Curvature of a function

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. 

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 second derivative of a function, denoted d2y/dx2, is defined like the first derivative but is applied to the function obtained by taking the first derivative. For example, the second derivative of the function x1 is the derivative of the function lx, which is the constant 2. Likewise, the second derivative of sin ax is —a2 sin ax. The second derivative is an indication of the curvature of the function. Where d2y/dx2 is positive, the graph has a (j shape where it is negative, the graph has a m shape. The greater the magnitude of d2y/dx2, the sharper the curvature of the graph. 

Radius of curvature as a function of (a) quenching distance and (b) equivalence ratio. 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

Most transport vesicles bud off as coated vesicles, with a unique set of proteins decorating their cytosolic surface. The coat has two major known functions. First, it concentrates and selects specific membrane proteins in a discrete portion of donor organelle membrane that will serve as origin to the transport vesicle. Second, the assembly of coat proteins into curved structures delineates the area of the forming transport vesicle. The size and curvature is a function of the coat composition. Thus, vesicles with similar vesicle coat have closely similar size and shape [3]. 

In this equation J7(r) is the electronic energy of the molecule (the total energy except for that associated with vibration or rotation), D, is the difference in energy of the minimum of the curve and the value for separated atoms, and a is a constant, which determines the curvature of the function near its minimum. The relation between the constants of the Morse function and the vibrational frequency of the molecule will be given below. 

The linear model, which may also be constructed from an approximate gradient, is simple but not particularly useful since it is unbounded and has no stationary point. It contains no information about the curvature of the function. It is the basis for the steepest descent method in which a step opposite the gradient is determined by line search vide infra). 

The quadratic model is an improvement on the linear model since it gives information about the curvature of the function and contains a stationary point. However, the model is still unbounded and it is a good approximation to fix) only in some region around xc. The region where we can trust the model to represent fix) adequately is called the trust region. Usually it is impossible to specify this region in detail and for convenience we assume that it has the shape of a hypersphere s <, h where h is the trust... 

If the second derivative, and hence the curvature of/, is negative at x, then / at x will be larger than the average of / at all neighbouring points, i.e. / concentrates at point x73. Therefore - V2p(r), which is the second derivative of a function depending on three coordinates x, y and z, has been called the Laplace concentration of the electron density distribution. Furthermore, the Laplacian of pir) provides the link between electron density p(r) and energy density Hir) via a local virial theorem (equation 8)67,... 

The first-order optimality conditions utilize information only on the gradients of the objective function and constraints. As a result, the curvature of the functions, measured by the second derivatives, is not taken into account. To illustrate the case in which the first-order necessary optimality conditions do not provide complete information, let us consider the following example suggested by Fiacco and McCormick (1968). 

In general, a locally convex domain D2(b),j(a> Fj) of a functional group F, relative to a reference curvature b, shows local shape complementarity with a locally concave domain Do( b),j(a, F2) of a complementing functional group F2, relative to a reference curvature of -b. The threshold values a and a are also likely to complement each other the shape complementarity between the higher electron density contours of one functional group and the lower electron density contours of the other functional group is relevant. 

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. 

Such a potential energy function gives rise to the famihar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmoitic contributions to the potential must be included for an accurate description. 

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