Linear criteria

Distribution Coefficients. Gel-permeation stationary-phase chromatography normally exhibits symmetrical (Gaussian) peaks because the partitioning of the solute between mobile and stationary phases is linear. Criteria more sophisticated than those represented in Figure 8 are seldom used (34). 

It is seen that errors in the smaller component can be as great as 12.5% (1.25% absolute) when the response index is 0.94. Yet on examining the curve for a response index of 0.94 in figure 2 the non-linearity is scarcely apparent. When the response index is 1.05 the error is 9.5% (0.95% absolute) and again the poor linearity is not obvious in figure 2. As already stated, to obtain accurate results without employing a correction factor, the response index should lie between 0.98 and 1.02. Most LC detectors can be designed to meet this linearity criteria. 

NOTE This protocol is based on the assumption that if the 10 unlabeled 2,3,7,8-substituted isomers provided in the EPA standard solutions meet linearity criteria, then the seven additional 2,3,7,8-substituted isomers and the cleanup standard in the CC3 solution may be assumed to have a sufficiently linear response to be used for quantitation. These eight RRFs cannot be used to determine percent relative standard deviation, but are used for percent difference determinations (as described in Paragraph 7.4.6.4) and quantitation of target analytes. 

Test for fit of data to linearity criteria for calibration curves or rate reactions... 

The requirements on the calibration curve are frequently confused with the dilutional linearity criteria. Dilutional linearity is the relationship between the... 

As defined in ICH-Q2A [ 1 ], linearity of an analytical procedure is its ability, within a defined range, to obtain results directly proportional to the concentrations of the analyte in the sample. This is sometimes misinterpreted to imply that the assay response should be a linear function of the analyte concentration. The linearity criteria should be applied only to the back-calculated (calibrated) results, that is, the back-calculated results should be directly proportional to the analyte concentrations of the sample within the range in which the study samples are expected to fall. 

If the linearity criteria is violated, a possible option is to increase W for the same value of B. Values of WIB up to 4 are permitted. The uniaxial tensile yield stress maximum load in the tension test, as described in Chap. 19. If the linearity and size criteria are met, then Kq is recorded as Kjc, that is, the plane strain value. The corresponding fracture energy, Qc can be obtained using ... 

This leads us to the selection of an appropriate pressure range for each adsorbent with the problem that the above "linearity criterion", which is essential, is unfortunately not self-sufficient, since several portions of the BET plot can fulfil this requirement. One can indeed consider, in Fig 1, several relative pressure ranges in which the plot is reasonably linear for instance 0.01-0.2, 0.02-0.05 and 0.05-0.15, the latter being the most commonly used. The resulting monolayer contents are 4.0, 4.5 and 5.2 mmol.g, respectively, i.e. a variation of up to 30% Hence the need of other criteria leading to the objective choice of a single linear portion of the BET plot. Those two selected hereafter are aimed to provide such a single choice ... 

Beyond the "linearity criterion" of the BET plot, two other criteria are found necessary, especially in the presence of micropores, to draw the specific advantage of the BET equation, le. to reach a single and reproducible value for the so-called "monolayer content". 

While the first and fourth of these are synnnetric with respect to particle iuterchauge and thereby satisfy the iudistiuguishability criterion, the other two are not and appropriate linear combinations must be fonned. Doing so, one finds the following four wavefimctions ... 

H-nmr chemical shifts of N-1—H and N-3—H signals have been used as a criterion for distinguishing between N-l-substituted and N-3-substituted hydantoin derivatives (22). They can often be related to electronic properties, and thus good linear correlations have been found between the shifts of N—H and Hammett parameters of the substituents attached to the aryl group of 5-arylmethylenehydantoins (23). 

Some of the tests and criterion used to define fire resistance may be found in the Hterature (9). Additionally, the compression—ignition and hot manifold tests as defined in MIL-H-19457 and MIL-H-5606, respectively the Wick test as defined by Federal Standards 791, Method 352 flash point and fire point as defined in ASTM D92 autoignition temperature as defined in ASTM D2155 and linear flame propagation rate are defined in ASTM D5306 are used. 

Overall Assessment of Properties. Table 3 shows the effect for each criterion of increasing plasticizer concentration, increasing the size of the plasticizer molecule, increasing the linearity of the plasticizer molecule, and changing the acid constituent of the ester. An I indicates improved performance for a particular property, a P indicates poorer performance. I and P in parentheses indicate that any changes tend to be marginal. 

