### Table 2: +(V ) and SQ for the representations in Theorem 2.5. By convention, !0 = 0 and !n = 0 for SLn.

### Table 2: Complexity of inference and Space Efficiency of Theorem Representations

1996

Cited by 26

### Table 2: Complexity of inference and Space Efficiency of Theorem Representations

1996

Cited by 6

### Table 1. Situations to which the Main Theorem applies G

"... In PAGE 6: ... 2. Main Theorem for Unitary Groups In this section we shall state and prove the Main Theorem corresponding to U(n + m) in the left column of Table1 . Concerning the representation theory of unitary groups, we use the following notation: The roots for U(N) are all nonzero linear functionals er?es in the dual h of the diagonal subalgebra with 1 r; s N.... In PAGE 21: ...his completes the proof of Lemma 2.6 and also Theorem 2.1b. 3. Main Theorem for Rotation Groups In this section we shall state and prove the Main Theorem corresponding to SO(n + m) in the left column of Table1 . The details will depend slightly on the parity of n and m as we shall see.... In PAGE 27: ...nd also Theorem 3.1b. 4. Main Theorem for Quaternion Unitary Groups In this section we shall state and prove the Main Theorem corresponding to Sp(n+m) in the left column of Table1 . We regard Sp(n+m) as the group of unitary matrices over the quaternions, and we write quaternions using the customary basis 1; i; j; k.... ..."

### Table 6. Representation of all integers. Note that although the three square theorem is commonly ascribed to Legendre, his \proof quot; depended on an unsubstantiated assumption only later established by Dirichlet, and the rst complete proof is due to Gauss. We nish by noting that in problems involving sums of two squares, meth- ods more e ective than the circle method can be brought into play (see especially Hooley (1981a,b) and Br udern (1987)).

### Table 1. Register Binding From now on, we will not use the auxiliary variable names p; q; r; : : : any more but replace them by register names r1; r2; r3; : : :. In each of the functions g0; g1; : : : the names r1; r2; r3; : : : are used to represent the register values before the evaluation of the function and r10; r20; r30; : : : are used to indicate the register values after the evaluation of the function. Variable renaming is performed by - conversion (see [Davi89]). The formal representation of the result of the register binding in theorem (4) is achieved by expansion of the operators and let- expressions and -reductions.

1995

Cited by 7

### Table 2: Branches explored and CPU time (seconds) used to find an optimal golomb ruler or prove that none exists. The variables were ordered lexicographically. The numbers of branches in the hidden representation are not given because they are always equal to the corresponding numbers in the non-binary representation. A * means that there was a cut off after 1 hour of CPU.

"... In PAGE 5: ... This gives a0 a3 a0 a22 a38a26a13a9a5a39a40 ternary constraints and a clique of binary `not equals apos; constraints. Table2 compares MGAC on the ternary representation to MAC on the hid- den and double representations. As Theorem 1 predicted, MGAC in the non-binary encoding explores the same num- ber of branches as MAC in the hidden.... ..."

### Table 2: Branches explored and CPU time (seconds) used to find an optimal golomb ruler or prove that none exists. The variables were ordered lexicographically. The numbers of branches in the hidden representation are not given because they are always equal to the corresponding numbers in the non-binary representation. A * means that there was a cut off after 1 hour of CPU.

"... In PAGE 5: ... This gives n(n ? 1)=2 ternary constraints and a clique of binary `not equals apos; constraints. Table2 compares MGAC on the ternary representation to MAC on the hid- den and double representations. As Theorem 1 predicted, MGAC in the non-binary encoding explores the same num- ber of branches as MAC in the hidden.... ..."

### Table 1 shows 15 situations in which the mode interaction condition (IC) holds and gives the theorems which applies to this situation. In Table 1, %2; %3; %4 are certain irreducible representations of various H, not all of which we de ne precisely. For H = D1 3 (or D2 3), %3 denotes the two-dimensional irreducible representation. For H = K1 6(K2 6; K3 6) %3 and %4 denote two nontrivial one-dimensional irreducible representations.

1993

"... In PAGE 18: ...16: IC cannot be satis ed. O { IC does not hold Table1 : Mode Interaction condition for D6... ..."

Cited by 1

### Table 2. All the eight matrices Xk in Theorem 4

2004

"... In PAGE 10: ... The formal series xdp+ has the 2-dimensional linear representation L, (Xk)7 k=0, C, where L = 1 1 , C = 1 0 gt; and Xk is given by (Xk)ij = 8 gt; lt; gt; : 1 T(k2 + k1 + j) if i = 0 and k2 k1 k0 = j ; T(k2 + k1 + j) if i = 1 and k2 k1 k0 = j ; 0 otherwise for i; j 2 f0; 1g, where k = k24+k12+k0 and T : f0; 1; 2; 3g ! R is the mapping T(0) = 0, T(1) = T(2) = 1 2 and T(3) = 1. (For completeness, all the matrices Xk are given in Table2 .) Thus, xdp+ is a rational series.... ..."

Cited by 1