Time-contour variables

Green s function belongs to a larger class of functions of two time-contour variables that includes a divergent term as well ... 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

Figure 11.7 compares the predictions of the MLD theory to steady-state shear data for a solution of nearly monodisperse polystyrene. In this comparison, the reptation time Tj and the plateau modulus have been taken as adjustable parameters. The theory used in this comparison is not the simplified toy model given by Eqs. 11.14 to 11.17, but a more complete theory with a contour variable, described in Mead et al [27]. The more complete version of the theory is able to include the effects of primitive path fluctuations as well as a more complete description of reptation and convective constraint release. Nevertheless, if the reptation time constant Tj is suitably adjusted to account phenomenologically for primitive path fluctuations (as discussed in Section 6.4.3 and 6.4-4.2), the predictions of Eqs. 11.14 to 11.17 are very similar to those of the full theory. 

An example of the results obtained in the form of a chromatoelectropherogram can be seen in Figure 9.6. The contour type data display showed the three variables that were studied, namely chromatographic elution time, electrophoretic migration time, and relative absorbance intensity. Peptides were cleanly resolved by using this two-dimensional method. Neither method alone could have separated the analytes under the same conditions. The most notable feature of this early system was that (presumably) all of the sample components from the first dimension were analyzed by the second dimension, which made this a truly comprehensive multidimensional technique. 

As we have seen in earlier sections, wave functions can be used to perform useful calculations to determine values for dynamical variables. Table 2.2 shows the normalized wave functions in which the nuclear charge is shown as Z (Z = 1 for hydrogen) for one electron species (H, He+, etc.). One of the results that can be obtained by making use of wave functions is that it is possible to determine the shapes of the surfaces that encompass the region where the electron can be found some fraction (perhaps 95%) of the time. Such drawings result in the orbital contours that are shown in Figures 2.3, 2.4, and 2.5. 

The Nyquist stability criterion that we developed in Chap. 13 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem which says that the dilTerence between the number of zeros and poles that a function has inside a dosed contour can be found by plotting the function and looking at the number of times it endrdes the origin. 

Figure 1. Comparison of univariate (one-variable-at-a-time) and simplex optimization of a simple response surface with two interdependent parameters (density and temperature). Ellipses represent contours of the response surface. See text for additional discussion.

The length of a linear polymer molecule may be described by its contour length, which equals nl, where n is the number of units of length t. The end-to-end distance, r, of a convoluted polymer molecule is much shorter. It is extremely variable, both with respect to time and to other molecules of the same contour length. To calculate the most probable root-mean-square end-to-end length,... 

The ordering operator Tc places the operators in the Taylor expansion of the Tc exponent to the left with a later-in-time variable on the contour y. The operators are taken in the interaction representation on the Keldysh contour. 

A complete picture of how the independent variables affect Tgl can be obtained from examining a contour plot of the predicted response surface of T. i at constant DEGM level DEGM was held constant because it had little effect on Tgl (Figure 9). It is evident from the contours that Tgi is increased not only by increased epoxy prereaction time but also by increased initiator concentration. The lowest values for Tgl are found with low epoxy prereaction times and low initiator concentrations, independent of DEGDM concentration. 

This conclusion may be totally wrong if there is an interaction between pH and T. The situation may well be as depicted in Fig. 2.3 where the variations in both variables are given and the function describing the yield variation will be described by a response surface over the plane spanned by the two variables. The topography of the surface is given by the isoresponse contours which show the levels of the yields for different settings of the experimental variables. It is seen that the optimum yield will be 97 %, and that the conditions for this will be at pH = 3.6 and T = 55 °C, which does not at all correspond to the results obtained by the one-variable-at-a-time study. 

FIGURE 2.2. One-variable-at-a-time optimization procedure for two factors, Xi and X2, in the presence of an interaction effect between the factors. Dotted lines = hypothetical contour plot of response to optimize. A = starting point B = best result after varying xi a first time C = best result after varying X2 a first time (= usually reported optimum) and D = best result after varying xi a second time (= real optimum). 

The power of FT NMR is that one is not confined to a single exciting pulse. One can have several pulses with various durations, delays and phases in order to edit a one-dimensional spectrum. Or one can have an array of pulses with a variable evolution time and then perform the Fourier transform with respect to both the evolution time and the decay of the FID, generating a two-dimensional spectrum whose output is a contour plot. With very powerful machines (> 600 MHz, H) it is even possible to perform the Fourier transform in three dimensions, with two evolution times. These pulse sequences are known by (usually arch) acronyms such as COSY, INADEQUATE, etc., and modern NMR machines are supplied with the hardware and software to perform the commoner experiments already installed. It is not necessary to understand fully the spin physics behind such sequences in order to use them, but the basic viewpoint used in their description is worth grasping. 

Now we will obtain asymptotic formulae for the field in the far zone (a 1). In deriving a formula we will deform the contour of integration in eq. 10.33 on the complex plane of variable m. However, such a procedure requires either the proof of absence of poles of the integrand or evaluation of their contribution to the integral value. The problem of determination of poles is extremely difficult because of the complexity of the integrand. At the same time sufficient agreement of results of calculations by asymptotic and exact formulae allows us to think that if there are poles in the upper half-plane of m, their contribution in a considered part of the spectrum is sufficiently small. Let us present integral in eq. 10.33 in the form ... 

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