Discretization grid

In the mathematical sense, the TB model is a discrete (grid) version of the continuous Schrodinger equation, thus it is routinely used in numerical calculations. 

It should be mentioned here that these concepts of interpolation are commonly applied in numerical analysis, where the values are functions defined at the nodes as a result of a domain discretization (grid). Hence, if the value of the function on a point between nodes i + 1 and i - 1 is needed, we can use a first (eqns. (7.6) or (7.7)) or a second order interpolation (eqn. (7.12)). For example, let us define the derivative of u (x) at x in terms of the discreet values at locations + 1/2 and i — 1/2, i.e ... 

The set of d-MEP-SIms, however, represents the mutual similarity of the discrete grid MEP functions. For the guanine molecule Petke showed that, in precise similarity study, one had to analyze statistically the distribution of d-MEP-SI(P) values [116]. If a single number is needed to measure the similarity one should use the average value of d-MEP-SI(P)s or the cumulative Petke MEP-SI [116]. 

In another popular definition of mutate and crossover, a binary representation of the unknown variables is required whereby the simulated DNA is converted into a concatenated sequence of binary numbers (0 s and l s). To obtain a binary representation of the ionic coordinates, the ions are constrained in that they can only sit on one of 2m discrete grid points across the unit cell (Fig. 1). For each grid point there is a unique binary number of length m. Note that the grid points can either be numbered 0 to 2m-l (000 to 111 for m=3) or, as shown in Fig. 1, have... 

Orthogonal collocation in the chemical engineering literature refers to the family of collocation methods with discretization grids associated to Gaussian quadrature methods [34, 204]. Spectral collocation methods for partial differential equations with an arbitrary distribution of collocation points are sometimes termed pseudo spectral methods [22]. 

Figure 20 Schematic view of the k spectrum sampled on a three-dimensional cubic grid (bottom) and on a skewed grid (top). The Fourier transform of a function / is contained in the sphere pQ. Sampling the function / on a discrete grid produces copies of f(K), each containing a sphere with radius Kmax. These spheres should be distinct for optimal sampling.

In the well-tempered regime (for a discrete grid approximation to the system coordinates), the DAFs yield comparable accuracy for both the wave function and its derivatives (27,29). 

The DAFs provide comparable accuracy for the wave function both on and off the discrete grid points (16,27,29). 

The described differential equations were solved by the finite difference method of Patankar and Spalding (26). The boundary-layer nature of the problem permits us to solve the equations by moving in the direction of flow for discrete grid points, starting with known initial conditions. 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

Value may be comparable with the hydrodynamic dispersion D. and even greater than it. When D., modeling results to a greater extent reflect the influence of numerical and not hydrodynamic dispersion. If in equation (4.4) is used Peclet number (Pe) of the discretization grid equal to... 

Digitization involves two processes sampling the image value at a discrete grid of points and quantizing the value at each of these points to make it one of a discrete set of gray levels. In this section we briefly discuss sampling and quantization. 

The results of all these tests can be summarized in a few simple statements. For a given geometry, with a fixed ratio R/a or h/a, the accuracy of the predictions critically depends upon the ratio kg (or Ka/n) of the grid spacing to the double layer thickness. For k < 1, the errors o n all the perturbation quantities (a, a, p,...) do no exceed a few percent of Kay. When discretization is too coarse (ko > 1), the predictions become grossly inaccurate, as could be expected since the double layers are overlooked by the discretization grid. 

Finite-difference and spectral methods Two different numerical approaches to solve a prediction model by representing the model variables at discrete grid points and by collection of continuous functions, respectively. 

Because digital images are sampled on a discrete grid, but T generally maps to continuous values, interpolation of intensities is required. The interpolation process can affect the effectiveness of the registration, so that the choice of an appropriate interpolation algorithm plays an important role in the development of the registration procedure. 

Impedance spectra are measured on a discrete grid of frequencies yielding vectors of measured impedance points. For the discrete vector distance, however the choice of measurement frequencies is of particular importance. Implicit weighting effects arise which are analyzed and explicit weighting factors are introduced. Furthermore, possible applications which require distance measures are proposed. 

Employing finite differencing on a set of grid points defining the discrete grid denoted by il , this elliptic differential equation is transformed into an algebraic matrix equation of the form... 

Poisson s equation is usually simplified by assuming the dielectric constant to be stepwise constant in the position space. It should be noted that this approximation does not preclude the possibility of having dielectric interfaces within the computational domain what is assmned here is that the dielectric constant changes abruptly at the interface of different materials. This assumption is completely natural when Poisson s equation is solved on a discrete grid by a finite differences scheme. 

The system of Eqs. [18], [54], and [55] is usually solved iteratively, with each iteration defined by the successive solution of the three equations. An initial guess is first supplied for the force field P in Eq. [54], which is then solved on a discrete grid to provide the components of the cnrrent density /. The divergence of J is then computed with the steady-state continuity equation (Eq. [55]) to obtain the charge distribution that, in turn, is used in the forcing function of Poisson s equation. From the gradient of the computed potential, one derives a new (better) approximation to the force P that is used to start... 

First, we guarantee maximum isotropy on the grid by imposing that the continuum and discrete (grid-restricted) linkage operators are identical for... 

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