Discrete variable technique

Table 3 Second and third resonance states, n = 2 m = 0, for the hydrogen Stark effect in dependence of the electric field strength. Comparison of our results obtained by discrete variable technique and finite element method with the results of C. Cerjan (1978)...

Discrete Variable techniques and Finite Element methods, if necessary combined with additional model dependent numerical techniques, turned out to be a useful, quick and accurate way for studying non-integrable quantum systems. By this methods we were... 

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

A recent numerical development is the introduction of the slow or smooth variable discretization (SVD) technique [101-103]. In the diabatic-by-sector method, the basis functions to expand the total wavefunction are fixed within each sector. In the SVD method, the hyperangular basis functions are constructed using the discrete variable representation (DVR) [104], The requirement is only that the total wavefunction be smooth in the adiabatic parameter p. By expanding the hyperradial wavefunctions using DVR basis functions, a new set of hyperangular basis functions are determined and they... 

Data are usually unmanageable in the form in which they are collected. In this section, the graphical techniques of summarizing such data so that meaningful information can be extracted from it is considered. Basically, there are two kinds of variables to which data can be assigned continuous variables and discrete variables. A continuous variable is one that can assume any value in some interval of values. Examples of continuous variables are weight, volume, length, time, and temperature. Most environmental data are taken from continuous variables. Discrete variables, on... 

Using whatever propagation method, one has to evaluate the action of the Hamiltonian operator on the wavefunction P(r). This is normally carried out by expanding P(f) in a suitable basis set and then evaluates the operator action on basis functions. One can use the FFT (fast Fourier transform) techniques (7,14), discrete variable representation (DVR) (15,16) techniques, or simply calculate matrix elements of the operator in a given basis set. 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

An alternative proposed in previous works (for example, Bdrard et al 2(X)0) consists in coupling a Discrete Event Simulator (DES) (in order to evaluate the feasibility of the production at medium term scheduling) with a master optimization procedure generally based on stochastic techniques such as GA (to take into account the combinatorial feature due to the large number of discrete variables in the optimization problem). These ideas... 

The finite difference method is one of the oldest methods to handle differential equations. Like the orthogonal collocation method discussed in Chapter 8, this technique involves the replacement of continuous variables, such as y and x, with discrete variables, that is, instead of obtaining a solution which is continuous over the whole domain of interest, the finite difference method will yield values at discrete points, chosen by the analyst. 

Another convenient and effective scheme for the approximate solution of a mathematical description of the polymerization reaction replaces the discrete variable of infinite range, polymer chain length, by a continuous variable. The difference-differential equations become partial differential equations. Barn-ford and coworkers [16,27,28] used this procedure in their analysis of vinyl (radical chain growth) polymerization. Zeman and Amundson [18,19] used it extensively to study batch and continuous polymerizations. Recently, Coyle et al. [4] have applied it to analysis of high conversion free radical polymerizations while Taylor et al. [3] used it in their modelling efforts oriented to control of high conversion polymerization of methyl methacrylate. A rather extensive review of the numerical techniques and approximations has been presented by Amundson and Luss [29] and later by Tirrell et al. [30]. 

Another real-space approach is to use a finite-element (FE) basis. " The equations that result are quite similar to the FD method, but because localized basis functions are used to represent the solution, the method is variational. In addition, the FE method tends to be more easily adaptable than the FD method a great deal of effort has been devoted to FE grid (or mesh) generation techniques in science and engineering applications. Other real-space-related methods include discrete variable representations, Lagrange meshes,and wavelets. ... 

Molecular Dynamics with Discrete Variable Representation Basis Sets Techniques and Application to Liquid Water. 

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