Relative maxima and minima

After the problem of the brachistochrone had been solved, James Bernoulli, brother of John, proposed another variety of problem—the so-called isoperimetrical problem—of which the fol- 

Example.—Find the curve of given length joining two fixed points so that the area hounded by the curve, the a-axis, and the ordinates at the fixed points may be a maximum. Here we have 

This is obviously the equation of a circular line. The limits are fixed, and therefore Ix - I0 — 0. The constants o, Glt and C2 can be evaluated when the fixed points and the length of the curve are known. 

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Most engineering students are well aware that the first derivative of a continuous function is zero at a maximum or minimum of the function. Fewer recall that the sign of the second derivative signifies whether the stationary value determined by a zero first derivative is a maximum or a minimum. Even fewer are aware of what to do if the second derivative happens to be zero. Thus, this appendix is presented to put finding relative maxima and minima of a function on a firm foundation. 

Relative maxima and minima of a function of several variables are found by solving simultaneously the equations obtained by setting all partial derivatives equal to zero. 

This algorithm represents a signal peak (e.g., from chromatography) as a series of equally spaced points and obtains a local smoothed value for each point in a fashion that preserves features such as relative maxima and minima as well as width, that are usually flattened by other averaging techniques. 

An interesting consequence of the relationship in Eq. [75] is that relative maxima and minima in protein stability occur at pHs for which the net charge of the denatured and native states are equal. Thus, the isoionic point is the pH of maximal stability only when this pH happens to be isoionic for the denatured state as well. 

The potential energy is often described in terms of an oscillating function like the one shown in Figure 10.9(a) where the minima correspond to the relative orientations in which the interactions are most favorable, and the maxima correspond to unfavorable orientations. In ethane, the minima would occur at the staggered conformation and the maxima at the eclipsed conformation. In symmetrical molecules like ethane, the potential function reflects the symmetry and has a number of equivalent maxima and minima. In less symmetric molecules, the function may be more complex and show a number of minima of various depths and maxima of various heights. For our purposes, we will consider only molecules with symmetric potential functions and designate the number of minima in a complete rotation as r. For molecules like ethane and H3C-CCI3, r = 3. 

The transmittance spectrum of a titania nanotube-film (transparent) on glass is shown in Fig. 5.33. The optical behavior of the Ti02 nanotube-arrays is quite similar to that reported for mesostructured titanium dioxide [133], The difference in the envelope-magnitude encompassing the interference fringe maxima and minima is relatively small compared to that observed in titania films deposited by rf sputtering, e-beam and sol-gel methods [134],... 

In the initial work135,136 on the sensitized isomerization of stilbene, it was found that quinone sensitizers gave photostationary states which were trans-rich relative to those predicted by energy transfer (Fig. 3). The original plot of sensitizer triplet energy versus isomer ratio of the photostationary state therefore showed several maxima and minima. Later work on the rates of energy transfer from various sensitizers to stilbene proved that the plot was in fact a smooth curve and that the photostationary states observed with quinones were not the true ones.137 Irradiation of benzene solutions of... 

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

Figure 5. Records of 5 0ct variation (given relative to the PBD standard) for speleothems from six different karst regions between 22° and 62° north latitude in North America and Bermuda after Harmon et al. (1978a). Vertical lines indicate suggested correlation between isotopic maxima and minima that are interpreted to denote paleoclimatic events. Small upward pointing arrowheads indicate U-series age determinations. Horizontal arrows labeled modern indicate the 5 0ct value of calcite presently forming a particular speleothem locality.

Figure 2-21 Diagram showing the energy maxima and minima as two substituted carbons connected by a single bond are rotated 360° relative to each other.

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