### Table 1.1. Parameterization of canonical simple ruled quadrics

### Table 2: Number of 107-\ ops quot; taken for computing the gradient signals in di erent state-space parameterization situations as well as the signals z(t; ; s) simulated by equation (6) for the signals s(t) present in expression (4).

"... In PAGE 10: ...nd 10-states system, i.e. p = m = 5 and n = 10. The number of data measurements is chosen identical to N = 1000. In Table2 , we have presented the numerical complexity (as given by the Matlab function \flops quot;) of the two method for computing the gradient signals in di erent parameteriza- tion situations; namely, in the full (F) and the canonical (C) parameterizations with four sets of state-space matrices, i.... In PAGE 11: ...) spent for computing the gradient signals in di erent state-space parameterization situations. in Table2 on a Sun Ultra 1/170E with 128 MB RAM by use of the Matlab platform for the matrix evaluations. When seconds are counted, the improvement performed by the present method takes a more real-life relevance.... ..."

### Table 3: Parameterizations of Altitude

"... In PAGE 11: ... This work expands upon parameterized sub-optimal minima found in that research and shows that by appropriate choice of parameterization better sub-optimal solutions can be found which achieve greater fuel savings. Table3 lists the various altitude pro les for each of the ve parameterizations considered in this work. 0.... ..."

### Table 1 . Model parameterizations.

"... In PAGE 1: ... 1996). A list of the parameterizations used for these simulations can be found in Table1 . The SCM has been used to validate GCM tests of several different cloud-radiation schemes with observa tional data (Lee et al.... ..."

### Table 6. Canonical redundancy analysis Canonical

2003

"... In PAGE 7: ... Although the first and second canonical functions are ignificant according to the above analysis, it is mmended that redundancy analysis be utilized to etermine which functions should be used in the terpretation [37]. Redundancy is defined as the ability of f independent variables, taken as a set, to explain the ariation in the dependent variables taken one at a time Table6 summarizes the redundancy analysis for the dent and independent variables for the two canonical unctions that were found to be significant by using the easure of model fit. The results indicate that the first onical function accounts for the highest proportion of otal redundancy (93.... ..."

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### Table 6. Canonical redundancy analysis Canonical

2003

"... In PAGE 7: ... Although the first and second canonical functions are ignificant according to the above analysis, it is mmended that redundancy analysis be utilized to etermine which functions should be used in the terpretation [37]. Redundancy is defined as the ability of f independent variables, taken as a set, to explain the ariation in the dependent variables taken one at a time Table6 summarizes the redundancy analysis for the dent and independent variables for the two canonical unctions that were found to be significant by using the easure of model fit. The results indicate that the first onical function accounts for the highest proportion of otal redundancy (93.... ..."

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### Table 2: Parameterization of k

1995

"... In PAGE 10: ... It was heartening to see that any possible size increase due to loss of precision was almost always overtaken by the reduction due to many-to-one object name mapping. Table2 summarizes our observations. Note that for deriv2, the size of alias solution and precision of function resolution su ered for a lower k.... ..."

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### Table 1: Parameterized vertices

"... In PAGE 9: ... How- ever, each vertex is given as the intersection of hyperplanes defined by the constraints with faces of A1BG. Figure 2 shows the general position of these intersection points, and Table1 presents them as a list. The third column of Table 1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope.... In PAGE 9: ... Figure 2 shows the general position of these intersection points, and Table 1 presents them as a list. The third column of Table1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope. Such conditions we will subsequently encounter in great numbers; they are formally introduced by the following definition.... In PAGE 10: ...(r)=0.4, I(s)=0.2 (0,0,1,0) (0,0,1,0) I(r)=0, I(s)=0 (0,0,1,0) Figure 1: Polytopes for different parameter values 3 5 2 4 8 6 7 1 Figure 2: General vertex positions problem statement for generating a complete parameterized vertex list can now be refined as follows: given input constraints BV, we have to find a list DACY BM APCY B4BD AK CY AK C5B5 (13) where each DACY is a parameterized vertex as in (7), and the APCY are lists of p-constraints, such that for every parameter instantiation C1 the set of vertices of A1B4C1B4BV B5B5 is just CUC1B4DACYB5 CY C1 satisfies APCYCV (where, naturally, C1 satisfies AP iff for every APCX AH D4CX AO BC BE AP: C1B4D4CXB5 AO BC). Table1 provides this list for the input constraints (9) and (10). To obtain a systematic method for generating such a list it is convenient to consider one by one the different faces of A1BEC3, in... In PAGE 12: ...icularly suitable method is fraction free Gaussian elimination (see e.g. [7]). This is a variant of Gaussian elimination that avoids divisions, which is useful for us, as otherwise we would have to divide by symbolic expressions that might be zero for some parameter values and nonzero for others, thereby requiring us to make a number of case distinctions. As an illustration for the working of the algorithm we retrace how vertex 8 in Table1 was generated. This vertex is the solution of the system (14)-(16) defined by CS BP BE, C0 BP CUBDBN BEBN BFCV and the (then mandatory) selection of both constraints CRBDBN CRBE for (16).... In PAGE 14: ... To illustrate the general method, we continue with our example, taking C8 B4BMBT CY BUB5 to be the target probability of the inference rule to be derived. The probability of BMBT given BU at the vertices listed in Table1 is evaluated by computing DABFBPB4DABDB7DABFB5, which leads to the values listed in Table 4. Note that the possible values of C8 B4BMBT CY BUB5 are still annotated with the parameter constraints on the vertices at which they are attained, and that for vertices 5 and 8 the new p-constraint D7 BO BD has been added.... In PAGE 21: ... Minimal irredundant sets of values for minimization and maximization of C8 B4BT CY BU CM BWB5 are indicated by the +-marks in the columns 8 and 9, respectively. The final bound functions we obtain now are C4B4D6BN D8BN D9BN DAB5 BP minCJD6BPDA BM AQ BN DA BQ BCCL (40) CDB4D6BN D8BN D9BN DAB5 BP maxCJBC BM D6 BP BCBN DA BP BDBN BD BM AQ BN D9 AK D6BN D6 BQ BCBN BD BM AQ BN D9 AK D8BN D8 AK DABN D8 BQ BCBN BD BM AQ BN D6 AK D9BN D9 AK D8BN D9 BQ BCBN BD BM AQ BN DA AK D8BN DA BQ BCBN D8BPD9 BM AQ BN D8 AK D9BN D9 BQ BCCLBM (41) where the p-constraints AQ suppressed in Table1 have been reinstated. Remembering the con- ventions min BN BP BDBN max BN BP BC, and taking into account that the conditions D6 AK D8BN D9 AK DA are taken for granted in (29), these functions can be seen to be the same as (30) and (31).... ..."

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### Table 1: Model parameterization

2003

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