One-dimensional improvement and two-dimensional and 3-dimensional treatments for stable oscillation condition, mode structure and power output in high-speed-flow lasers

For high-speed-flow lasers, the one-dimensional and first-order approximate treatment in[1] under approximation of geometrical optics is improved still within the scope of approx-imation of geometrical optics. The strict accurate results are obtained, and what is more,two- and three-dimensional treatments are done. Thus for two- and three-dimensional cases, thestable oscillation condition, the formulae of power output and analytical expression of modesunder approximation of geometrical optics (in terms of gain function) are derived. Accord-ing to the present theory, one-and two-dimensional calculations for the typical case of Gerry'sexperiment are presented. All the results coincide well with the experiment and are better thanthe results obtained in [1].In addition, the applicable scope of Lee's stable oscillation condition given by [1] is ex-panded; the condition for the approximation of gcometrical optics to be applied to mode con-structure in optical cavity is obtained for the first time and the difference between thiscondition and that for free space is also pointed out in the present work.

Mode coupling between first- and second-order whispering-gallery modes in coupled microdisks

The mode characteristics for two coupled microdisks are investigated by the finite-difference time-domain technique. In the two coupled micodisks, mode coupling between the same order whispering-gallery modes (WGMs) results in coupled WGMs with split mode wavelengths. The numerical results show that the split mode wavelengths of the coupled first- and second-order WGMs can have a crossing point in some cases, which can induce anticrossing mode coupling between them and greatly reduce the mode Q factor of the coupled first-order WGMs. The time variation of mode field pattern shows the transformation between the coupled first- and second-order WGMs. (C) 2007 Optical Society of America

First and second order Raman scattering spectroscopy of nonpolar a-plane GaN

Nonpolar a-plane [(1120)] GaN samples have been grown on r-plane [(1102)] sapphire substrates by low-pressure metal-organic chemical-vapor deposition. The room-temperature first and second order Raman scattering spectra of nonpolar a-plane GaN have been measured in surface and edge backscattering geometries. All of the phonon modes that the selection rules allow have been observed in the first order Raman spectra. The frequencies and linewidths of the active modes have been analyzed. The second order phonon modes are composed of acoustic overtones, acoustic-optical and optical-optical combination bands, and optical overtones. The corresponding assignments of second order phonon modes have been made. (c) 2007 American Institute of Physics.

First- and second-order phase transitions in the adlayer of biadipate on Au(111)

The adsorption of biadipate on Au(111) was studied by cyclic voltammetry and chronocoulometry. The biadipate adlayer undergoes a potential-driven phase transition. It is shown that the phase transition can be either of the first- or second-order depending on the biadipate concentration. At low surfactant concentrations, the first-order transition is characterised by a discontinuity in the charge density-potential curve and by the presence of very sharp peaks in the voltammetric response. At higher concentrations, these peaks are no longer observed but a discontinuity in the capacity curve is still noticeable, in agreement with a second-order transition. © the Owner Societies.

Enhancement Factor for Gas Absorption in a Finite Liquid Layer. Part 2: First- and Second-Order Reactions in a Liquid in Plug Flow

Gas absorption accompanied by an irreversible chemical reaction of first-order or second-order in a liquid layer of finite thickness in plug flow has been investigated. The analytical solution to the enhancement factor has been derived for the case of a first-order reaction, and the exact solution to the enhancement factor has been obtained via numerical simulation for the case of a second-order reaction. The enhancement factor in both cases is presented as a function of the Fourier number and tends to deviate from the prediction of the existing enhancement factor expressions based on the penetration theory at Fourier numbers above 0.1 due to the absence of a well-mixed bulk region in the liquid layer. Approximate enhancement factor expressions that describe the analytical and exact solutions with an accuracy of 5?% and 9?%, respectively, have been proposed.

Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition

This paper considers various asymptotic approximations in the near-integrated firstorder autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous time approximation of Perron (1991). We assess how these alternative methods provide or not an adequate approximation to the finite-sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991) continuous time approximation performs very well while the others only offer improvements when the initial condition is zero.

Second-order negative-curvature methods for box-constrained and general constrained optimization

A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.

