Taylor generation

P.K. Benicewicz, J.P. Roberts, A.J. Taylor, Generation of 39 fs pulses and 815 nm with a synchronously pumped mode-locked dye laser. Opt. Lett. 16, 925 (1991)... 

The family of hierarchical elements are specifically designed to minimize the computational cost of repeated computations in the p-version of the finite element method (Zienkiewicz and Taylor, 1994). Successive approximations based on hierarchical elements utilize the derivations of a lower step to generate the solution for a higher-order approximation. This can significantly reduce the... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

R. A. Crewdson, W. F. Martin, Jr., D. L. Taylor, and K. Bakhtar,M Evaluation of the Technology and Economics of Extracting Energyfrom Magma Resources for Electric Power Generation, report by Mine Development and Engineering Co., Bakersfield, Calif., 1991. 

When fluid flows around a curwe in a duct, or when fluid is confined between differentially rotating cylinders, secondary flows called Taylor vortices are generated. If a membrane is mounted on a rotating... 

Finally, there is an interesting article" that shows how to use Taylor s series to generate shortcut methods from established theory. Examples are given for developing a criterion for replacing log mean temperature differences with average differences and for estimating the effect of temperature on reaction rate. 

The above formulas combined with Eqs. (74) and (75) taken at zero charge density yield Eq. (54) for the differential capacitance. Eq. (82) can be used recursively to generate the derivatives of the differential capacity at zero charge density to an arbitrary order, though the calculations become rather tedious already for the second derivative. Thus, in principle at least, we can develop capacitance in the Taylor series around the zero charge density. The calculations show that the capacitance exhibits an extremum at the point of zero charge only in the case of symmetrical ions, as expected. In contrast with the NLGC theory, this extremum can be a maximum for some values of the parameters. In the case of symmetrical ions the capacitance is maximum if + — a + a, < 1. We can understand this result... 

More elaborate approaches make use of multiple substitutions to generate additional information. Consider a reaction in which substitutions can be made at two sites, X and y, so that the observed response is a function of both substituents, f x,y). Expanding this in a Taylor s series gives... 

Taylor has reported a number examples of intramolecular variations directed towards heterocyclic systems.The following two reactions are representative. Intramolecular addition of triazine 79, after loss of nitrogen afforded 80. Alternatively, triazine 81 generated bicyclic systems 82 which could be oxidized to 83. 

Since the machine performs only arithmetic operations (and these only approximately), iff is anything but a rational function it must be approximated by a rational function, e.g., by a finite number of terms in a Taylor expansion. If this rational approximation is denoted by fat this gives rise to an error fix ) — fa(x ), generally called the truncation error. Finally, since even the arithmetic operations are carried out only approximately in the machine, not even fjx ) can usually be found exactly, and still a third type of error results, fa(x ) — / ( ) called generated error, where / ( ) is the number actually produced by the machine. Thus, the total error is the sum of these... 

A similar convergence is found for the third harmonic generation process at the lower of the two frequencies, 671.5 nm. At the higher frequency, 476.5 nm, the Taylor approximations for the third harmonic generation hyperpolarizability converge only very slowly, even with a tenth-order Taylor approximation a one-percent accuracy is not obtained. This accuracy, however, is still achieved with a [1,2] Fade approximant calculated from the dispersion coefficients up to sixth order. 

Three paths can be advanced (1) expansion, e.g., Taylor series (2) trial and error, e.g., generating curves on the plotter and (3) simplex optimization algorithm. (See Section 3.1.)... 

Hui, P.H., Taylor, I.K., Taylor, G.W., Rubin, P., Kesterton, J., Barnes, N.C. and Barnes, P.J. (1991). Effect of a 5-lipoxygenase inhibitor on leukotriene generation and airway responses after allergen challenge in asthmatic patients. Thorax 46, 184-189. 

To evaluate L ip) from equation (F.2), we first factor out in the generating function and expand the remaining exponential function in a Taylor series... 

Reactors which generate vortex flows (VFs) are common in both planktonic cellular and biofilm reactor applications due to the mixing provided by the VF. The generation of Taylor vortices in Couette cells has been studied by MRM to characterize the dynamics of hydrodynamic instabilities [56], The presence of the coherent flow structures renders the mass transfer coefficient approaches of limited utility, as in the biofilm capillary reactor, due to the inability to incorporate microscale details of the advection field into the mass transfer coefficient model. 

Transmetalation with Zn on a chiral lithiated alkyl carbamate had been reported by Taylor via addition of a solution of ZnCl2 to an in situ generated chiral organolithium (Scheme 8.10) [15]. 

Y. S. Tang. Ph.D has more than 35 years of experience in the field of thermal and fluid flow. His research interests have covered aspects of thermal hydraulics that are related to conventional and nonconventional power generation systems, with an emphasis on nuclear reactor design and analysis that focuses on liquld-meta -cooled reactors. Dr. Tang is co-author of Radioactive Waste Management published by Taylor 8 Francis, and Thermal Analysis of Liquid Metal Fast Breeder Reactors, He received a B5. from National Central University In China and an MS. in mechanical engineering from the University of Wisconsin. He earned his Ph.D. 

J. F. Knifton, R. J. Taylor, and P.E. Dai, Diisopropyl ether one-step generation from acetone-rich feedstocks, Chemical Industries (Dekker) 85 Catal. Org. React.) 131-141 (2002). 

---