New tools in non-linear modelling and prediction

In this paper we give an account of a new change of perspective in non-linear modelling and prediction as applied to smooth systems. The core element of these developments is the Gamma test a non-linear modelling and analysis tool which allows us to examine the nature of a hypothetical input/output relationship in a numerical data-set. In essence, the Gamma test allows us to efficiently calculate that part of the variance of the output which cannot be accounted for by the existence of any smooth model based on the inputs, even though this model be unknown. A key aspect of this tool is its speed: the Gamma test has time complexity O( ), where M is the number of data-points. For data-sets consisting of a few thousand points and a reasonable number of attributes, a single run of the Gamma test typically takes a few seconds. Around this essentially simple procedure a new set of analytical tools has evolved which allow us to model smooth non-linear systems directly from the data with a precision and confidence that hitherto was inaccessible. In this paper we briefly describe the Gamma test, its benefits in model identification and model building, and then in more detail explain and motivate the procedures which facilitate a Gamma analysis. We briefly report on a case study applying these ideas to the practical problem of predicting level and flow rates in the Thames valley river basin. Finally we speculate on the future development and enhancement of these techniques into areas such as datamining and the production of complex non-linear models directly from data via graphical representations of process charts and automated Gamma analysis of each input-output node.     

Liquid-liquid phase separation of a surfactant-solubilized membrane protein

Solubilization of membrane proteins requires surfactants, whose structural properties play a crucial role in determining the protein phase behavior. We show that ionization of a pH-sensitive surfactant, lauryldymethylamino-N-oxide, bound to the bacterial photosynthetic Reaction Center, induces protein phase segregation in micrometric "droplets." Liquid-liquid phase separation takes place in a narrow pH range, is promoted by increasing temperature, and vanishes by adding salt. After a fast initial droplet growth, the nearly arrested kinetics at a later stage leaves the system in a finely divided, long-lasting emulsified state.     

Palladium-containing perovskites: recoverable and reuseable catalysts for Suzuki couplings

Palladium-containing perovskites (LaFe0.57Co0.38Pd0.05O3) have been exploited as recoverable and reuseable catalysts in Suzuki coupling reactions; residual levels of Pd after removal of the catalyst by filtration are low (2 ppm) despite evidence that the reaction is occurring via a homogeneous process.     

Determination of glyphosate and its metabolites in plant material by reversed‐polarity CE with indirect absorptiometric detection

A simple CE method for simultaneous determination of glyphosate and its metabolites (i.e. aminomethylphosphonic acid, glyoxylate, sarcosine and formaldehyde) in plants is reported here. A BGE of pH 7.5, 10% ACN, 7.5 mM phthalate, containing 0.75 mM hexadecyltrimethylammonium bromide as an electro‐osmotic flow modifier, an applied voltage of –20 kV and absorptiometric monitoring at 220 nm were the optimal chemical and instrumental parameters. The method, with development time 20 min, shows linear calibrations within the range 5–500 μg/mL (for all target analytes) with correlation coefficients between 0.999 and 0.998. It has been validated by application to samples of Lolium spp. The electroinjection mode hinders most interferents to enter the capillary, thus providing a clean electropherogram and making unnecessary long sample‐preparation steps.     

On a Two-Variable Class of Bernstein–Szegő Measures

The one-variable Bernstein–Szegő theory for orthogonal polynomials on the real line is extended to a class of two-variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.      

Near holomorphy of some automorphic forms at critical points

Oblatum 1-IV-1989 &; 27-IX-1989     

Performance of lateral vorticity probe in a turbulent wake

A useful experimental check of the accuracy of measured lateral vorticity components is provided by comparing the vorticity-velocity cross products with the lateral derivative of the Reynolds shear stress. The check has been carried out in the self-preserving turbulent wake of a cylinder.     

Vorticity characteristics of the turbulent intermediate wake

Measurements of the lateral components _j (j=2 and 3) of the vorticity fluctuation vector have been made, using a vorticity probe consisting of two X-wires, in the intermediate wake of a circular cylinder. The effect of the spatial resolution of the probe on the measurement of _j has been studied. As the spatial resolution impairs, the variance and flatness factor of _j decrease whereas the skewness of _j increases. Reasonably accurate values of _j ² can be obtained by applying spectral corrections for the spatial resolution effect.Near the beginning of the intermediate wake, the variance of ₂ is larger than that of ₃ due to the significant contribution from ribs which connect consecutive spanwise roll vortices. This difference decreases with downstream distance. Also, the presence of the rolls is reflected by a local extremum in the skewness of ₃ on each side of the wake centerline. The magnitude of the extremum decreases with downstream distance.The support of the Australian Research Council is gratefully acknowledged.     

Effect of different initial conditions on a turbulent round free jet

Velocity measurements were made in two jet flows, the first exiting from a smooth contraction nozzle and the second from a long pipe with a fully developed pipe flow profile. The Reynolds number, based on nozzle diameter and exit bulk velocity, was the same (䏪,000) in each flow. The smooth contraction jet flow developed much more rapidly and approached self-preservation more rapidly than the pipe jet. These differences were associated with differences in the turbulence structure in both the near and far fields between the two jets. Throughout the shear layer for x<3d, the peak in the v spectrum occurred at a lower frequency in the pipe jet than in the contraction jet. For x́d, the peaks in the two jets appeared to be nearly at the same frequency. In the pipe jet, the near-field distributions of f(r) and g(r), the longitudinal and transverse velocity correlation functions, differed significantly from the contraction jet. The integral length scale L_u was greater in the pipe jet, whereas L_v was smaller. In the far field, the distributions of f(r) and g(r) were nearly similar in the two flows. The larger initial shear layer thickness of the pipe jet produced a dimensionally lower frequency instability, resulting in longer wavelength structures, which developed and paired at larger downstream distances. The regular vortex formation and pairing were disrupted in the shear layer of the pipe jet. The streamwise vortices, which enhance entrainment and turbulent mixing, were absent in the shear layer of the pipe jet. The formation of large-scale structures should occur much farther downstream in the pipe jet than in the contraction jet.     

The effect of Reynolds number on a turbulent far-wake

The turbulent far-waked generated by a circular cylinder is investigated for two values (1350 and 4600) of the Reynolds number Re _θ (based on the free stream velocity and the momentum thickness). Two arrays of sixteen X-wires, eight in the (x,?y)-plane and eight in the (x,?z)-plane, are used to capture the main features of the large-scale motion in two orthogonal planes. Both the magnitude of the measured Reynolds stresses and the size of the two-point velocity and vorticity correlation contours increase with Reynolds number. The probability density function and spectra of the velocity signals also exhibit differences with Re _θ. A comparison of centerline turbulence intensities with those in the literature suggests that the Reynolds number dependence may disappear for Re _θ?5000.