Normal form techniques

Using center-manifold theorem and normal form techniques [65,66], we have explicitly reduced our reaction-diffusion system (3) to the Hopf normal form (64) of the single-front solution. Technically we have used the normal form reduction method proposed by Coullet and Spiegel [110] (see also [111]). We refer the reader to [62] and [104], where this lengthy calculation has been carried out step by step. For the sake of simplicity, we will skip this technical part here, and we will focus on the theoretical prediction so obtained (on the critical surface n = 0) for the coefficient k in Equation (64). Our purpose is actually to compare the prediction for the value of Re K, with the measurement of the same quantity from direct simulations of the reaction-diffusion system (3). The numerical estimate is easily obtained from the amplitude of oscillation of the single-front solution in the ( , ") direction (Figure 20). If we write 2 = p e, the real part of Equation (64) yields ... 

The physicochemical basis behind the technique of radiocarbon dating is the isotopic abundances of carbon s three isotopes 12C is the normal form and constitutes 98.9 per cent of all naturally occurring carbon. 13 C is the other naturally occurring isotope, with an abundance of about 1 per cent. 14C does not occur naturally, but tiny amounts of it are formed when high-energy particles from space collide with gases in the upper atmosphere, thus causing radiochemical modification. 

Fluidized-bed combustion systems use a heated bed of sandlike material suspended (fluidized) within a rising column of turbulent air to burn many types and classes of waste fuels. The vendor claims that this technique results in improved combustion efficiency of high moisture content fuels and is adaptable to a variety of waste -type fuels. The scrubbing action of the bed material on the fuel particle is said to enhance the combustion process by stripping away the carbon dioxide and char layers that normally form around the fuel particle. This allows oxygen to reach the combustible material much more readily and increases the rate and efficiency of the combustion process. 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

Because the rates of this reaction are very rapid, normal sampling techniques were not satisfactory and an infrared technique was used. This esterification reaction was shown to be about 100 times faster than the disproportionation reaction and inter-intra-molecular assistance was also found to be important. This assistance seems to be a common pattern in acid-catalysed processes of oligosiloxanols in inert solvents. In dioxane solvent the redistribution kinetics can be interpreted in terms of an unzipping mechanism. The ratedetermining step is terminal silanol cleavage by water forming dimethylsilanediol which rapidly reacts with other substrate silanols (Scheme 4). 

CVD is a synthesis process in which the chemical constituents react in the vapor phase near or on a heated substrate to form a solid deposit (Pierson 1999). The reactions happened in the CVD system can be divided into homogeneous gas phase reactions and heterogeneous substrate surface reactions. Normally, CVD technique is utilized to make thin films. CVD is also defined as a process whereby a thin solid film is synthesized from the gaseous phase by a chemical reaction (Hitchman and Jensen 1993). The CVD apparatus arrangement is dependant on the particular application. The apparatus is made up with three major components precursors and... 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

Next we put into effect the visualization techniques we discussed in Section V.B. Our purpose is to show how the complex dynamics of the pair of trajectories in Figs. 11 and 12 is simplified by the normal-form transformation. 

Usually, the liver contains 0.8-1.5% of its wet weight in the form of extractable, finely dispersed structural fats, which cannot be detected by normal histological techniques. Under the light microscope, the liver fat, which is mainly made up of small droplets of triglycerides, only becomes visible when an increase to >2-3% occurs. Above this value, hepatocytes register this event as a pathological process per se. 

Crystallization of D-fructose is usually effected at ice-box temperature from an ethanol solution, and further purification and recrystallization is performed at room temperature. The spherulitic aggregates of fine needles obtained by this technique are those of the hemihydrate, as shown by Young and coworkers. This possibility had been put forward by Honig and. lesser, but sufficient proof was lacking at that time. The anomalous x-ray diffraction data reported by Wolfrom and Thompson for their preparation of i fructose, in comparison with those of the normal form of D-fructose, may be explained by hemihydrate formation. Indeed, purification with ethanol at 25° results in dehydration, with formation of the... 

Both the CNF and the QNF have the form of power series where each successive term is constructed via an iterative technique. Consequently, in general, we will only be able to compute a finite number of terms of the normal form. Therefore the obvious question that arises is how many terms of the normal form are required in order that the quantities derived from the normal form are accurate A discussion, as well as some references with specific examples, was given in Section 2.5 for the CNF. In this section we are concerned with the behavior of the QNF, for which there has been essentially no work from this point of view. 

The normal experimental technique is to scan rapidly through the velocity range and repeat this scan many times imtil data of the required accuracy has been accumulated. The Doppler motion is provided by an electromechanical drive system controlled by a servo -amplifier. Usually, the source is attached to the drive shaft and driven either in a saw-tooth or a triangular constant acceleration wave form. The transducer is coupled to a multichannel analyser operating in the multiscaler mode, and the servo-amplifier is controlled by the channel advance frequency. The dwell time in each channel, corresponding to a specific velocity increment, is 100 ps, and while the channel gate is open it accepts pulses from the detector. 

With the mixed-variable generating functions, after Birkoff [54], Gustav-son [55] developed an elegant technique to extract the new Hamiltonian to avoid divergence by assuming that the new Hamiltonian is expandable in normal form if complete inversion of the variables is not required, the procedure to calculate the new Hamiltonian can be rather straightforward. 

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