### Table 3. Performance of checking races for transactional Multiset

"... In PAGE 9: ... This commit action was inserted in the synchronization event list where the first lock release in a transaction would have been inserted if we were not explicitly considering transactions. We measured, for different numbers of threads sharing a multi- set of size 10, the runtime with race checking enabled for the trans- actions as described in Sections 4 and 5 ( Table3 ). The table also reports the number of shared variable accesses and the number of transactions in the executions.... ..."

### Table 6: Lagrangean algorithm and CPLEX results - Instance set III

"... In PAGE 17: ... Actually, the largest instances of this set overestimate the size of real cellular networks. Table6 reports the results of the Lagrangean algorithm and CPLEX. The meaning of each column is the same previously defined for table 2.... ..."

### Table 3 Upper Multi-set Bounds for Single Input Case

1999

"... In PAGE 11: ...9,marg=0.2,ct=1} Figure 12 Occurrence graph of Single Input Initial Marking for Untimed Net Table3 shows the upper multi-set bounds for the CP net. While the integer bound provides information about the maximum and minimum number of tokens in a place, the multi-set bound provides information about the value that the tokens can carry.... ..."

Cited by 1

### Table 3 Upper Multi-set Bounds for Single Input Case

1999

"... In PAGE 11: ...9,marg=0.2,ct=1} Figure 12 Occurrence graph of Single Input Initial Marking for Untimed Net Table3 shows the upper multi-set bounds for the CP net. While the integer bound provides information about the maximum and minimum number of tokens in a place, the multi-set bound provides information about the value that the tokens can carry.... ..."

Cited by 1

### Table 2: Computation Time of TS in Seconds for SMALL Problem Instances (Exact Optimal Solutions Were Obtained in All Cases)

1999

"... In PAGE 22: ...lgorithms in Section 6.4. 6.1 Results for Problem Set SMALL When applied to instances in SMALL, algorithm TS obtained optimal solutions for all the tested instances within a few seconds, as shown in Table2 . There are five instances for each size, and each instance was solved five times using different random seeds.... ..."

Cited by 17

### Table 1: Simiiariiies and differences stable sets vs. stable multi-sets

"... In PAGE 19: ... In particular, properties of the associated polytope were identified. In Table1 , the obtained results are compared with their analogues for the stable set polytope. Note that Table 1 does not show all conditions for the model inequalities to define facets.... In PAGE 19: ... In Table 1, the obtained results are compared with their analogues for the stable set polytope. Note that Table1 does not show all conditions for the model inequalities to define facets. The table shows that many of the results obtained for stable sets have their counterpart for stable multi-sets.... ..."

### Table 1. Comparison of the factorial HMM on four problems of varying size. The negative log likelihood for the training and test set, plus or minus one standard deviation, is shown for each problem size and algorithm, measured in bits per observation (log likelihood in bits divided by NT) relative to the log likelihood under the true generative model for that data set.7 True is the true generative model (the log likelihood per symbol is de ned to be zero for this model by our measure); HMM is the hidden Markov model with KM states; Exact is the factorial HMM trained using an exact E step; Gibbs is the factorial HMM trained using Gibbs sampling; CFVA is the factorial HMM trained using the completely factorized variational approximation; SVA is the factorial HMM trained using the structured variational approximation. M K Algorithm Training Set Test Set

1997

"... In PAGE 14: ... This provides a measure of how well the model generalizes to a novel observation sequence from the same distribution as the training data. Results averaged over 15 runs for each algorithm on each of the four problem sizes (a total of 300 runs) are presented in Table1 . Even for the smallest problem size (M = 3 and K = 2), the standard HMM with KM states su ers from over tting: the test set log likelihood is signi cantly worse than the training set log likelihood.... ..."

Cited by 279

### Table 1. Comparison of the factorial HMM on four problems of varying size. The negative log likelihood for the training and test set, plus or minus one standard deviation, is shown for each problem size and algorithm, measured in bits per observation (log likelihood in bits divided by NT) relative to the log likelihood under the true generative model for that data set.7 True is the true generative model (the log likelihood per symbol is de ned to be zero for this model by our measure); HMM is the hidden Markov model with KM states; Exact is the factorial HMM trained using an exact E step; Gibbs is the factorial HMM trained using Gibbs sampling; CFVA is the factorial HMM trained using the completely factorized variational approximation; SVA is the factorial HMM trained using the structured variational approximation. M K Algorithm Training Set Test Set

1997

"... In PAGE 14: ... This provides a measure of how well the model generalizes to a novel observation sequence from the same distribution as the training data. Results averaged over 15 runs for each algorithm on each of the four problem sizes (a total of 300 runs) are presented in Table1 . Even for the smallest problem size (M = 3 and K = 2), the standard HMM with KM states su ers from over tting: the test set log likelihood is signi cantly worse than the training set log likelihood.... ..."

Cited by 279

### Table 3. Examples of conformable and not conformable multisets. From De nition 3.1 it follows that if is -conformable then so is each multiset ; 6= 0 . In particular, the set sp( ) = f j 2 g is itself -conformable.

"... In PAGE 8: ...1 De nition 3.1 (illustrated in Table3 ) means that each value v contained in one of the original types is contained in by (a) (then it is and remains weak) or, if it is strong, by (b) does not intersect with any value w from or any other set (including itself, if is contained twice or more in ) in and hence, is safely combinable with other strong values. The last condition implies, by the use of multiset minus (?) rather than n, that the types with membership in exceeding 1 must be completely weak.... ..."