### Table 1: Correspondence between particular con gurations in the original and dual com- binatorial maps

1999

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### Table 1. The main goal of the third step is to store the data base in an application friendly way. To this end, we provide two representations of the data base. An explicit version of the data base contains one point set for each planar order type, all in 16-bit integer representation.

in Abstract Order Type Extension and New Results on the Rectilinear Crossing Number (Extended Abstract)

"... In PAGE 2: ... Table1 : Number of order types of cardinality n = 11. Supporting the reliability in the construction of our data base, all algorithms to generate the complete data base of abstract order types are of purely com- binatorial nature.... ..."

### Table 5: Average fraction of available surplus obtained by producers in each net- work for the three protocols.

2000

"... In PAGE 29: ...Table 5: Average fraction of available surplus obtained by producers in each net- work for the three protocols. Table5 shows the average fraction of available surplus obtained by producers, respectively, in each network, for the three protocols. Recall that in the com- binatorial auction, extra surplus (available as specified by the strategic bids) not taken by producers is distributed evenly among all consumers.... ..."

Cited by 35

### Table 1. Compressed pattern matching.

1999

"... In PAGE 1: ... 1 Introduction Compressed pattern matching is one of the most interesting topics in the com- binatorial pattern matching, and many studies have been undertaken on this problem for several compression methods from both theoretical and practical viewpoints. See Table1 . One important goal of compressed pattern matching is to achieve a linear time complexity that is proportional not to the original text length but to the compressed text length.... ..."

Cited by 12

### Table 1 summarizes the relationship of familiar vector calculus operators to the combinatorial graph theoretic operators de ned above and Table 2 summarizes the relationship of equations in both domains. With these de nitions in place, we can determine how to solve for the harmonic function that interpolates values on free (\interior quot;) nodes between values on xed (\boundary quot;) nodes. A combinatorial formulation of the Dirichlet integral (1) is

2003

"... In PAGE 8: ...Vector calculus Combinatorial Gradient r A Divergence r AT Curl r r K Laplacian r r AT A Beltrami r C r AT CA Table1 : Correspondence between continuum di erential operators and com- binatorial di erential operators on graphs. C represents a constitutive ma- trix relating ux to ow, e.... ..."

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### Table 1: Benchmarks

"... In PAGE 5: ... We compare phase-based adaptation with hardware interval-based adaptation. Our test suite is given in Table1 . We pick programs from differ- ent sets of commonly used benchmarks to get an interesting mix, representing common computation tasks in signal processing, com- binatorial optimization, structural and unstructured N-body simula- tions, compiler, and database.... ..."

### Table 1: Number of function evaluations required to minimize the linear minimum surface (LMINSURF) and Broyden tridiagonal (BROYDN3D) functions in various dimensions (from [28]).

2006

"... In PAGE 14: ... We also note that it is not crucial to solve the com- binatorial problem of Step 2 exactly: we may, for instance, use a simple greedy algorithm to identify a suitable index set I. Does this work in practice? Table1 shows the performance in terms of complete function evalua- tions of the method just described (under the head- ing PS , for partial separability), compared to the same algorithm without the exploitation of problems structure (under the heading no st. ).... ..."

### Table 2 Comparison of DSM-based GA research on process sequencing

"... In PAGE 5: ...arded. In contrast, Whitfield et al. H2085144H20852 mainly investigated and compared GA crossover and mutation operators for sequencing DSMs. Table2 summarizes the DSM/GA research. These previ- ous publications recognized the power of GAs in the field of com- binatorial problems but did not exploit its entire repertoire.... ..."

### Table 2. Tabular representation of the model space by explicitly listing each network variant

"... In PAGE 7: ...odels. Figure 4 displays the structure of this model variant space. Here, the models related by one elementary generalization or spe- cialization step are linked by an arrow. In the rows of Table2 these model variants are listed. Fortunately, the model space of all com- binatorially possible models contains only few models biologically meaningful.... ..."

### Table 2. Elimination lemmas

2004

"... In PAGE 3: ... Note that the de- pendency graph of the constructions must be cycle free. To eliminate a point from the goal we need to apply one of the elimination lem- mas shown on Table2 on page 5. This table can be read as follows: To eliminate a point Y , choose the line corresponding to the way Y has been constructed, and apply the formula given in the column corresponding to the geometric quantity in which Y is used.... In PAGE 4: ... We rst translate the goal (A0B0 k AB) into its equivalent using the signed area: SA0B0A = SA0B0B Then we eliminate compound points from the goal starting by the last point in the order of their construction. The geometric quantities containing an oc- currence of B0 are SA0B0B and SA0B0A, B0 has been constructed using the rst construction on Table2 with = 1 2: SA0B0A = SAA0B0 = 1 2SAA0A + 1 2SAA0C = 1 2SAA0C and SA0B0B = SBA0B0 = 1 2SBA0A + 1 2SBA0C The new goal is SAA0C = SBA0A + SBA0C Now we eliminate A0 using: SCAA0 = 1 2SCAB + 1 2SCAC = 1 2SCAB SABA0 = 1 2SABB + 1 2SABC = 1... In PAGE 11: ... This tactic (called eliminate_all) rst searches the con- text for a point which is not used to build another point (a leaf in the dependency graph). Then for each occurrence of the point in the goal, it applies the right lemma from Table2 by nding in the context how the point has been constructed and which geometric quantity it appears in. Finally it removes the hypotheses stating how the point has been constructed from the context.... In PAGE 12: ...this classical reasoning step. As noted before, the elimination lemmas given in Table2 on page 5, do eliminate an occurrence of a point Y only if Y appears only one time in the geometric quantity (A,B,C and D must be di erent from Y ). If Y appears twice in S, this is not a problem because then the geometric quantity is zero, and so already eliminated by the simpli cation phase.... ..."

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