Curves Parametric

In a given motion, a particular material particle will experience a strain history The stress rate relation (5.4) and flow rule (5.11), together with suitable initial conditions, may be integrated to obtain the eorresponding stress history for the particle. Conversely, using (5.16) instead of (5.4), may be obtained from by an analogous ealeulation. As before, may be represented by a continuous curve, parametrized by time, in six-dimensional symmetric stress spaee. 

Fig. 10.9. The neutral stability curve in the n n plane below the curve parametrized by tr(J) = 0 the uniform state is unstable to perturbations of appropriate spatial form...

Fig. 10.10. The neutral stability curve for a system with k < 0.0279, showing curves parametrized by tr(J) = 0 and det(J) = 0. Within the latter, non-uniform profiles may be stable .

The estimate for rj tor Z < Zg remains the same, but note the theoretical results predict that log(jj/ / should be plotted versus log (9)2 for (cp Z) > q>2Z)e in order to produce families of curves parametric in Z,... 

Chou et compared Karplus curves parametrized by using experimental V couplings with those derived by best-fitting the results of DFT calculations carried out on conformers of valine and threonine dipeptide analogues. The experimentally parametrized Karplus curves agreed well with the curves obtained from DFT calculations. 

Fig. 8. itp contributions versus phenomenology in (a) Dj.ib) and(c) Dj. The curves labeled 2it Paris and 27t Bonn represent the predictions by the Paris and Bonn model, respectively, when only the contributions from n, 2it, and cu are taken into account. Adding the phenomenological short-range potential yields the dotted Paris curve (parametrized Paris potential [14]). Adding the up contributions (Fig. 9) yieids the solid 2it + np Bonn curve (Bonn full model [15]). 

Surface Integrity, Fig. 11 Curve parametrization and characteristic values of residual stress-depth gradients (acc. to de Leon 2009)... 

Beyond the cubic polynomial, there are two main approaches to fitting the term structure parametric and non-parametric curves. Parametric curves are based on term-structure models such as those discussed in chapter 4. As such, they need not be discussed here. Non-parametric curves, which are constructed employing spline-based methods, are not derived from any interest rate models. Instead, they are general approaches, described using sets of parameters. They are fitted using econometric principles rather than stochastic calculus, and are suitable for most purposes. 

Keywords deterministic methods, STOllP, GllP, reserves, ultimate recovery, net oil sands, area-depth and area-thickness methods, gross rock volume, expectation curves, probability of excedence curves, uncertainty, probability of success, annual reporting requirements, Monte-Carlo simulation, parametric method... 

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x =f t), y = g(t), z = h t) in the parameter t. 

For banks of staggered tubes, the friction factor for isothermal flow is obtained from Fig. (6-42). Each fence (group of parametric curves) represents a particular Reynolds number defined as... 

For distillations, it is often of more interest to ascertain the effect of entrainment on efficiency than to predic t the quantitative amount of liquid entrained. For this purpose, the correlation shown in Fig. 14-26 is useful. The parametric curves in the figure represent approach to the entrainment flood point as measured or as predicted by Fig. 14-25 or some other flood correlation. The abscissa values are those of the flow parameter discussed earher. The ordinate values y are fractions of gross hquid downflow, defined as follows ... 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

Note that the expression can be rearranged to obtain AP in terms of other variables. The expression can therefore be used to generate a family of curves of HP versus AP with GPM as a parametric variable. 

The angle bending in H9O occurs without breaking any bonds, and the electron correlation energy is therefore relatively constant over the whole curve. The HF, MP2 and MP4 bending potentials are shown in Figure 11.14, where the reference curve is taken from a parametric fit to a large number of spectroscopic data. ... 

Log normal distribution, the distribution of a sample that is normal only when plotted on a logarithmic scale. The most prevalent cases in pharmacology refer to drug potencies (agonist and/or antagonist) that are estimated from semilogarithmic dose-response curves. All parametric statistical tests on these must be performed on their logarithmic counterparts, specifically their expression as a value on the p scale (-log values) see Chapter 1.11.2. 

The domain of the stable flow is located to the right of the boundary PeL(i ) (the shaded region in the graph). To the left of this curve is the domain in which stable flows in a heated capillary cannot occur. From the relation between the parameters and Ja, the parametric plane Pep — may be subdivided into two domains (1) < Ja, and (2) > Ja. Within the first of these the stable flows cannot occur... 

Fig. 23—A schematic force curve plotted as a function of sliding velocity. A viscous friction forms the background of the force curve upon which the frictions from superharmonic and parametric resonance are superposed.

Figure 5. Concordia diagram similar to Figure 4 illustrating the concordia curve for initial = 150 (appropriate for marine samples), with age in ka depicted parametrically along concordia. Also illustrated are continuous uranium gain/loss model curves for samples with primary ages of 80 ka (dashed) and 150 ka (thin solid curve). See text for discussion of this model and related models (after Cheng etal. 1998).

Note that the curvature is independent of the kind of parameterization s of the curve, which indicates that, in the case of a stream line, the curvature can be made to depend on the location of this point in the vector field and not on the parametrization along the stream line itself. By virtue of Eqs. (3.3), (3.4), and (3.5) and some manipulations, the curvature of a stream line at a location x can be expressed as ... 

In their work [58], GY demonstrated that a standard Lennard-Jones model grossly over-predicted the well-depth of rare gas-halide ion dimer potential energy curves when they were parametrized to reproduce the neutral rare gas-halide dimer curves. They further showed that the OPNQ model performed just as badly when the charge dependence of the expressions were ignored, but the potential energy curves for both the neutral and ionic dimers could be simultaneously be reproduced if the charge dependence is considered. 

Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos < >, y = a sin < > where 0 is the parameter. 

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