Maxima/minima

Figure B-3 Maxima, Minima, and Inflection Points of a Function of a Single Variable...

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. 

Figure 6.12 Classification of all types of extremum or critical point that can occur in one-, two-, and three-dimensional functions a one-dimensional function can possess only a maximum or a minimum a two-dimensional function has maxima, minima, and one type of saddle point a three-dimensional function may have maxima, minima, and two types of saddle point. The arrows schematically represent gradient paths and their direction. At a maximum all gradient paths are directed toward the maximum, whereas at a minimum all gradient paths are directed away from the minimum. At a saddle point a subset of the gradient paths are directed toward the saddle point, whereas another subset are directed away from the saddle point (see Box 6.2 for more details).

Once the total entropy of a composite system has been formulated as a function of the various extensive parameters of the subsystems, the extrema of this total entropy function may in principle be located by direct differentiation and classified as either maxima, minima or inflection points from the sign of the second derivative. Of these extrema, only the maxima represent stable equilibria. 

Find all of its stationary points and determine if they are maxima, minima, or inflection (saddle) points. Sketch the curve in the region of... 

A global multiresponse non-linear regression was performed to fit Eq. (57) to all the runs with both 2% and 6% v/v 02 feed content to obtain the estimates of the kinetic parameters (Nova et al., 2006a). Figure 37 (solid lines) illustrates the adequacy of the global fit of the TRM runs with 2 and 6% 02 the MR rate law can evidently capture the complex maxima-minima NO and N2 traces (symbols) at low T at both NH3 startup, that a simple Eley-Rideal (ER), approach based on the equation... 

Recall further that sub- (super-) harmonic functions have no inner maxima (minima) and that by the Hopf theorem the inward derivative of a sub-(super-) harmonic function at the points of the boundary maxima (minima) are strictly negative (positive). Since tp has no more than a finite number of extrema, one of the following two statements must be true ... 

With the help of (12.89), it is now possible to establish some simple theorems concerning the possibility of stationary points (e.g., maxima, minima, or horizontal inflections) in thermodynamic phase diagrams. In each case, we suppose that Rh Rj are chosen from any set of/+l intensive variables (spanning at least/ — 1 dimensions), and that iy are... 

Understand the significance of higher-order derivatives and identify maxima, minima and points of inflection... 

We seek extrema (maxima, minima, or saddle points) of/, subject to these two conditions. We shall show that there exist two constants, defined as a and ft (these two are known as the Lagrange multipliers), such that the system of n + 2 equations... 

None of the approximation methods investigated appears to be very reliable for application, in the room-temperature region, to the (nearunity) isotope-exchange equilibrium constants for the systems considered here. The inadequacy of the approximation methods in the majority of the cases studied arises, for the most part, from anomalies in the temperature dependences of the exact equilibrium constants. It has been shown (22) that whereas the logarithms of partition-function ratios are always smooth monotonic functions of temperature, plots of the logarithms of ratios of partition-function ratios—Le., isotope-exchange equilibrium constants—vs. T (or vs. log T) may exhibit maxima, minima, and inflection points. In addition, equilibrium constants may exhibit the... 

Now we can visualize evolutionary optimization as a hill-climbing process on a landscape that is given by an extremely simple potential [Eqn. (11.15)]. This potential, an ( — 1 )-dimensional hyperplane in n-dimensional space, seems to be a trivial function at first glance. It is linear and hence has no maxima, minima, or saddle points. However, as with every chemical reaction, evolutionary optimization is confined to the cone of nonnegative concentration restricts the physically accessible domain of relative concentrations to the unit simplex (xj > 0, X2 > 0,..., x > 0 Z x = 1). The unit simplex intersects the (n — 1 )-dimensional hyperplane of the potential on a simplex (a three-dimensional example is shown in Figure 4). Selection in the error-free scenario approaches a corner of this simplex, and the stationary state corresponds to a corner equilibrium, as such an optimum on the intersection of a restricted domain with a potential surface is commonly called in theoretical economics. 

The electron density p of a molecule is a physical quantity which has a definite value p(r) at each point of coordinates r in three-dimensional space. The topological properties of this electronic charge distribution can be summarized in terms of its critical points maxima, minima and saddles. Figure 8.1 displays the electronic charge density in three planes of the ethylene molecule. 

Figures 2a and 2b show how the predicted solvent-solute pair correlation function, gAB varies with density. At the highest density (p = 0.6) the structure is liquid-like with hrst, second, third,. .. maxima/minima oscillating about 1.0. The size of the solvent-solute cluster for this state was calculated to be about —1 solvent molecule the presence of one solute molecule at this state excludes about one solvent molecule.

Table 1. Average, maxima, minima (189 samples) and representative modal and petrophysical data of Serraria sandstones... 

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