Taylor’s method

The fact that none of these reports has emphasized the physical aspects of electrophilic substitution in the series reflects the paucity of quantitative studies, and the low reactivity of these compounds in the presence of electrophiles. Few kinetic studies have been reported and the regio-chemical effects of substituents have seldom been quoted in quantitative form. The present chapter brings together those quantitative results that are available, and collates data on substituent effects. One worthwhile field of study would appear to be the application to the azines of Taylor s method involving thermolysis of esters [75JCS(P2)277, 75JCS(P2) 1783]. 

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. 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

Eng 20, 470-477 (1919) (Description of ammonia oxidation process beginning with Kuhl-mann s method of 1839 and ending with the cyanamide process at Muscle Shoals) 7) C.L. Parsons, 1EC 11,541 (1919) (Oxidation of ammonia to nitric acid as well as the prepn of nitric acid from Chile saltpeter) 8) F.C. Zeis-berg, ChemMetEng 24, 443-45 (1921) (Manuf of nitric acid from Chilean saltpeter brief description) 9) G.B. Taylor, IEC 26,1217-19 (1922) (Some economic aspects of ammonia oxidation) 10) Ministry of Munitions, Manufacture of Nitric Acid from Nitre and Sulfuric Acid , London (1922) (Book No 7 of Technical Records of Explosives Supply, 1915—1919)... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

In Newton s method for a set of nonlinear equations, each equation is expanded in a truncated Taylor series. The result is a set of linear equations in corrections to previous estimates. Repetition of the process ultimately may converge to correct roots provided initial estimates are sufficiently close. 

As discussed in Section (8.2), Equations (8.64) and (8.65) is a set of (n + m) nonlinear equations in the n unknowns x and tn unknown multipliers A.. Assume we have some initial guess at a solution (x,A). To solve Equations (8.64)-(8.65) by Newton s method, we replace each equation by its first-order Taylor series approximation about (x,A). The linearization of (8.64) with respect to x and A (the arguments are suppressed)... 

Given an initial guess x0 for x, Newton s method is used to solve Equation (8.84) for x by replacing the left-hand sidex>f (8.84) by its first-order Taylor series approximation at x0 ... 

A set of nonlinear equations can be solved by combining a Taylor series linearization with the linear equation-solving approach discussed above. For solving a single nonlinear equation, h(x) = 0, Newton s method applied to a function of a single variable is the well-known iterative procedure... 

The determination of many thermodynamic solution properties of these trihalides, especially enthalpies of solution, requires the use of anhydrous materials. These are by no means always readily available for this group of compounds. Therefore we precede our discussion of thermochemical results with a section on preparative methods, in which we deal summarily with many of the relevant papers published since Taylor s extensively referenced review, published in 1962 (2). 

The preparations of rare-earth trihalides can be found in various books (2-8) and in Taylor s review (2 ). This review, however, did not include the preparation of scandium and yttrium trihalides, and only covered the preparation of the trifluorides very briefly. We have reviewed the preparation of all the trihalides (including scandium and yttrium) from Taylor s review up to June 1979 and have also included some methods and references missed by Taylor. Although we have mentioned all the methods available for the preparation of the trihalides, emphasis has been placed on the methods used since Taylor s review, and these have been referenced fully, whereas for the other methods, Taylor s review is recommended as a source of references. 

An alternative approach to our problem is Newton s method. The idea behind this method is illustrated in Fig. 3.5. If we define g(x) = f (x). then from a Taylor expansion g(x I It) = g(x) + hg (x). This expression neglects terms with higher orders of h, so it is accurate for small values of h. If we have evaluated our function at some position where g(x) A 0, our approximate equation suggests that a good estimate for a place where the function is zero is to define x = x I h = x g(x)/g x). Because the expression from the Taylor expansion is only approximate, x does not exactly define a place where g(.ri) = 0. Just as we did with the bisection method, we now have to repeat our calculation. For Newton s method, we repeat the calculation starting with the estimate from the previous step. 

