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A fundamental problem with the standard expansion techniques used to derive hydrodynamic equations is that any truncation after a finite number of terms, in general, leads to coarse-grainedequations that do not possess the proper structure of the timeevolution equations established for general nonequilibrium systems In the course of time, higher-order terms do arise so that an arbitrary truncation after a finite number of terms is inconsistent with the time evolution Although such expansions may be systematic to a certain order in an expansion parameter, thermodynamic consistency can make it necessary to keep certain higher-order terms We here show how a moment expansion with GENERIC structure can be constructed In 6, we have shown how coarse-graining can be carried out without destroying the GENERIC structure of beyond-equilibrium thermodynamics.

Here, however, we are not dealing with coarse-graining; we rather try to solve the thermodynamically admissible Boltzmann equation, and we want the approximate or truncated solution to be thermodynamically admissible, too For that purpose we can consider the solution procedure as a transformation to the parameters of the trial solution In the space of trial solutions,' the transformation from the variables of the original problem to the parameters of the trial solutions can be considered as a one-to-one transformation, and the formulas of Section 124 lead us to a GENERIC solution Now, how should we choose the trial solutions of the Boltzmann equation in terms of moments From a GENERIC perspective, Grad's choice (7123) is not ideal for the time-structure invariance of reversible dynamics This can be seen after recalling the arguments of Section 61.

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6, in particular, concerning the natural transformation behavior of the variables on different levels of description under space transformations Although the action of space transformations on single-particle distribution functions is not needed to construct the Poisson bracket on the Boltzmann level, it is important for the extended set of hydrodynamic variables constructed from the single-particledistribution function because, as moments, these variables inherit their space-transformationbehavior from the distribution functions, so that the construction.

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The deviations are calculated as the difference between the threshold (zero) and the monthly return, although positive differences are treated as a deviation of zero. The deviations are squared and summed. The monthly downside semivariance is the sum divided by 11 (12 months less 1). The annualized downside semivariance is 12 times the monthly semivariance. The downside deviation is the square root of the downside semivariance. 7.23 The Sharpe ratio and Sortino ratio can be calculated from the monthly data or annualized values. The arithmetic average is used in the numerator because the arithmetic average is used in the standard deviation calculation. The monthly Sharpe ratio is: 1.73% 5%/12 = .301 4.36% The Sharpe ratio based on annualized performance is:

Listing 7-2 (continued)

special state-space model, as was shown in Sec. 31.2, then there should also exist a correspondence between the RLS array algorithms of this chapter and the Kalman filtering array algorithms. This correspondence was developed by Sayed and Kailath (1993,1994b) and it is detailed in 31, Apps. 35.A and 37.A, and Probs. VIII.12 and VIII.13 -it is also covered in Haykin (1996,2000).

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coach the Six Sigma expert or consultant who sets a schedule, de nes results of a project, and who mediates con icts or deals with resistance to the program.

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