Smooth curves

The accuracy of the conversion depends on the smoothness of the D 86 curve. Errors affect essentially the points in the low % distilled ranges. Average error is on the order of 5°C for conversion of a smooth curve. 

Let the problem of focusing laser radiation into the smooth curve L have a smooth solution function (p, rf)e.C (G). Then the inverse image of each point M ff) EiL is a certain segment F (ff) S G. ... 

Figure A2.5.7. Constant temperature isothenns of reduced Helmlioltz free energy A versus reduced volume V. The two-phase region is defined by the line simultaneously tangent to two points on the curve. The dashed parts of the smooth curve are metastable one-phase extensions while the dotted curves are unstable regions. (The isothenns are calculated for an unphysical r = 0.1, the only effect of which is to separate the isothenns...

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The first step is the collection of data showing corresponding values of the variables under consideration. From a scatter diagram, a plot of Y (ordinate) versus X (abscissa), it is often possible to visualize a smooth curve approximating the data. For purposes of reference, several types of approximating curves and their equations are listed. All letters other than X and Yrepresent constants. 

To complete an approximate sketch of the titration curve, we draw separate straight lines through the two points before and after the equivalence point (Figure 9.35e). Finally, a smooth curve is drawn to connect the three straight-line segments (Figure 9.35f). 

Let the mid-surface of the Kirchhoff-Love plate occupy a domain flc = fl Tc, where C is a bounded domain with the smooth boundary T, and Tc is the smooth curve without self-intersections recumbent in fl (see Fig.3.4). The mid-surface of the plate is in the plane z = 0. Coordinate system (xi,X2,z) is assumed to be Descartes and orthogonal, x = xi,X2)-... 

Let C be a bounded domain with the smooth boundary L, which has an inside smooth curve Lc without self-intersections. We denote flc = fl Tc. Let n = (ni,ri2) be a unit normal vector at L, and n = ( 1,1 2) be a unit normal vector at Lc, which defines a positive and a negative surface of the crack. We assume that there exists a closed continuation S of Lc dividing fl into two domains the domain fl with the outside normal n at S, and the domain 12+ with the outside normal —n at S (see Section 1.4). By doing so, for a smooth function w in flc, we define the traces of w at boundaries 912+ and, in particular, the traces w+ and the jump [w] = w+ — w at Lc. Let us consider the bilinear form... 

Let c be a bounded domain with a smooth boundary F, and Fc C be a smooth curve without selfintersections. Assume that Fc contains... 

If U varies along the tube length or the stream temperature profile is not a smooth curve, then divide the entire tube length into a number of small heat-exchange elements, apply steps (2) through (8) to each element, and sum up the resulting area requitements as follows ... 

The coUigative properties of antifreeze chemicals may also result in boiling point elevation. As the chemical is added to water, the boiling point of the mixture increases. Unlike the freeze depression, the boiling elevation does not experience a maximum the boiling point versus concentration curve is a smooth curve that achieves its maximum at the 100% antifreeze level. The boiling point elevation can be another important characteristic for antifreeze fluids in certain heat-transfer appHcations. 

Approximately 0.016 (kg-mol)/s [126 (lb-mol)/li] of vapor is absorbed with an energy liberation of about 198,000 W (670,000 Btii/b), 20 percent of which is removed by the intercooler on stage 7. The temperature profile departs from a smooth curve at stages 4 and 7, where secondary oil enters and heat is removed respectively... 

Draw a smooth curve through these data. 

Figure 4-12. Stopped-flow study of the pyridine-catalyzed hydrolysis of acetic anhydride, showing the formation and decay of the acetylpyridinium ion intermediate. Initial concentrations were 0.087 M pyridine, 2.1 x im M acetic anhydride the pH was 5.5 ionic strength, 1.0 M temperature, 25 C. Five hundred data points tabsorbance at 280 nm) were measured in I s. The smooth curve is a ht to Eq. (3-27). Source Data of D. Khossravi and S.-F. Hsu, University of Wisconsin.

It should be emphasized that the estimation methods presented previously apply to any hazard paper and, in addition, to a nonparametric fit to the data obtained by drawing a smooth curve through data on any hazard paper. 

When a set of data does not plot as a straight line on any of the available papers, then one may wish to draw a smooth curve through the data points on one of the plotting papers, and use the curve to obtain estimates of distribution percentiles and probabilities of failure for various given times. With such a nonparametric fit to the data, it is usually unsatisfactory to extrapolate beyond the data because it is difficult to determine how to extend how to extend the curve. Nonparametric fitting is best used only if the data contain a reasonably large number of failures. 

Obviously the regularity expressed in the qualitative form (a) is far less informative than any one of the quantitative presentations, (b), (c), or (d). The relative merits of the expressions (b), (c), and (d) depend upon the use. Table l-II tells in most detail exactly how much is known about the pressure-volume behavior of oxygen gas (from this experiment). In the graphical presentation of Figure 1-8 the trend of the data is shown by the smooth curve drawn to pass near as many points as possible. Uncertainties caused... 

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