INDEX problem

Attention is drawn to a special feature of this Volume—inclusion of a generalized Cumulative Index to Volumes 1-10. It is believed that use of this Cumulative Index, in conjunction with the detailed Cumulative Subject Index to Volumes 1-5 (in Volume 6) and the detailed, individual Subject Indexes to Volumes 6-10, will afford the reader speedy access to sources of detailed information in the various Volumes. This solution to the indexing problem was chosen for economic reasons. 

This example illustrates some of the theoretical difficulties with profile optimization problems. Again, we see that care must be applied to the discretization of the DAE system. Also, if the discretization cannot handle high index problems that are encountered in the optimization, we may be forced to use a suboptimal parameter optimization approach. 

The constant-pressure formulation results in a high-index problem even with implementation of instead of A as the dependent variable. This results from the absence of any time derivatives for u. Attempts to solve the methane-ignition problem with the constant-pressure formulation were generally unsuccessful, except by significantly relaxing the error control on A and u. Even then, while the solutions appear generally correct, they exhibit unstable behavior near fast transients, particularly on A. 

The basic idea of the active constraint strategy is to use the Kuhn-Tucker conditions to identify the potential sets of active constraints at the solution of NLP (4) for feasibility measure ip. Then resilience test problem (6) [or flexibility index problem (11)] is decomposed into a series of NLPs with a different set of constraints (a different potential set of active constraints) used in each NLP. 

Floudas and Grossmann (1987b) have shown that for HENs with any number of units, with or without stream splits or bypasses, and with uncertain supply temperatures and flow rates but with constant heat capacities, the active constraint strategy decomposes the resilience test (or flexibility index) problem into NLPs which have a single local optimum. Thus the resilience test (or flexibility index) also has a single local optimum solution. 

The preceding theorem describes an operability test for class 2 HENs. Similarly, by omitting the energy recovery constraint from flexibility index problem (15) or resilience index problem (19), an operability index could be defined for class 2 HENs. 

Step 6. Apply the active constraint strategy to the flexibility index (F) at the stage of structure (without the energy recovery constraint). The form of this flexibility index problem is described in a later section, (a) If F a 1, then the HEN is operable in the specified uncertainty range. Stop, (b) If F< 1, then add the critical point for operability as another period of operation and return to step 5. 

The dynamic formulation of the model equations requires a careful analysis of the whole system in order to prevent high-index problems during the numerical solution (144). As a consequence, a consistent set of initial conditions for the dynamic simulations and suitable descriptions of the hydrodynamics have to be introduced. For instance, pressure drop and liquid holdup must be correlated with the gas and liquid flows. 

Pantelides [27, 28], in his work with Sargent, defined the index in a manner that exposes its potential to cause problems in initialization as well as in the integration error. They too showed that the index problem can be eliminated by differentiation. Noting that only some of the equations need to be differentiated, they use a method based on the structural properties of the equations to discover these equations. They cite several examples in which the index problem is almost certain to occur in setting up and solving dynamic simulation models, e.g., calculations of flash dynamics and problems in which the trajectory of a state variable is specified. 

Finally, one should be concerned about the index problems for mixed sets of PDEs and algebraic equations. How many problems have been solved where some derivable independent equations have been missed ... 

Searchers need not become expert indexers of chemical literature, but the better they understand indexers problems and answers, the shorter the path to information needed from an index. 

It is the custom of CA to publish an Introduction to its Subject Indexes. The indexes are built to stand on their own feet. The introduction is not essential to the ready and effective use of the CA Subject Index. Nevertheless, for the best results in the use of any index the user must meet the index maker part way in understanding the indexing problems and nomenclature in particular. Use of the information in our Subject Index Introductions is recommended to the searcher who is doing more than incidental searching. 

A consideration of the indexing problems associated with the completed abstract throws considerable light on the philosophy of abstracting itself. 

We conclude this section with a simple notion it is impossible to solve the crystal structure of a material using an incorrect unit cell. Thus, proper indexing of the experimental powder diffraction pattern is of utmost importance, and in this chapter we shall consider various strategies leading to the solution of the indexing problem and how to find the most precise unit cell dimensions. 

Solving the indexing problem becomes a matter of identifying the differences that result in whole numbers when divided by a common divisor and c, respectively). The expected whole numbers are shown in Table 5.14 through Table 5.16 for several small h, k and /. It only makes sense to consider these small values because successful indexing is critically dependent on the availability of low Bragg angle peaks, which usually have small values of indices. 

The indexing problem is essentially a puzzle (. . . ). It would be quite an easy puzzle if errors of measurement did not exist. 

The most common and widely used indexing programs are ITO, TREOR " and DICVOL91. All three classic programs are present in the indexing Crysfire suite. Their approach to the indexing problem is different and will be briefly described. 

The indexing problem is usually solved in a few minutes if (a) the symmetry is not lower than monoclinic (b) the cell volume is less than 2000 A (c) the cell parameters are less than 20 A. More computing time is required for triclinic symmetry indeed the main drawback of the McMaille approach is the high request of computing time in the case of low crystal symmetry. 

Now, these equations have been written with formal integration, but of course in the numerical implementation only the evaluation of the integrand at the grid points is required. Therefore, it is evident that the derivatives lyl, (Vpo)M and (Vpp)tyl can be evaluated on the grid independently of the indices pv, and so the four-index problem is decomposed into two independent two-index procedures, avoiding the potential computational bottleneck. By comparison, the resolution of the identity technique proposed by Komornicki and Fitzgerald [66] gives an approximate result in terms of a product of one two-index and two three-index quantities. 

J. C. Geniesse, A Study of the ASTM Viscosity-Index Problems, Proceedings of the 5th World Petroleum Congress, Section V-Paper 30A, pp. 407-409 (1959). 

The integration of a DAE system can be performed by transformation in an ODE system. It is worthy to note that this operation might be confronted with the index problem. Index is the minimum number order of differentiation needed to transform a DAE system into a set of first-order ODEs. Problems of index one can be solved by means of standard differentiation methods. When the index is higher than one then the DAE system needs a special treatment. Modem codes have capabilities for automatic detection of index higher-than-one, diagnose the problem and suggest modifications. 

To illustrate the index problem let s consider the following simple DAE system ... 

However, in the context of a practical simulation with a large number of variables the mathematical diagnosis of an index problem is not trivial (Pantelides Barton, 1992). The user might prevent such troubles by examining carefully the problem, particularly the definition of specifications, which can be different as those met in steady state simulation. 

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