### Table A.2: Driver routines for linear equations, Ax = b Type of Matrix Routine

1999

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### Table 2. Symmetry Reduction for D4 Without Fixed Points Let us now summarize our method of symmetry reduction: Suppose that a discretization of a boundary integral equation with geometric symmetries leads to a linear system of equations Ax = b with the (full) system matrix of size (n; n) which is equivariant under a group of permutations ? obtained from a group of symmetries (orthogonal transformations). Assume further that the group of permutations ? acts xed point free. Then the method consists of the following steps: Method 1.

"... In PAGE 9: ... Note again that we still have 16 unknowns in this example. Table2 describes the symmetry reduction in a schematic way. The number n of elements has to... In PAGE 13: ... All blocks of (21) are invertible if and only if A is invertible. Going back to our simple example of Figure 4, the symmetry reduction analogous to Table2 is shown in Table 3. Total number of original unknowns n = 16 Total number of transformed unknowns n = 16 Irreducible representation Size of transformed unknowns producing a subsystem in this subsystem r2 ( n=8 ? 1 ; 1 ) = ( 1 ; 1 ) r3, r4 ( n=8 ; 1 ) = ( 2 ; 1 ) r1 ( n=8 + 1 ; 1 ) = ( 3 ; 1 ) r5 ( n=4 ; 2 ) = ( 4 ; 2 ) Table 3.... ..."

### Table 1: AX.

"... In PAGE 2: ... Thus R1 _ R2 is again a test, that succeeds when we can proceed with either an R1- or an R2-step. Table1 contains a nite set of axioms, AX, that is intended to completely axiomatise equational validity in dynamic relation algebras. We write ` t1 = t2 if this equation is derivable from the equations in AX and the rules of equational logic.... ..."

### Table 6 Performance comparison for N-body

"... In PAGE 7: ... We ran the simu- lations on 2-, 4- and 8-machines, respectively. The results are shown in Table6 . In the case of 2-machines, eight tasks are created.... In PAGE 8: ... Given an initial assignment, our method converges to the same assignment as that produced by the batch method in all the three cases of 2-, 4- and 8-machines. Table6 shows that a near-linear speedup can be achieved as we increase the number of machines in the simulations. A classic Gaussian elimination algorithm has been used to solve linear equations (AX b).... ..."

### Table 1: Model coefficients for the AX transport equation

### Table A.3 Equation of a linear plane that was fit using a least-squares regression through the elevation of the water table in permanent monitoring wells during each of eighteen rounds of quarterly sampling. The plane is in an x,y,z coordinate system where x increases toward the east, y increases toward the north, and z increases with elevation above mean sea level. The equation is in the form Ax+By+C+z where x and y are the grid location in UTM meters and z is the elevation of the water table in feet.

### Table A.4 Equation of a linear plane that was fit using a least-squares regression through the elevation of the water table in permanent monitoring wells during each of fourteen rounds of monthly sampling. The plane is in an x,y,z coordinate system where x increases toward the east, y increases toward the north, and z increases with elevation above mean sea level. The equation is in the form Ax+By+C+z where x and y are the grid location in UTM meters and z is the elevation of the water table in feet.

### Table 2: Quadratic curve estimation for y = ax + bx2 (y = number of active pairs, x = number of words)

2006

"... In PAGE 6: ... For reasons of space, we only display the data from DDT, but the PDT data exhibit very similar patterns. Both treebanks are represented in Table2 , where we show the result of fitting the quadratic equation y = ax + bx2 to the data from each condition (where y is the num- ber of active words and x is the number of words in the sentence). The amount of variance explained is given by the r2 value, which shows a very good fit under all conditions, with statistical significance beyond the 0.... In PAGE 6: ...001 level.4 Both Figure 2 and Table2 show very clearly that, with no constraints, the relationship between words and active pairs is exactly the one predicted by the worst case complexity (cf. section 4) and that, with each added constraint, this relationship becomes more and more linear in shape.... ..."

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