Standard nonlinearity

Unless very few experimental data are available, the dimensionality of the problem is extremely large and hence difficult to treat with standard nonlinear minimization methods. Schwetlick and Tiller (1985), Salazar-Sotelo et al. (1986) and Valko and Vajda (1987) have exploited the structure of the problem and proposed computationally efficient algorithms. 

It is clear that pulse sequences may not only be designed by analytical means, they may also be designed numerically (see, e.g., reviews on numerical aspects of solid-state NMR in [54, 65, 66]) using standard nonlinear optimization to well-defined analytical expressions [67, 68], by optimizing pulse sequences directly on the spectrometer [69], or by optimal control procedures [70-72] to name but a few of the possibilities. We will in this review restrict ourselves to optimal control design procedures that recently in analytical and numerical form have formed a new basis for efficient NMR experiment design. 

The application of the standard nonlinear programming techniques of constrained optimization on analyzing the mean and variance response surfaces has been investigated by Del Castillo and Montgomery [34]. These techniques are appropriate since both the primary and secondary responses are usually quadratic functions. 

Unfortunately, constraint functions g,y are not necessarily monotonic. However, each gti has at most one stationary point with respect to flow rates w , and Wj (Saboo et al., 1987b). Standard nonlinear programming can be used to locate each stationary point. If the stationary point of g,y lies outside the uncertainty range or if the stationary point is a maximum, then gii.min occurs at a corner point. 

Apart from N (an integer) the optimisation problem is a standard Nonlinear Programming (NLP) problem, with the inner optimisation problem providing the values of the outer objective function and constraints. In the outer problem Mujtaba... 

The relative ease with which large-scale versions of these algorithms could be produced is an encouraging development. It means that we now have the capability of solving realistic engineering problems by standard nonlinear programming algorithms, without the need for clever exploitation of particular features of the problem to make the computations practicable. 

When a = 0 and P = 1, the standard nonlinear Burgers equation results ... 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

Values of the kinetic and adsorption parameters in the above tables were extracted from a standard nonlinear regression technique employed to select the best fits of the experimental data. The regression methods used were based on a minimization of the squares of the difference between experimentally observed and predicted results. 

Additionally, in a potential regulatory framework, a method of constructing 90% confidence interval for F must also be prespecified in order to control the Type I error. Several methods are possible. First, standard nonlinear regression packages provide asymptotic standard errors of parameter estimates. This readily leads to the construction of asymptotic confidence intervals, based on the normality assumption. Other methods include likelihood profile and bootstrapping. We considered these three methods briefly. 

Alternatively, all thej model equations may be listed singly, and then solved simultaneously using a standard nonlinear equation solver, such as a spreadsheet program. For the two-component system, the equations include the van Laar equations for both components in each liquid phase ... 

Chaotic dynamics are by definition aperiodic dynamics in deterministic systems with sensitive dependence on initial conditions. Although many refer to chaotic dynamics in Eq. (3), this does not conform to standard nonlinear dynamics terminology since such systems have a finite state space and must necessarily cycle. However, in Eq. (5), chaotic dynamics are possible and have been demonstrated numerically and analytically in some example networks [42, 46]. We have no way to predict whether any particular logical structure is capable of generating chaotic dynamics for some set of parameters. A necessary condition for chaotic dynamics is that there is a vertex that lies on at least two... 

The system of N(M + 1) equations is then solved by using standard nonlinear solvers such as the Newton-Raphson method or the conjugate-gradient minimization algorithm, both of which are described in Press et al. (1992). 

F ff is the peak isometric force developed by the ith musculotendinous actuator, a quantity that is directly proportional to the physiological cross-sectional area of the ith muscle. Equation (6.12) expresses the n relationships between the net actuator torques T " , the matrix of actuator moment arms R(q), and the unknown actuator forces F". Equation (6.13) is a set of m equations that constrains the value of each actuator force to remain greater than zero and less than the peak isometric force of the actuator defined by the cross-sectional area of the muscle. Standard nonlinear programming algorithms can be used to solve this problem [e.g., sequential quadratic programming (Powell, 1978)],... 

The conventional optimization problem so defined and first stated for systems subject to random vibrations (Nigam, 1972) can be transformed into a standard nonlinear programming one that is stated as ... 

Keywords Impedance standard, nonlinearity, simulation, four point measurment, HELIOS... 

The Michaelis-Menten model was fitted to the experimental data using standard nonlinear regression techniques to obtain estimates of and K (Fig. 4.1). Best-fit values of and K of corresponding standard errors of the estimates plus the number of values used in the calculation of the standard error, and of the goodness-of-fit statistic are reported in Table 4.3. These results suggest that succinate is a competitive inhibitor of fumarase. This prediction is based on the observed apparent increase in Ks in the absence of changes in Vmax (see Table 4.1). At this point, however, the experimenter cannot state with any certainty whether the observed apparent increase in Ks is a tme effect of the inhibitor or merely an act of chance. A proper statistical analysis has to be carried out. For the comparison of two values, a two-tailed t-test is appropriate. When more than two values are compared, a one-way analysis of variance (ANOVA),... 

The Michaehs-Menten model was fitted to the experimental data using standard nonlinear regression techniques to obtain estimates of K (Fig. 4.2). Best-fit values of and K, corresponding standard errors of the estimates plus the number of values used in the calculation of the standard error, and goodness-of-fit statistic are reported in Table 4.5. 

To obtain estimates of Ki and ki, a k versus [Iq] data set has to be created. A plot of these k versus [lo] data would yield a rectangular hyperbola (Fig. 5.3). With the aid of standard nonlinear regression procedures, the values of A, and ki can be obtained. 

In the EFISH technique (Electric Field Induced Second Harmonic) a solution of the (dipolar) chromophore under consideration is put into a uniform static electric field, obtained for example with parallel metal electrodes. In practice, instead of putting metal electrodes in the solution, the solution is put in a wedge shaped cell bounded by glass windows at which metal electrodes are applied [8]. Owing to the dipolar nature of the chromophores, they statistically orient under the electric field and the whole solution becomes a second order NLO-active polar medium. When a laser radiation crosses the solution, second harmonic radiation is generated and collected. By comparison with a standard nonlinear medium (usually a quartz slab) it is possible to extract the value of the dot product of the chromophore. As we have... 

Clearly, Eq. [43] is the modified MD equation of motion the first term is the standard nonlinear interatomic force, the second term is the impedance force discussed above, and the third term is a stochastic force representing the exchange of thermal energy between the MD region and the surrounding eliminated fine-scale degrees of freedom. In other words, this last term acts... 

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