Variational characterization

We follow here the derivation by Benguria and Depassier [34, 35]. The starting point for the variational principle is the ordinary differential equation for the RD equation in the frame comoving with the front (4.2). Without loss of generality, we assume that the front connects the states p = 0 and p = 1, i.e., lim oo p = 0 and limj ooP = 1- Since the front is monotonic, we define q p) = —p 0. Monotonic fronts are solutions of 

This lower bound for the front velocity is valid for any F(p) 0 on (0,1) and F(0) = F(l) = 0, i.e for a front propagating into unstable states (reaction terms of Case A) [34]. To show that (4.54) represents indeed a variational principle, we must establish that there exists a function, namely g, for which the equality holds in (4.54). Equality holds if g satisfies hq = Fg/ Dq), i.e., 

Therefore, if n 2-Jdf (0), we have /J gdp oo and /J Fghdp oo. In summary, we have proven that there exists a positive, continuous, and monotonically decreasing function g, for which the integrals in (4.54) exist and which maximizes the lower bound in (4.54) in such a way that the equality holds. In summary. 

We have used /J ftpdp = /J gdp, which follows from integration by parts. Finally, 

Note that the variational characterization given in (4.58) only holds if F 0 on (0,1) and for fronts propagating into unstable states. To derive a variational result valid if F 0 for some values of p, we need to extend these results [35]. To do so, we multiply (4.49) by g. Integrating between p = 0 and p = 1, we obtain after integration by parts 

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We illustrate the power of the variational characterization (4.68) by solving some examples for cases A and B. To do so, we will consider the trial function... 

Pulled-Pushed transition. The variational characterization can account for the pulled-pushed transition for fronts propagating into unstable states. We consider the Ben-Jacob case [37], where F p) = p( - p)( + ap) with a > 0. Equation (4.70) yields... 

Therefore, the stated above results have confirmed again that D, values distribution is the main reason of microgels stracture variation, characterized by its fractal dimension Dp Dp change at reaction duration growth is well described quantitatively within the frameworks of aggregation mechanism cluster-cluster. Fractal space, in which curing reaction proceeds, is formed by the stracture of the largest cluster in system [55],... 

The variational characterization (A) of equilibrium states is appropriate to a fiber held at fixed tension T° in a "soft device", i.e., bearing a dead load of magnitude T°. The characterization (B) is appropriate to a fiber in a "hard device" that holds the fiber at length -t in that variational problem the numoer T° arises as the Lagrange multiplyer associated with the constraint on -t. 

A the time of eddy-current testing, we exploit information given at distance by the variation of the impedance [L,R] of a solenoid near a conducting piece to defect and characterize the defects. 

The variation impedance analysis must permit to characterize, to localize and getting dimensions of defects. 

We present two optical methods for characterizing wire surfaces. These methods allow us to measure the roughness and the correlation length of the surface. It is also possible to identify qualitatively, at a glance, the variations of the roughness along a wire or among its different zones. 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The mass-spectrometric fragmentation of 2-aminothiazole-3-oxides is characterized by the abstraction of O and OH out of the molecule ion. Variations observed in the mass spectra suggest an equilibrium between tautomers 354a and 354b in the gas phase (Scheme 203). 

Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

In writing this result, we have assumed that the same value of characterizes all classes and have indicated the variation in r by attaching the subscript i to values of j3. The summation is carried out over the entire spectrum of relaxation... 

Since the development of a method for polymer characterization has been spread over several sections and since the literature contains several variations in the manner data is displayed, a summary of some pertinent definitions and relationships will be helpful at this point ... 

Photomultipliers are used to measure the intensity of the scattered light. The output is compared to that of a second photocell located in the light trap which measures the intensity of the incident beam. In this way the ratio [J q is measured directly with built-in compensation for any variations in the source. When filters are used for measuring depolarization, their effect on the sensitivity of the photomultiplier and its output must also be considered. Instrument calibration can be accomplished using well-characterized polymer solutions, dispersions of colloidal silica, or opalescent glass as standards. 

Variational inequality characterizing an interaction between the punch and the plate can be written in the form... 

Then it follows that the solution of the variational inequality (2.165) is characterized by the equilibrium equation... 

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