Variables technique

In order to investigate whether the wavefunction can indeed be written in this way, we use the separation of variables technique and so write a wavefunction of the form... 

Equation (8.4.3) is a linear first-order differential equation of concentration and reactor length. Using the separation of variables technique to integrate (8.4.3) yields... 

An example of this change-of-variable technique where is not monotonic arises when we choose a quadratic device, (x) = x2. Equation (3-42) is valid for any , so we can write... 

The pure variable technique can be applied in the column space (wavelength) as well as in the row space (time). When applied in the column space, a pure column is one of the column factors. In LC-DAD this is the elution profile of the compound which contains that selective wavelength in its spectrum. When applied in the row space, a pure row is a pure spectrum measured in a zone where only one compound elutes. 

Application of State Variable Techniques to the Control of a Polystyrene Reactor... 

The present paper applies state variable techniques of modern control theory to the process. The introduction of a dynamic transfer function to manipulate flow rate removes much of the transient fluctuations in the production rate. Furthermore, state variable feedback with pole placement improves the speed of response by about six times. 

Fig. 4.16. The A dynamics method for alchemical transformations was developed by Guo and Brooks [57] for rapid screening of binding affinities. In this approach the parameter A is a dynamic variable. Techniques like ABF or metadynamics [34] can be used to accelerate this type of calculation. A dynamics was used by Guo [57] to study the binding of benzamidine to trypsin. One simulation is sufficient to gather data on several benzamidine derivatives. Substitutions were made at the para position C5 (H, NH2, CH3 and Cl). The hydrogen atoms are not shown for clarity...

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

A special case of the gamma distribution is obtained by replacing a. by n+1, where n is an integer, and x by fix (this can be easily achieved using the change of variable technique discussed below). The resulting distribution f(x) is... 

In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation. 

This equation is solved with heat transfer boundary condition (15.11) by a standard separation-of-variables technique, to give ... 

Unconstrained optimization deals with situations where the constraints can be eliminated from the problem by substitution directly into the objective function. Many optimization techniques rely on the solution of unconstrained subproblems. The concepts of convexity and concavity will be introduced in this subsection, as well as discussing unimodal versus multimodal functions, singlevariable optimization techniques, and examining multi-variable techniques. 

Partial differential equations may be written directly using an infinitesimal generator technique, called the random-variable technique, given in Bailey [387]. For intensity functions of the form (9.33), we define the operator notation... 

The separation of variables technique docs not work in this case since the medium is infinite. But another clever approach that converts the partial differential equation into an ordinary diSerendal equation by combining the Bvo independent variables x and t into a single variable rj, called the similarity variable, works well. For transient conduction in a semi-infinite medium, it is defined as... 

Consider that one of the main advantages of the Laplace transform technique is that it can be used for time dependent boundary conditions, also. The separation of variables technique cannot be directly used and one has to use DuhameFs superposition theorem[l] for this purpose. Consider the modification of example 8.7 ... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

Most of the crack problems that have been solved are based on two-dimensional, linear elasticity (i.e., the infinitesimal or small strain theory for elasticity). Some three-dimensional problems have also been solved however, they are limited principally to axisymmetric cases. Complex variable techniques have served well in the solution of these problems. To gain a better appreciation of the problems of fracture and crack growth, it is important to understand the basic assumptions and ramifications that underlie the stress analysis of cracks. 

Applying the separation-of-variables technique to solve the preceding partial differential equation, the Nusselt number for the constant heat rate case in the fully developed region can be shown [6] to be given by the following equation ... 

O. Rockinger, Pixel-Level Fusion of Image Sequences Using Wavelet Frames, in Proceedings in Image Fusion and Shape Variability Techniques, Leeds, UK, (K. V. Mardia and C. A. Gill and I. L. Dryden, Eds.). Leeds University Press, 1996, pp. 149-154. 

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... 

Another approach is to use latent variable techniques, e.g. PLS or PCR, in which the selection of the dimension replaces the selection of variables. Although variable selection seems more natural, and is more commonly used in typical applications of DOE than latent variable methods, neither of the approaches have been proved generally better. Therefore, it is good to try out different apvproaches, combined with proper model validation techniques. 

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