### Table 2: Branching equations and signs of eigenvalues for primary bifurcation branches on the hexagonal lattice; ; b1; : : : ; b6 are coe cients in the bifurcation equation (16). See [9, 10] for more details. Planform and branching equation Signs of non-zero eigenvalues

2000

"... In PAGE 15: ... Note that if is O( ) then the t1-time scale is unnecessary. This is the situation of interest here, since we know from Table2 that when is O(1) there are no stable small-amplitude steady states. However, in the following computation we must retain both time scales to compute the general form of the cubic coe cient b2.... In PAGE 19: ... In short, the quadratic nonlinearities from F2(u; v) and G2(u; v) completely disappear from the bifurcation coe cients b1 and h( ), and appear in b2 solely through the term v7 2: (38) This further implies that all -dependence disappears from the bifurcation equations, and that rhombs and super hexagons are always unstable at the bifurcation point, for the degenerate problem. The signs of the eigenvalues for rhombs and super hexagons are given in Table2 . The rst two eigenvalues for rhombs depend on the signs of b1+b4 and b1 ? b4, which cannot both be negative.... In PAGE 19: ... Stripes and hexagons may bifurcate stably; Figure 2 gives their stability assignments in the ( ; )-parameter plane. Note that, for the cubic truncation of the bifurcation problem, hexagons are only known to be neutrally stable, since the eigenvalue ? x + O(x3) in Table2 depends on higher-order terms, which are not computed here. For the corresponding square lattice computation, stripes are stable for lt; 0; all other solutions are unstable for = 0.... In PAGE 26: ... Figure 2 for = 0, lt; 0). The results summarized by Table2 may be used to determine certain aspects of the hexagonal bifurcation problems near the degeneracy = 0. In section 4.... ..."

Cited by 3

### Table 1. Runtimes for the lattice computation

2000

"... In PAGE 11: ... It might be of interest to compare the performance of the described eae algorithm with the traditionally used cyclic extension code. Table1 gives runtimes (seconds on a 200MHz PentiumPro under Linux) for a set of arbitrarily selected solvable groups. The eae code was implemented by the author, for cyclic extension the standard GAP library function LatticeByCyclicExtension was used.... ..."

### Table 2. The Lattice MCFQRD POS B Equations.

"... In PAGE 8: ...in Table 1 can now be carried out in a forward manner. The rotation angles Q(i) (k + 1) are obtained through Q(i) (k + 1) i 1(k + 1) 0 = i(k + 1) f(N i+2)(k + 1) (35) and the joint estimation is performed according to e(i) q1 (k + 1) d(N i+1) q2 (k + 1) = Q(N i+1) (k + 1) e(i) q1 (k + 1) 1=2d(N i+1) q2 (k) (36) In order to adequate the equations of steps 1 to 3 of the algorithm as in Table 1 to this formulation, it su ces to observe that they can be easily split up into M M blocks that will be executed recursively as shown in Table2 . It is worth mentioning that, for the sake of simpli cation due to space constraints, we have used matrix notation in the single loop operations as shown in Table 2.... In PAGE 8: ... The rotation angles Q(i) (k + 1) are obtained through Q(i) (k + 1) i 1(k + 1) 0 = i(k + 1) f(N i+2)(k + 1) (35) and the joint estimation is performed according to e(i) q1 (k + 1) d(N i+1) q2 (k + 1) = Q(N i+1) (k + 1) e(i) q1 (k + 1) 1=2d(N i+1) q2 (k) (36) In order to adequate the equations of steps 1 to 3 of the algorithm as in Table 1 to this formulation, it su ces to observe that they can be easily split up into M M blocks that will be executed recursively as shown in Table 2. It is worth mentioning that, for the sake of simpli cation due to space constraints, we have used matrix notation in the single loop operations as shown in Table2 . However, when implementing these equations, it is straightforward to reduce the simple Givens rotations matrices into scalar operations.... ..."

