Discrete variable

Periodic boundary conditions force k to be a discrete variable with allowed values occurring at intervals of lull. For very large systems, one can describe the system as continuous in the limit of i qo. Electron states can be defined by a density of states defmed as follows ... 

Deuflhard P and Wulkow M 1989 Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable Impact of Computing in Science and Engineering vol 1... 

Luckhaus D 2000 6D vibrational quantum dynamics generalized coordinate discrete variable representation and (a)diabatic contraction J. Chem. Phys. 113 1329—47... 

Bacic Z, Kress J D, Parker G A and Pack R T 1990 Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. 4. discrete variable representation (DVR) basis functions and the analysis of accurate results for F + Hg d. Chem. Phys. 92 2344... 

Bade Z and Light J C 1986 Highly exdted vibrational levels of floppy triatomic molecules—a discrete variable representation—distributed Gaussian-basis approach J. Chem. Phys. 85 4594... 

Colbert D T and Miller W H 1992 A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method J. Chem. Phys. 96 1982... 

Leforestier C and Museth K 1998 Response to Comment on On the direct complex scaling of matrix elements expressed in a discrete variable representation application to molecular resonances J. Chem. Phys. 109 1204... 

Manthe U 1996 A time-dependent discrete variable representation for mulitconfigurational Hartree methods J. Chem. Phys. 105 2646... 

In the basis set formulation, we need to evaluate matrix elements over the G-H basis functions. We can avoid this by introducing a discrete variable representation method. We can obtain the DVR expressions by expanding the time-dependent amplitudes a (t) in the following manner ... 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

Henrici, Peter, Discrete Variable Methods in Ordinary Differential Equations,... 

First of all, we perform an important reordering, introducing the new discrete variables... 

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

The unit power to mix can be generally described as a function of the rotor surface area, the RPM used during mixing, the time required to reach dump temperature for the first pass mix, and a treatment for the type of antidegradant used. Again the discrete variable of antidegradant was treated continuously by assigning the values of 0, 1, and 2 to the conditions of Ctrl, PPD, and QDI. 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

The last example differs from the previous examples in this section in that they involved discrete variables, while pressure and temperature are continuous functions. The same problem could also arise in the discrete case. For instance, although the initial design might favor crystallization over extraction, if the sequence of processing steps were changed the extractive process might be preferable. 

While there is always this possibility of a blind spot, for discrete variables the one-at-a-time procedure is still frequently used. For continuous variables other procedures, which follow, should be used. 

A mixed DVR (discrete variable representation)43 for all the radial coordinates and basis set representations for the angular coordinates are used in the wavepacket propagation.44... 

The results for this scenario were obtained using GAMS 2.5/CPLEX. The overall mathematical formulation entails 385 constraints, 175 continuous variables and 36 binary/discrete variables. Only 4 nodes were explored in the branch and bound algorithm leading to an optimal value of 215 t (fresh- and waste-water) in 0.17 CPU seconds. Figure 4.5 shows the water reuse/recycle network corresponding to fixed outlet concentration and variable water quantity for the literature example. It is worth noting that the quantity of water to processes 1 and 3 has been reduced by 5 and 12.5 t, respectively, from the specified quantity in order to maintain the outlet concentration at the maximum level. The overall water requirement has been reduced by almost 35% from the initial amount of 165 t. 

The results for scenario 2 were obtained using GAMS 2.5/DICOPT. The NLP and MILP combination of solvers selected for DICOPT were MINOS5 and CPLEX, respectively. The overall formulation involves 421 constraints, 175 continuous variables and 36 discrete variables. Only 2 nodes were explored in the branch and... 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

For continuous variable optimization we consider (3-84) without discrete variable y. The general NLP problem (3-85) is presented here ... 

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