### Table 1: Some mathematical symbols used in the SMA formulation.

### Table 3.1: Structural comparison of different techniques on the level of mathematical program- ming formulations.

### Table 1 Notation

"... In PAGE 4: ...2. Mathematical formulation of the problem Table1 deFFnes the variables used hereafter in the mathematical formulation of the problem and in... ..."

### Table 3 demonstrates that our reduced mathematical formulations do speed up the process of multi-domain clock skew schedule. Especially, in benchmark circuits S3384, S6669, S9234 and S13207, the mathematical solver cannot solve the original formula- tions within twelve hours, but the mathematical solver can solve our reduced formula- tions within twelve hours. However, on the other hand, in benchmark circuits S15850 and S38584, although a lot of redundant binary variables and a lot of redundant con- straints can be pruned, the corresponding problem complexity still cannot be solved within twelve hours.

"... In PAGE 17: ...1 The Minimization of Max_Overlap Here, our objective function is to minimize the value of max_overlap under the con- straint of given clocking domains. Table3 demonstrates the advantage of exploiting the ASAP and ALAP schedule of each register to prune the redundancies. The column Origi- nal Formulation describes the original mathematical formulations proposed by [14].... In PAGE 17: ... We use the notation t/o to denote that the problem complexity cannot be solved within twelve hours. Table3 . The advantage of pruning redundancies for the minimization of max_overlap.... In PAGE 18: ... The column CPU time gives the CPU time (in seconds) spent in both the proce- dure ZONE_BASED_SCHEDULING and the mathematical solver. On the other hand, from Table3 , we can find the results of whole-circuit scheduling, which solves the whole circuit as a single zone. Clearly, in each benchmark circuit, the zone-based scheduling is much faster than the whole-circuit scheduling.... ..."

### Table 1: Symbols and Their De nitions 3.3 Theorems We postulate that the user has an \ideal quot; vector ~ q in mind, and that the distance of the sample vectors xi from this ideal vector ~ q is an generalized ellipsoid distance. Our goal is to \guess quot; ~ q and M to minimize the penalties. Obviously important samples (i.e., samples with high goodness scores vi) should have small distance from ~ q. Thus, the problem is mathematically formulated as follows:

"... In PAGE 8: ...1 0 1 0 1 q q q ellipsoid distance generalized weighted Euclidean Euclidean Figure 2: Isosurfaces for Distance Functions 3.2 Method Table1 gives a list of symbols used in the following discussion. The proposed distance function is D(~x; ~ q) = (~x ? ~ q)TM(~x ? ~ q); (2) or, equivalently D(~x; ~q) = n Xj n Xk mjk(xj ? qj)(xk ? qk); (3) where ~ q = [q1; : : :; qn]T is the \ideal quot; point, an n-d query vector and ~ x = [x1; : : :; xn]T is a fea- ture vector that corresponds to an entry in a database and apos;T apos; indicates matrix transposition.... ..."

### Table 1: Symbols and Their De nitions 3.3 Theorems We postulate that the user has an \ideal quot; vector ~ q in mind, and that the distance of the sample vectors xi from this ideal vector ~ q is an generalized ellipsoid distance. Our goal is to \guess quot; ~ q and M to minimize the penalties. Obviously important samples (i.e., samples with high goodness scores vi) should have small distance from ~ q. Thus, the problem is mathematically formulated as follows:

"... In PAGE 8: ...1 0 1 0 1 q q q ellipsoid distance generalized weighted Euclidean Euclidean Figure 2: Isosurfaces for Distance Functions 3.2 Method Table1 gives a list of symbols used in the following discussion. The proposed distance function is D(~x; ~ q) = (~x ? ~ q)TM(~x ? ~ q); (2) or, equivalently D(~x; ~q) = n Xj n Xk mjk(xj ? qj)(xk ? qk); (3) where ~ q = [q1; : : :; qn]T is the \ideal quot; point, an n-d query vector and ~ x = [x1; : : :; xn]T is a fea- ture vector that corresponds to an entry in a database and apos;T apos; indicates matrix transposition.... ..."

### Table 3: Bounds and formulations.

1996

"... In PAGE 24: ... While we have focused there on the physical meaning of these relations, we show in this section how they can be used to provide performance bounds for MQNETs by solving appropriate mathematical programming problems. We shall consider in what follows a linear cost function c#28x#29= X j2N c j x j ; and denote by Z the minimum cost achievable under the appropriate class of policies #28dynamic stable or static, nonidling and stable#29 policies, Z = min 8 #3C : X j2N c j x j j x 2X 9 = ; : Wehave summarized in Table3 several lower bounds and their corresponding mathematical pro- gramming formulations, obtained by selecting appropriate subsets of the constraints developed in previous sections.... ..."

### Table 1: The classical formulation used with the rst data set.

"... In PAGE 15: ... Since the two formulations are mathematically equivalent, we have been able to test and compare the classical Tsai-Lenz method with the two methods developed in this paper. Table1 , Table 2, and Table 3 summarize the results obtained with the three data sets mentioned above. The lengths of the translation vectors thus obtained are: ktXk = 93mm and ktY k = 681mm.... ..."

### Table 9 Relationship of Presence of Accountability System to Improvements in NAEP Mathematics Performance

"... In PAGE 39: ... The latter concentrates on the period of most activity in accountability but relies on the growth formulation with possible explicit measures of state differences to isolate the effects of accountability systems. Table9 presents the basic estimates of the effects of accountability systems on growth in student achievement. The simplest version (columns 1 and 5) looks at whether the state has some form of accountability system in place during the period of observation.... ..."

### Table 1. Notations

"... In PAGE 2: ...2. Formulation of Dynamic Segmentation Refer to Table1 for mathematical notations. Dynamic seg- mentation can be formulated by using the framework of Dy- namic Programming (DP).... ..."