### TABLE I Quadratic placementand#0C-regularization compared against optimal linear programming results. Total wirelength

### Table 1. Inversion formula Inversion algorithm Algorithm for solving

1995

"... In PAGE 2: ...Boros-Kailath-Olshevskyalgorithm [BKO] Three-term Verde{Star formula [VS] Calvetti{Reichel Heinig-Hoppe-Rost Vandermonde and algorithm [CR] algorithm [HHR] and matrices Gohberg{Olshevsky Higham algorithm formula [GO2] [Hig1], [Hig2] All the algorithms in Table1 are called fast algorithms, because their complexity of O(n2) op- erations compares favorably with the O(n3) operations of general purpose algorithms like Gaussian elimination. Now inversion and fast solution of a linear system are two classical applications of the concept of displacement structure, see [KS2].... In PAGE 2: ...1), but also certain natural generalizations thereof, which we shall call polynomial Vandermonde-like matrices. We shall see that the concept allows us to nicely unify and extend the results in Table1 . The main reason is that the displacement structure is essentially preserved under inversion, see [KKM] and [KS2].... In PAGE 4: ... However, a natural question is how these three new classes of matrices relate to each other? We show that no matter which of the three displacement operators is chosen, all three de nitions lead in fact to the same class of matrices, which we shall call the class of polynomial Vandermonde-like matrices. Furthermore we show that all the results in the bottom line of Table1 can be carried over to this wider class of matrices. In particular, we derive (a) two inversion formulas; (b) a structured implementation of Gaussian elimination with partial pivoting, and (c) an inversion algorithm for polynomial Vandermonde-like matrices.... ..."

Cited by 14

### Table 4. Inversion formula Inversion algorithm Algorithm for solving

1995

"... In PAGE 4: ...All the algorithms in Table4 have complexity O( n2) operations, where is the displacement rank of a matrix. 0.... ..."

Cited by 14

### Table 1: Semigroups satisfying x

"... In PAGE 2: ... 3. Is the congruence #18 m;n on A #03 decidable? In other words: Is the free word #28iden- tity#29 problem for the variety de#0Cned by #28associativityand#29x m+n = x n recursively solvable? The top-left case #28m =1,n =0#29of Table1 is a trivial algebra. Thenextcaseof the top row#28m; n = 1#29, the #28free#29 idempotent semigroups, are called #28free#29 bands.... In PAGE 3: ... Green and Rees #5B 1952 #5D proved that x m = 1 is #0Cnite if and only if the semigroup x m+1 = x is. So columns 1 and 2 of Table1 haveidentical #0Cniteness properties. In particular, bands #28xx = x#29 are #0Cnite.... ..."

### Table 1. Sizes of some investigated semigroups.

2003

Cited by 3

### Table 1: Examples of intermediate free semigroups and bases of identities for V

### Table 1: Summary of polyomino inversion relations

"... In PAGE 13: ... We shall generalize these results to polygons of any width. Table1 summarizes the self- reciprocity properties we have established. Most of them can be proved in various ways.... In PAGE 15: ... This very general result will be derived in Section 4. Likewise, the inversion relations for three-choice polygons and staircase polygons with a staircase hole, given in Table1 , are also special cases of more general formulae which will be derived in Section 5.... In PAGE 22: ... Theorem 4.1 then gives Hm(1=y; 1=q) = ? 1 yqm Hm(y; q); which implies G(x; y; q) + yG(xq; 1=y; 1=q) = 0: (33) Note that the rst two self-reciprocity relations of Table1 depend quadratically on the variable q: for this reason, they only yield an inversion relation for q = 1. 5 Self-reciprocity via Stanley apos;s general results 5.... ..."

### Table 1. Number of PMUL, SMUL, and inversion of HECADD and HECDBL

"... In PAGE 6: ... For these pairs which are expressed in the Table with the symbol , we can use the proposed nite eld multiplication PMUL, but for other cases, we compute sequentially via SMUL not in parallel via SSE2. We compare the non-parallel formulae [30] with our parallel formulae regarding calculation cost in Table1 . M, and I are the time required for multiplication, inversion of the binary eld, respectively.... ..."

Cited by 1

### Table 1: Computational formulas for the new annealing schedule.

"... In PAGE 2: ... 3. Applications The computational formulas for the new annealing schedule are listed in Table1 . During the initialization Paper 22.... In PAGE 6: ... 4. Practical considerations We observe from Table1 that our annealing schedule uses floating point computations heavily. To reduce the number of floating point operations, we approximate our annealing schedule by computing the new inverse temperature after every ten steps instead of one: This speeds up our annealing schedule by a few percents without degrading the quality of the final solutions.... ..."