There is no natural way to generalize the one-dimensional bisection method to solve this multidimensional problem. But it is possible to generalize Newton s method to this situation. The one-dimensional Newton method was derived using a Taylor expansion, and the multidimensional problem can be approached in the same way. The result involves a 3/V x 3/V matrix of derivatives, J, with elements 7y = dg, /dxj. Note that the elements of this matrix are the second partial derivatives of the function we are really interested in, E(x). Newton s method defines a series of iterates by... 

Using the methods of Taylor s analysis as the basis of machine computation, Lutzky has computed "Spherical Taylor Wave for several high explosives (TNT, Pentolite, 65/35 60/40-Cyclotols, TNEtB, RDX, Tetryl and NGu) with the equation of state ... 

The calculation of AH° and AS° values from the pK-temperature data in each solvent mixture was performed by the nonempirical method of Clarke and Glew (26) as simplified by Bolton (27). In this method the thermodynamic parameters are considered to be continuous, well-behaved functions of temperature, and their values are expressed as perturbations of their values at some reference temperature 0 by a Taylor s series expansion. The basic equation is ... 

Bums, J., Chan, V. T.-W., Joasson, J. A., Fleming, K. A., Taylor, S., and McGee, J. O D. (1985) A sensitive method for visualizing biotinylated probes hybridized m situ rapid sex determination on intact cells. J Clin Pathol. 38, 1085-1092... 

The linearisation method employed in Section 7.5.2 considers the dependent variable (volumetric flowrate of liquid leaving a tank Q ) being perturbed about its steady-state value Q in response to changes in the level of liquid z in the tank about its steady-state level z. Thus, Qx can be expressed as a function of z in terms of a Taylor s series, viz. ... 

A new approach to the application of group theory in the study of the physical properties of crystals, which is more powerful than the direct method described in Section 15.2, has been developed by Nowick and is described fully in his book Crystal Properties via Group Theory (Nowick (1995)). A brief outline of Nowick s method will be given here. The equilibrium physical properties of crystals are described by constitutive relations which are Taylor expansions of some thermodynamic quantity Yt in terms of a set of thermodynamic variables Xj. Usually, only the first term is retained giving the linear relations... 

FIGURE 1.12 Top an electron microscopy image of an epithelial tight junction. The position of the tight junction at the apical end of the lateral membrane is marked with an arrowhead, whereas the position of a desmosome is indicated with an asterisk. Bar = 200 nm. (From Matter, K. and Baida, M.S., Methods, 30, 228, 2003. With permission.) Bottom a schematic location of the tight junctions, intermediate junctions, and actin cytoskeleton in the epithelial cell. (From Ballard, S.T., Hunter, J.H., and Taylor, A.E., Ann. Rev. Nutr., 15, 35, 1995. With permission.)... 

A.U. Hazi, H.S. Taylor, Stabilization method of calculating resonance energies Model problem, Phys. Rev. A 1 (1970) 1109. 

Taylor S, Walsh ME. Optimization of an analytical method for determining white phosphorus in contaminated sediments. Cold Regions Research and Engineering Laboratory, Hanover, NH. NTIS/AD-A261 333/9. 

Walsh ME, Taylor S. 1993. Analytical method for white phosphorus residues in munitions-contaminated sediments. Analytica Chimica Acta 282(N1) 55-61. 

In the Herzberg-Teller method [22] for accounting for induced vibronic intensities, the transition moment M°e between the ground e" and excited e electronic states is expanded as a Taylor s series in the normal coordinate Q for anitsymmetric mode k... 

When harvested, these roots are scraped and then cut into pieces, which are then either chewed (the Tonga method) or crushed between rocks (the Fiji method). In Norman Taylor s Narcotics, there appears a charming account of these preparations—which result in a grayish brown or whitish liquid that most find to be soapy, spicy and numbing to the mouth. In Hawaii, according to Hoffer and Osmond,... 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

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