Order sensitivity

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Anomalous X-ray diffraction or resonant scattering refers to the modification of its intensity due to absorption processes involving interactions between the X-ray beam and the atoms in the sample. This interaction combines the chemical and short-range order sensitivity of absorption with the long-range order sensitivity of diffraction. This results in a chemical selectivity, i.e. it is possible to differentiate elements with close atomic numbers or even cations with the same number of electrons like Rb+ and Sr2+... 

The reaction set presented in Table XV also was subjected to a full sensitivity analysis in order to determine the rank order of the reactions in the mechanism. For this, normalized first-order sensitivity gradients (Sy) were calculated along the flame using the following definition ... 

Kramer, M. A., Calo, J. M Rabitz, H., and Kee, R. J., AIM The Analytically Integrated Magnus Method for Linear and Second Order Sensitivity Coefficients. SAND82-8231, Sandia National Laboratories, Livermore, August, 1982b. 

We (K1) attempted to develop a noncompetitive assay based on the anti-idiotype antibodies for a conjugated bile acid metabolite, ursodeoxycholic acid 7-A-acetyl-glucosaminide (UDCA 7-NAG), which is expected to serve as a diagnostic index for an autoimmune disease, primary biliary cirrhosis. In our assay, the hapten UDCA 7-NAG, a /3-type antibody, and a biotin-labeled a-type antibody were simultaneously added to a microtiter plate coated with an F(ab )2 fragment of a specific anti-UDCA 7-NAG antibody, then incubated at room temperature for 8 h. Bound biotin was then detected with HRP-labeled streptavidin, whose enzyme activity was measured using o-phenylenediamine/H202 as a substrate. This noncompetitive assay system provided a subfemtomole-order sensitivity (detection limit 118 amol) that was 7 times lower than the competitive immunoassay using the same anti-hapten antibody (K2), even with a common colorimetric detection (Fig. 13). Somewhat improved specificity was also obtained namely, better... 

Gao, D W. R. Stockwell, and J. B. Milford, First-Order Sensitivity and Uncertainty Analysis for a Regional-Scale Gas-Phase Chemical Mechanism, J. Geophys. Res., 100, 23153-23166 (1995). 

Furthermore, the coherence order (sensitivity to twisting of a coherence by a gradient) is ambiguous, at least with respect to the sign... 

This parameter controls the option of calculating the first-order sensitivity functions. 

Similar effects have also been observed for films of the pressure-sensitive polymers poly(vinylidene fluoride) and poly(tetrafluoroethylene). Thus, repeated measurements of the investigated polymer are recommended in order to confirm this phenomenon and to avoid misinterpretations. For samples without conformation and state of order sensitive absorption bands, this phenomenon is not relevant. 

Kinetic-Order Sensitivities— The ratio of relative change in a dependent concentration X, to relative change in a kinetic-order parameter, gj or /lyt, can be determined by differentiation of the explicit solution with respect to the parameter in question. 

Seefeld, S. and W.R. Stockwell First-order sensitivity analysis of models with time-dependent parameters An application to PAN and ozone, Atmoi. Environ. 33 (1999) 2941-2953. 

Moving Beyond First-Order Sensitivity Analysis... 

A second-order sensitivity coefficient describing the cooperative/antico-operative effects of two model parameters—and —in affecting an ensemble-averaged property <0> has the general form ... 

The formulas involving thermodynamic properties, however, are somewhat different because the formula for calculating a thermodynamic quantity differs from that for calculating an ensemble-averaged quantity. For example, the first-order sensitivity coefficient relating the Helmholtz free energy A of a biomolecular system to a potential parameter is expressed in the form... 

Because first-order sensitivity coefficients are easier to calculate than higher order sensitivity coefficients, it is likely that the former may be used more frequently in guiding molecular design. However, first-order sensitivity theory can provide reliable predictions only when the sensitivities of the properties of interest are approximately linear with respect to the model parameters. This linear response limit is satisfied when the perturbations of model parameters are small. For certain applications, such as in protein engineering where one amino acid is mutated into another, the linear response approximation may fail to reliably predict the change in the properties of a protein resulting from a point mutation. It is therefore useful to examine in more detail how well first-order sensitivity theory performs in guiding such predictions. 

The two-dimensional square lattice protein folding model discussed earlier provides a simple basis for probing this issue. The model has the advantage of allowing one to carry out many exact calculations to check the predictions from first-order sensitivity theory. Unlike molecular dynamics or Monte Carlo simulations, there are no statistical errors or convergence problems associated with the calculations of the properties, and their parametric derivatives, of a model polypeptide on a two-dimensional square lattice. 

It is clear from the data in Table 3 that first-order sensitivity theory works best when e. and Ae, are both small. When e. and Ae, are of the order of 2 k, the predictive reliability decreased to 75%. Therefore, first-order sensitivity theory does not always give correct predictions. However, since first-order sensitivity coefficients can usually be calculated more easily than higher order sensitivity coefficients in (bio)molecular simulations, first-order sensitivity coefficients can be used as a preliminary screening tool for suggesting a small number of modifications to a (bio)molecule that may lead to the desired biological effect. More sophisticated (but usually more expensive) calculations and/or suitable experimental studies can then be carried out to sort out from this small number of suggestions those that are more likely to achieve the desired biological effects. If experimentation is easier, the predictions can be tested in the laboratory. 

An obvious extension of first-order sensitivity theory is to develop higher order theories utilizing higher order sensitivity coefficients. For example, some... 

Table 3. Predicted Results of Mutations for a Two-Dimensional Square Lattice Model of Protein Folding Using First-Order Sensitivity Theory"...

Although first-order sensitivity theory is not always reliable in predicting the properties of a structurally modified (bio)molecule, it may be useful as a preliminary classification tool for suggesting a small number of modifications... 

Representation of human-automation integration requires functions of attentional control and concurrent task performance. Distributed attention and attention switching refer to an operator s ability to perform multiple tasks simultaneously. In many cases, a second task can be added to the performance of a primary task with little or no impact to the performance of the first task. In other cases, the performance of two tasks simultaneously has a disastrous interaction. Such context-and order-sensitive effects are determined in the scheduling and agenda management function provided in the MIDAS model. Attention capture functions are represented through a preattentive filter mechtmism that responds to physical characteristics of environmental stimuli (e.g., color, blinking, auditory characteristics). 

ABSTRACT We present an algorithm named EASI that estimates first order sensitivity indices from given data, hence allowing its use as a post-processing module for pre-computed model evaluations. Ideas for the estimation of higher order sensitivity indices and the computation of regression curves are also discussed. 

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