The Curvature of a Function

The second derivative determines the curvature of a function. Higher derivatives were defined. Derivatives are useful in applying the rule of I Hopital ... 

Some derivative algorithms can be used to calculate derivatives of different orders. For example, the derivative of a first derivative is a second derivative. The second derivative of a function measures its concavity, which is a measure of the direction of the curvature of a function. You can visualize the concavity of a spectrum by thinking about pouring water on it. The places where the water collects are concave up like a bowl. The places from which the water drains are concave down like a hillside. Concavity can also be thought of as a measure of the change in slope of a function. The second derivative of an absorbance feature is shown in Figure 3.19. 

The Hessian matrix is a generalization in R of the concept of curvature of a function. The positive-definiteness of the Hessian is a generalized notion of positive curvature. Thus, the properties of H are very important in formulating minimum-seeking algorithms. 

Methods that use analytic-derivative information clearly possess more information about the smooth objective function. Gradient methods can use the slope of a function, for example, as the direction of movement toward extremum points. Second derivative methods can also incorporate curvature information from the Hessian to find the regions where the function is convex. 

We have made the estimations for the case where the non-uniform electric field is produced by the charge distributed on the surface with a rather moderate curvature. The field near an apex can allow to control the motion of a functionalized wall by applying smaller voltage. Thus, our estimations show the possibility of the feasibility method of nanotube-based NEMS control. 

The value of [OH] may be calculated from the flame composition and a plot of log [OH]/ against 1/T should yield a straight line whose slope is the dissociation energy of X2OH. In fact, a curve results, which tends to a limiting slope (Fig. 3.13). The curvature is a function of height in the flame, becoming less as the flame is ascended. 

Another way to interpret the procedure described is that the curvature along an arbitrary direction, in the surface a = r -f- wr, is a quadratic function of the values of i and m. Diagonalizing this quadratic form, subject to the constraint that a is a unit vector, is mathematically equivalent to the minimization/maximization of k. Thus, the extremal curvatures, Ka and Kb are determined by the extremal values of I and w, which we denote as I and m. For directions on the surface close to these extremal values, the expansion of the curvature as a function of ( — ) and (m — m ) has no linear terms since... 

Other workers have employed gaussian-type functions for fitting the curvature as a function of the reaction coordinate. In many A -t- BC reactions, curvature is not a simple function of s and the reaction probabilities are very sensitive to it. Attempts to approximate the curvature in such cases led to large quantitative errors however, Carrington eT used their RPH to calculate the lifetime of vinyl-... 

The stability of the system to phase separation is determined by the curvature of AFroix/NkT versus x. Figure 25.5 show-s AFmix/NkT versus x from Equation (15.14) for two different temperatures (different values of xab)- This function is concave upward when the temperature T is high (xab 0) (see Figure 25.5(a)). The free energy function is concave downward when T is low ( chiAB 0) (see Figure 25.5(b)). Where AFmix x)/NkT is concave upward, systems are miscible. Where it is concave downward, systems are immiscible. Here s w hy the curvature of this function determines the stabihty. 

Unslaving of the harmonics The width of the sidebands grows either when the control parameter B increases or if the curvature, as a function of the wavenumber, at the maximum of the linear critical frequency becomes smaller, for instance as the ratio DaiDh decreases. Then overtones of the critical modes, that were otherwise slaved, progressively rejoin the active set of modes and their resonant interactions with the fundamental modes must be taken into account. This effect also grows more important as the space dimension of the pattern increases. Indeed for a periodic structure in one dimension the first harmonic to become excited lies at k = 2kc, whereas for patterns of hexagonal symmetry that are periodic in two dimensions the first overtone is at A = /3 kc- Further, for 3D patterns they appear... 

Fig. 7a-c. Finite element analysis of the effect of clamping on the curvature of a cantilevered substrate, (a) Deflection (b) slope and (c) curvature as a function of the length-to-width ratio for A=1 and with the Poisson ratio V as a parameter. From [2000Dah]. 

Now in order to determine [a] as a function of z, consider the strain e(z) at any depth across the section. It will be made up of an in-plane component (e) plus a bending component z/R) which is normally written as z-k, where k is the curvature of bending. 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

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