study the performance of imposing the three-index constraints on local clusters of 2 × 2 and 3 × 3 sites. We subsequently derive new constraints which extend these cluster constraints to incorporate the open-system nature of a cluster on a larger lattice. The feasibility of implementing these new constraints is demonstrated by performing a proof-of-principle calculation on the 6 × 6 lattice. It is shown that a large portion of the three-index result can be recovered using these extended cluster constraints, at a fraction of the computational cost.     

On iterative solutions of the LTE model atmosphere problem

We discuss iterative methods for solving the coupled radiative-transfer and energy-balance equations in the LTE model atmospheres problem including isotropic coherent scattering. We show that iterative solution (e.g. by SOR techniques) of the grand matrix encountered in such problems is vastly more efficient than a direct solution, and is easily vectorized. The final computational effort is linear in the number of depths and frequencies considered, and thus this approach opens the door for the computation of both static and dynamic line-blanketed models using large numbers of depth-points and huge numbers of frequencies.The iterative methods discussed here can be applied to line-formation problems with complete redistribution and to certain classes of problems with partial redistribution (e.g. Compton scattering problems in the Fokker-Planck approximation).     

Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation

We describe the use of upwind monotonic interpolation methods for the solution of the time-dependent radiative transfer equation in both optically thin and thick media. These methods, originally developed to solve Eulerian advection problems in hydrodynamics, have the ability to propagate sharp features in the flow with very little numerical diffusion. We consider the implementation of both explicit and implicit versions of the method. The explicit version is able to keep radiation fronts resolved to only a few zones wide when higher order interpolation methods are used. Although traditional implementations of the implicit version suffer from large numerical diffusion, we describe an implicit method which considerably reduces this diffusion.