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5.4 CUBIC SPLINE INTERPOLATION The previous application shows several ways of plotting a curve using MFC functions when the curve function is given. It is also possible to draw a curve when its function is not given. In this case, the curve can be generated from a set of points either through interpolation or approximation. In interpolation, the curve is drawn so that it passes through all the given points. In approximation, the curve may pass through some or none of the points. A relationship is derived from these terminologies: interpolation is a form of approximation in which the curve passes through all the given points. A curve drawn through interpolation or approximation is a generalization of its close relationship to the points. Curve interpolation can be performed numerically using many classical techniques such as the Lagrange, Newton, and Hermite methods, which produce polynomials. In approximation, several methods based on the least-squares method are applied. The least-squares method performs approximation by minimizing the sum of the squares of the errors to produce low-degree polynomials. Interested readers can refer to Burden and Faires [1] for details on these techniques. An interesting curve that finds a lot of applications in mathematics is the spline. A spline is a set of piecewise continuous functions obtained through the interpolation of several points. A spline of degree n is defined as follows: Sj(x) = aj + bj(x xj) + cj(x xj)2 + . . . + dj(x xj)n (5.4)

(q) (q 1)

where [I] represents an identity matrix. The above relation must be true for all Vi s. That happens when

Thus:

markings is much faster. Indeed, the number of successors of a symbolic marking is signi cantly lower than that for an ordinary marking [CHI 93b]. Most of these methods have been implemented in the GreatSPN tool [CHI 95]. 10.2. Markovian aggregation All aggregation techniques substitute a simpler system, supposed to re ect the original system with regard to some criteria, for a more complex system. In the performance evaluation area, this criterion is most often the fact that a synthesis of the stationary indices of the original system may be computed from the stationary indices of the reduced system. In the previous chapter, we saw how to approximate the stationary throughput of a stochastic marked graph (SMG) from the iterative evaluation of reduced SMG. In this case, aggregation lies in the model which generates the stochastic process. Conversely, we can look for aggregation conditions of the stochastic process leading to exact results. Let us look at Figure 10.2. On the left, we have an excerpt of a discrete time Markov chain (DTMC). States are grouped in two subsets E k = {ek , ek } and E h = 1 2 {eh , eh , eh }. We note that: 1 2 3 Pr Xn+1 E h | Xn = ek = Pr Xn+1 E h | Xn = ek = 1 2 5 6

Table 7.3 shows the steps for finding the minimum spanning tree of the graph using Kruskal s algorithm starting at v1. The problem requires seven steps for converging to its solution using this algorithm. Prim s Algorithm Prim s algorithm is another common method for solving the minimum spanning tree problem. The algorithm begins with an empty graph T. The strategy in Prim s

Third: Translate Only What Matters Working with corporate and local market specialists, you should determine exactly what people are look-

In Part II it was assumed that an RMP, the SHAMPU (Shape, Harness, And Manage Project Uncertainty) process, is initiated toward the end of a project s plan stage The rationale behind this was that toward the end of the plan stage is the easiest place to start for rst-time users of an RMP, with the best chance of success By this stage of a project a fair amount of information about the project is available, but scope for signi cant contributions to project performance by pursuit of formal risk management is likely to remain If formal risk management is introduced in an organization on a voluntary basis, a rst-time application to a project at this stage offers the best organizational learning opportunities It also makes sense to choose a project to learn on that has three characteristics: 1 the project has been very well managed to date; 2.

As a developing technology, pharmacogenomics presently is caught between these two approaches and is likely to remain in that uncomfortable middle ground for the foreseeable future (Foster et al, 2001) On the one hand, individualized treatment (arrived at by progressively narrowing the observable genetic characteristics until they comprise an individual genotype) is the long-term goal of pharmacogenomics However, neither the technology nor the clinical practice nor the ethical protections are yet suf ciently matured to make individualized medical treatment a reality Consequently, much of pharmacogenomic research and, presumably, initial clinical applications (once those become available) are and will be phrased in terms of racial, ethnic, and other social groups of people who are the more likely suspects to respond to a particular drug.

Interviewers are most impressed by candidates who can actively demonstrate knowledge of their company. You can really stand out by asking questions that show you ve done your research and want to delve even further into understanding the organisation. However, devising questions that are totally pertinent requires more effort on your part. The trick is to mention in passing the source of your knowledge before asking your question. Look at these examples for how to drop your sources into conversation: