Horizontal asymptote

Plot the percent of index strength vertically versus the time to failure horizontally. The vertical intercept where the curve becomes asymptotic horizontally can be taken as the creep endurance limit. 

Similarly, for T —> oo, it can be shown that the curves have contact of infinite order with their asymptotes, i.e., the horizontals through the limiting densities. 

Each curve therefore consists of three parts an initial and a final portion which are nearly horizontal for a finite part of their lengths, and an intermediate portion which slopes down comparatively rapidly from left to right. This means that the dissociation with rise of temperature is slow at first, then increases very rapidly, and then becomes increasingly slower as it approaches asymptotically to the limiting value for T = oo. The general form of curve so predicted corresponds exactly with the experimental curves, as will be seen from Fig. 66, which was drawn by Horstmann from the results of Wiirtz with amylene hydrobromide ... 

Fig. 39.17. Schematic illustration of Michaelis-Menten kinetics in the absence of an inhibitor (solid line) and in the presence of a competitive inhibitor (dashed line), (a) Plot of initial rate (or velocity) V against amount (or concentration) of substrate X. Note that the two curves tend to the same horizontal asymptote for large values of X. (b) Lineweaver-Burk linearized plot of 1/V against l/X. Note that the two lines intersect at a common intercept on the vertical axis.

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

On the magnitude plot, the low frequency (also called zero frequency) asymptote is a horizontal line at Kp. On the phase angle plot, the low frequency asymptote is the 0° line. On the polar plot, the zero frequency limit is represented by the point Kp on the real axis. In the limit of high frequencies,... 

We choose -180° (and not 0°) because we know that there must be a phase lag. On the magnitude log-log plot, the high frequency asymptote has a slope of-2. This asymptote intersects the horizontal K line at co = 1/x. 

On the magnitude plot, the low frequency asymptote is a horizontal line at Kc. The high frequency... 

Abstract. We present the results of a spectroscopic analysis of bright stars in the Carina dSph galaxy. We collected low-resolution FORS2 VLT spectra of ss 200 stars. Our spectroscopic targets have been selected among the evolved Carina stars, in particular we selected low-mass, old Red Giant, Asymptotic, and Horizontal Branch stars, as well as intermediate-age stars of the Red Clump. We present preliminary estimates concerning the radial velocities of old and intermediate-age populations. 

Fig. 4.8. Colour-magnitude (HR) diagram of the globular cluster Messier 68 with [Fe/H] —2. In order of successive evolutionary stages, MS (sd) indicates the main sequence occupied by cool subdwarfs, with the position of the Sun shown for comparison, SGB indicates the subgiant branch, RGB the red giant branch, HB the horizontal branch including a gap in the region occupied by RR Lyrae pulsating variables and AGB the asymptotic giant branch. Adapted from McClure etal. (1987).

Figure 5.5 Least-square mixing hyperbola for the isotopic data on Heard Island of Barling and Goldstein (1989). Data from Table 5.10. The 87Sr/86Sr value of the MORB source ( 0.7025) lies below the horizontal asymptote. Asthenosphere and oceanic lithosphere are unlikely source components of Heard Island basalts.

At low 7t, the denominator simplifies to unity in each case and both models are linear in n. For sufficiently high n, the parenthesis in the denominator approaches Ketv, the initial rate for the dual-site model then approaches zero, and that of the single-site model approaches a constant value. Thus the plot of the experimental data will indicate that the dual-site model is preferable if a maximum exists in the data, or that the single-site model is preferable if a horizontal high-pressure asymptote exists. Hence, for the data of Franckaerts and Froment (FI) shown in Fig. 2, the dual-site model is preferred over the single-site model. 

The high-frequency asymptote intersects the L = 0 line at breakpoint frequency. The log modulus is "flat (horizontal) out to this point and then begins to drop off. 

If followed in experimenrtally accessible dilute solutions, Henry s law would be manifested as a horizontal asymptote in a plot such as Figure 19.3 as the square of the molality ratio goes to zero. We do not observe such an asymptote. Thus, the modified form of Henry s law is not followed over the concentration range that has been examined. However, the ratio of activity to the square of the molality ratio does extrapolate to 1, so that the data does satisfy the definition of activity [Equations (16.1) and (16.2)]. Thus, the activity clearly becomes equal to the square of the molality ratio in the limit of infinite dilution. Henry s law is a limiting law, which is valid precisely at infinite dilution, as expressed in Equation (16.19). No reliable extrapolation of the curve in Figure 19.2 exists to a hypothetical unit molality ratio standard state, but as we have a finite limiting slope at = 0, we can use... 

The horizontal asymptote equals the distribution constant of CuAj, i.e., Koc. 

Equation (4.41) is valid only when the complexes MA ° can be neglected in the aqueous phase. Comparing Eqs. (4.37b) and (4.41c), it is seen that no horizontal asymptote is obtained even at high concentrations of AT or HA and H. Thus, for very large distribution constant of the uncharged complex (i.e., >1000) a straight line with slope -z is experimentally observed, as in the case for the Cu(II)-thenoyltrifluoroacetone (HTTA) system (Appendix D 5g). 

Fig. 4 a Mean-field result (solid line) for the rescaled brush free energy per polymer as a function of the inverse interaction parameter 1/F- The infinite stretching resnlt is indicated by a horizontal dotted line, the broken straight line denotes the infinite stretching result with the leading correction dne to the finite end-point distribntion entropy. b Rescaled lateral pressnre within mean-field theory (solid line) compared with the asymptotic infinite-stretching result (dotted line)... 

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