Function smoothly varying

Table 1.2 The excess inter-electronic repulsion for T configurations (column 4), relative to a smoothly varying baseline function, g(n), drawn through the formulae for f, f and f. ...

Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

In the PP theory, the valence electron wave function is composed of two parts. The main part is the pseudo-wave function describing a relatively smooth-varying behavior of the electron. The second part describes a spatially rapid oscillation of the valence electron near the atomic core. This atomic-electron-like behavior is due to the fact that, passing the vicinity of an atom, the valence electron recalls its native outermost atomic orbitals under a relatively stronger atomic potential near the core. Quantum mechanically the situation corresponds to the fact that the valence electronic state should be orthogonal to the inner-core electronic states. The second part describes this CO. The CO terms explicitly contain the information of atomic position and atomic core orbitals. 

While the direction of the axial velocity does not change in many of the stagnation flows, in some it does. Certainly the opposed flows Section 6.10) have both positive and negative velocities. So the convective difference formulas must change depending on the velocity direction. A sigmoid function can be used to switch the difference formula in a smoothly varying way as... 

On the other hand, it may be expected that, after the transient period (1.3), an approximation method could be based on the idea that P is a smoothly varying function of X with a width of order Q. Accordingly we now choose as our variable the intensive quantity x = X/Q and write the... 

In general, the SDF at low frequency is a superposition of a 8-function peak at the frequency Q, the zero-frequency peak, and a broad, smoothly varying background (at cox -C 1), which is proportional to I) and is small if I) is small. 

Suppose we have some smoothly varying function y = f x) which might look like Figure 2.1 when graphed (y = fix) on the vertical axis, x on the horizontal axis). 

There is a formal definition of a smoothly varying function , but for our purposes, what we mean is that the curve has no breaks or kinks. 

Much of the first semester of calculus is devoted to understanding what is meant by a smoothly varying function, and finding the derivatives of various functions. For... 

Beyond a chosen cutoff radius, the all-electron and pseudofunctions (potential and wavefunction) are identical, while inside the core region both the pseudopotential and pseudowavefunction are smoothly varying. After the construction of these pseudofunctions for a single atom and ensuring that their scattering properties are almost identical to those of the all-electron functions, they can be used in any chemical environment. 

Seen from the point of view of quantum chemistry, this model divides the interelectronic repulsion effects into two categories the intra-atomic effects which are approximated by an analytic function corresponding to a smoothly varying valence state energy as function of ionicity, and interatomic effects which are represented by the Madelung potential valid for spherically symmetric, non-overlapping ions. 

We have indicated that intensity dependent phenomena may be useful in at least two distinct ways. One is to obtain something approaching a "threshold detector" resist response. To obtain a threshold development response in typical positive resists is difficult, since the development rate is in general a smoothly varying function of the photochemical reaction progress. The application of a layer of polymer with the bleaching characteristics shown in Figure 5 provides a way to obtain such threshold response with conventional resists, provided an excimer laser is used in the illumination system. 

In the general case of a tensor A (r, ), we can assume that the product A (r) E " (r) is a smoothly varying function of the coordinates, and it can be taken outside the integral over the anomalous domain D. As a result, we obtain from the TQL equation (9.80)... 

This approximation is particularly appropriate if the background field is a smoothly varying function of the coordinates it forms the basis of localized nonlinear approximations (Habashy et al., 1993). 

P E, hv) is the distribution of photoelectron energy, E, excited by photon of energy, hv, T E) is the transmission function and D E) is the escape function. Both T E) and D(E) are smoothly varying functions. Therefore the structures in the observed intensities are almost entirely determined by P E, hv). The photoelectron energy distribution function is given by. 

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