First- and second-order stimulus length selectivity in New World monkey striate cortex

Motion is a powerful cue for figure-ground segregation, allowing the recognition of shapes even if the luminance and texture characteristics of the stimulus and background are matched. In order to investigate the neural processes underlying early stages of the cue-invariant processing of form, we compared the responses of neurons in the striate cortex (V1) of anaesthetized marmosets to two types of moving stimuli: bars defined by differences in luminance, and bars defined solely by the coherent motion of random patterns that matched the texture and temporal modulation of the background. A population of form-cue-invariant (FCI) neurons was identified, which demonstrated similar tuning to the length of contours defined by first- and second-order cues. FCI neurons were relatively common in the supragranular layers (where they corresponded to 28% of the recorded units), but were absent from layer 4. Most had complex receptive fields, which were significantly larger than those of other V1 neurons. The majority of FCI neurons demonstrated end-inhibition in response to long first- and second-order bars, and were strongly direction selective, Thus, even at the level of V1 there are cells whose variations in response level appear to be determined by the shape and motion of the entire second-order object, rather than by its parts (i.e. the individual textural components). These results are compatible with the existence of an output channel from V1 to the ventral stream of extrastriate areas, which already encodes the basic building blocks of the image in an invariant manner.

Transmission comparison of ultra-long Raman fibre laser based amplification with first and dual order Raman amplification using 10×118 Gbit/s DP-QPSK

Experimental investigations of 10×118 Gbit/s DP-QPSK WDM transmission using three types of distributed Raman amplification techniques are presented. Novel ultra-long Raman fibre laser based amplification with second order counter-propagated pumping is compared with conventional first order and dual order counter-pumped Raman amplification. We demonstrate that URFL based amplification can extend the transmission reach up to a distance of 7520 km in comparison with 5010 km and 6180 km using first order and dual order Raman amplification respectively. © 2014 IEEE.

Linear binocular combination of responses to contrast modulation:contrast-weighted summation in first- and second-order vision

Binocular combination for first-order (luminancedefined) stimuli has been widely studied, but we know rather little about this binocular process for spatial modulations of contrast (second-order stimuli). We used phase-matching and amplitude-matching tasks to assess binocular combination of second-order phase and modulation depth simultaneously. With fixed modulation in one eye, we found that binocularly perceived phase was shifted, and perceived amplitude increased almost linearly as modulation depth in the other eye increased. At larger disparities, the phase shift was larger and the amplitude change was smaller. The degree of interocular correlation of the carriers had no influence. These results can be explained by an initial extraction of the contrast envelopes before binocular combination (consistent with the lack of dependence on carrier correlation) followed by a weighted linear summation of second-order modulations in which the weights (gains) for each eye are driven by the first-order carrier contrasts as previously found for first-order binocular combination. Perceived modulation depth fell markedly with increasing phase disparity unlike previous findings that perceived first-order contrast was almost independent of phase disparity. We present a simple revision to a widely used interocular gain-control theory that unifies first- and second-order binocular summation with a single principle-contrast-weighted summation-and we further elaborate the model for first-order combination. Conclusion: Second-order combination is controlled by first-order contrast.

Viability Kernels of Higher Order

AMS subject classification: Primary 34A60, Secondary 49K24.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Alkaline hydrothermal treatment of titanate nanostructures