A more practical approach for quantifyiag the conditions required for fracture uses a stress intensity criterion instead of an energy criterion. Using linear elastic theory, it has been shown that under an appHed stress, when the stress intensity K,... 

The technique is based on the methods of linear algebra and the theoiy of games. When the problem contains many multibranched decision points, a computer may be needed to follow all possible paths and hst them in order of desirability in terms of the quantitative criterion chosen. The decision maker may then concentrate on the routes at the top of the list and choose from among them by using other, possibly subjective criteria. The technique has many uses which are weh covered in an extensive hterature and wih not be further considered here. 

The development of an SC procedure involves a number of important decisions (1) What variables should be used (2) What equations should be used (3) How should variables be ordered (4) How should equations be ordered (5) How should flexibility in specifications be provided (6) Which derivatives of physical properties should be retained (7) How should equations be linearized (8) If Newton or quasi-Newton hnearization techniques are employed, how should the Jacobian be updated (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds (10) What convergence criterion should be applied ... 

Following this procedure urea can be determined with a linear calibration graph from 0.143 p.g-ml To 1.43 p.g-ml and a detection limit of 0.04 p.g-ml based on 3o criterion. Results show precision, as well as a satisfactory analytical recovery. The selectivity of the kinetic method itself is improved due to the great specificity that urease has for urea. There were no significant interferences in urea determination among the various substances tested. Method was applied for the determination of urea in semm. 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Introduction of the local Archimedes criterion helped to clarify nonisothermal jet design procedure. Grimitlyn suggested critical local Archimedes number values, Ar , below which a jet can be considered unaffected by buoyancy forces (moderate nonisothermal jet) Ar, 0.1 for a compact jet, Ar, < 0.15 for a linear jet. 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

The terms that are linear in the stresses are useful in representing different strengths in tension and compression. The terms that are quadratic in the stresses are the more or less usual terms to represent an ellipsoid in stress space. However, the independent parameter F,2 is new and quite unlike the dependent coefficient 2H = 1/X in the Tsai-Hill failure criterion on the term involving interaction between normal stresses in the 1- and 2-directions. 

The analysis of stresses in the laminae of a laminate is a straight-fonvard, but sometimes tedious, task. The reader is presumed to be familiar with the basic lamination principles that were discussed earlier in this chapter. There, the stresses were seen to be a linear function of the applied loads if the laminae exhibit linear elastic behavior. Thus, a single stress analysis suffices to determine the stress field that causes failure of an individual lamina. That is, if all laminae stresses are known, then the stresses in each lamina can be compared with the lamina failure criterion and uniformly scaled upward to determine the load at which failure occurs. 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

Equation (6-19) was said to provide a fit as good as or better than those with other equations. The parameters were evaluated by fixing C and carrying out a linear least-squares regression of In k on T C was then altered and the procedure was repeated. The residual sum of squares was taken as a criterion of best fit. 

The Hammett equation is said to be followed when a plot of log k against a is linear. Most workers take as the criterion of linearity the correlation coefficient r, which is required to be at least 0.95 and preferably above 0.98. A weakness of r as a statistical measure of goodness of fit is that r is a function of the slope p if the slope is zero, the correlation coefficient is zero. A slope of zero in an LEER is a chemically informative result, for it demonstrates an absence of a substituent... 

A more general criterion for linearity can be derived noting that the denominator of the Langmuirian adsorption isotherms must approach 1, and consequently ... 

Calculation of TMB flowrates To calculate TMB flowrates, linear behavior of the adsorption isotherms for a feed concentration of 1 g is assessed. To check this point, we will use the criterion given in Equation (10). 

The relationship between a criterion variable and two or more predictor variables is given by a linear multivariate model ... 

The criterion which has been extensively employed to distinguish A-l mechanisms from A-2 or A-SE2 mechanism has been the linearity of plots of reaction rate coefficient versus the acidity function h0. The acidity function (see Volume 2, p. 358) is a measure of the proton-donating ability of a medium (as measured by its tendency to protonate a base B) which is given by equation (11), viz. 

Solving such a myopic deconvolution problem is much more difficult because its solution is highly non-linear with respect to the data. In effect, whatever are the expressions of the regularization terms, the criterion to minimize is no longer quadratic with respect to the parameters (due to the first likelihood term). Nevertheless, a much more important point to care of is that unless enough constraints are set by the regularization terms, the problem may not have a unique solution. 

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