Since its initial proposal in 1998, alkaline hydrothermal processing has rapidly become an established technology for the production of titanate nanostructures. This simple, highly reproducible process has gained a strong research following since its conception. However, complete understanding and elucidation of nanostructure phase and formation have not yet been achieved. Without fully understanding phase, formation, and other important competing effects of the synthesis parameters on the final structure, the maximum potential of these nanostructures cannot be obtained. Therefore this study examined the influence of synthesis parameters on the formation of titanate nanostructures produced by alkaline hydrothermal treatment. The parameters included alkaline concentration, hydrothermal temperature, the precursor material‘s crystallite size and also the phase of the titanium dioxide precursor (TiO2, or titania). The nanostructure‘s phase and morphology was analysed using X-ray diffraction (XRD), Raman spectroscopy and transmission electron microscopy. X-ray photoelectron spectroscopy (XPS), dynamic light scattering (non-invasive backscattering), nitrogen sorption, and Rietveld analysis were used to determine phase, for particle sizing, surface area determinations, and establishing phase concentrations, respectively. This project rigorously examined the effect of alkaline concentration and hydrothermal temperature on three commercially sourced and two self-prepared TiO2 powders. These precursors consisted of both pure- or mixed-phase anatase and rutile polymorphs, and were selected to cover a range of phase concentrations and crystallite sizes. Typically, these precursors were treated with 5–10 M sodium hydroxide (NaOH) solutions at temperatures between 100–220 °C. Both nanotube and nanoribbon morphologies could be produced depending on the combination of these hydrothermal conditions. Both titania and titanate phases are comprised of TiO6 units which are assembled in different combinations. The arrangement of these atoms affects the binding energy between the Ti–O bonds. Raman spectroscopy and XPS were therefore employed in a preliminary study of phase determination for these materials. The change in binding energy from a titania to a titanate binding energy was investigated in this study, and the transformation of titania precursor into nanotubes and titanate nanoribbons was directly observed by these methods. Evaluation of the Raman and XPS results indicated a strengthening in the binding energies of both the Ti (2p3/2) and O (1s) bands which correlated to an increase in strength and decrease in resolution of the characteristic nanotube doublet observed between 320 and 220 cm.1 in the Raman spectra of these products. The effect of phase and crystallite size on nanotube formation was examined over a series of temperatures (100.200 �‹C in 20 �‹C increments) at a set alkaline concentration (7.5 M NaOH). These parameters were investigated by employing both pure- and mixed- phase precursors of anatase and rutile. This study indicated that both the crystallite size and phase affect nanotube formation, with rutile requiring a greater driving force (essentially �\harsher. hydrothermal conditions) than anatase to form nanotubes, where larger crystallites forms of the precursor also appeared to impede nanotube formation slightly. These parameters were further examined in later studies. The influence of alkaline concentration and hydrothermal temperature were systematically examined for the transformation of Degussa P25 into nanotubes and nanoribbons, and exact conditions for nanostructure synthesis were determined. Correlation of these data sets resulted in the construction of a morphological phase diagram, which is an effective reference for nanostructure formation. This morphological phase diagram effectively provides a .recipe book�e for the formation of titanate nanostructures. Morphological phase diagrams were also constructed for larger, near phase-pure anatase and rutile precursors, to further investigate the influence of hydrothermal reaction parameters on the formation of titanate nanotubes and nanoribbons. The effects of alkaline concentration, hydrothermal temperature, crystallite phase and size are observed when the three morphological phase diagrams are compared. Through the analysis of these results it was determined that alkaline concentration and hydrothermal temperature affect nanotube and nanoribbon formation independently through a complex relationship, where nanotubes are primarily affected by temperature, whilst nanoribbons are strongly influenced by alkaline concentration. Crystallite size and phase also affected the nanostructure formation. Smaller precursor crystallites formed nanostructures at reduced hydrothermal temperature, and rutile displayed a slower rate of precursor consumption compared to anatase, with incomplete conversion observed for most hydrothermal conditions. The incomplete conversion of rutile into nanotubes was examined in detail in the final study. This study selectively examined the kinetics of precursor dissolution in order to understand why rutile incompletely converted. This was achieved by selecting a single hydrothermal condition (9 M NaOH, 160 °C) where nanotubes are known to form from both anatase and rutile, where the synthesis was quenched after 2, 4, 8, 16 and 32 hours. The influence of precursor phase on nanostructure formation was explicitly determined to be due to different dissolution kinetics; where anatase exhibited zero-order dissolution and rutile second-order. This difference in kinetic order cannot be simply explained by the variation in crystallite size, as the inherent surface areas of the two precursors were determined to have first-order relationships with time. Therefore, the crystallite size (and inherent surface area) does not affect the overall kinetic order of dissolution; rather, it determines the rate of reaction. Finally, nanostructure formation was found to be controlled by the availability of dissolved titanium (Ti4+) species in solution, which is mediated by the dissolution kinetics of the precursor.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.