### Table 1. Computation of Symmetries of Large MCNC Benchmarks.

2003

"... In PAGE 5: ...c). Table1 summarized the experiments conducted using the largest MCNC benchmark circuits. The program was run on a 933MHz Pentium III PC under MS Windows 2000.... In PAGE 5: ... The memory requirements of the algorithm amounted to less than 10% of the memory allocated by the CUDD package to construct the shared BDDs of the benchmark functions. The following notation is used in Table1 . The first four columns show the benchmark parameters: the name, the number of inputs and outputs, and the number of BDD nodes after reading and reordering by the sifting algorithm [22].... In PAGE 6: ... The fact that the symmetry information computed using all the available algorithms is identical was used to test the correctness of the proposed implementation. The right part of Table1 compares the runtimes of several symmetry computation algorithms. Column reading gives the CPU time needed to read the benchmark file, construct the BDD and perform the reordering.... ..."

Cited by 9

### Table 1 : Evolution of demands on the computational resources.

"... In PAGE 1: ... Specifically, this is because of the increase of computational requirements, of the data volume and of the intensity of electronic utilization of the data. Table1 provides an overview of trends in these three areas. During the development phase of an algorithm, the only aspect that is of relevance are the computational demands of a given algorithm.... In PAGE 13: ... 6 Conclusion As we have discussed and demonstrated using examples, there are several ways how HPC can be used in medical image analysis. There is the global trend to computationally more demanding algorithms, more data and more people interested in this type of work (see Table1 ) and there is the evolution of individual algorithms from conception to routine clinical use (see Table 2). Both of these trends open opportunities for the application of HPC.... ..."

### Table II The evolution equations for i = g2 i =4 are generally given up to two-loop order by

### Table 4: Computation times for nonlinear cloth simulation

"... In PAGE 20: ... 7 Results and Applications We have performed several cloth simulations using the parameters computed from the KES curves of section 6. Table4 summarizes the performance of our algorithm on an Intel Pentium 4-2Ghz.... ..."

### Table 1: Evolution of scheduling algorithms with parallel and distributed computing systems

2006

"... In PAGE 4: ... Looking back at such efforts, we find that scheduling algorithms are evolving with the architecture of parallel and distributed systems. Table1 captures some important features of parallel and distributed systems and typical scheduling algorithms they adopt. ... ..."

### Table 1: Summary of the lattice parameters used and relative values of C0 and Ci obtained as discussed in section 3. The a?1 values are taken from [19, 20] and are computed through the string tension.

"... In PAGE 4: ...ymanzik equation, can only depend on the bare coupling g0[18]. Eqs.(7) and (8) show that A0 (x) = C(g0)A (x) + a2 ZW W R (x) (9) so that, up to terms truly of order a2: A0 (x) = C(g0)A (x) (10) For Green apos;s functions insertions we have, therefore, in general: h: : : A0 (x) : : :i h: : : A (x) : : :i = C(g0) (11) We have numerically checked some consequences of eq.(11) by measuring on di erent lattices, whose characteristics are reported in Table1 , the following Green functions for SU(3) in the Landau gauge with periodic boundary conditions: hA0A0i(t) 1 V 2 X x;y T rhA0(x; t)A0(y; 0)i (12) hAiAii(t) 1 3V 2 X i X x;y T rhAi(x; t)Ai(y; 0)i (13) 1We are assuming that only one operator W (x) is present, in order to simplify the presentation. In general several dimension 3 operators exist and their mixing has to be taken into account, but the conclusions remain unchanged.... In PAGE 5: ... The proportionality factor, C(g0), may depend on the direction , if the lattice breaks cubic symmetry. In our case, as shown in Table1 , two of the lattices have a time extension di erent from the spatial one, so that we have a coe cient C0(g0) relating A0 0 to A0 and a di erent one, Ci(g0), connecting A0 i to Ai. It is worth noting that hA0A0i(t), when evaluated through A (x), should be con- stant in t con guration by con guration, in virtue of the Landau gauge condition which, together with periodic boundary conditions, implies @0A0 = 0.... In PAGE 5: ... For both correlators, the error is just due to uctuation of their constant (in t) value, con guration by con guration. In Table1 we report the t of the ratio: hA0 iA0 ii hAiAii C2 i (g0) (17) as a constant in time. In Fig.... In PAGE 5: ... (10). As shown in Table1 , C0(g0) and Ci(g0) coincide, within the errors, for the symmet- ric lattices W58 and W60a , while they have a di erent value for W60b and W64. This is probably due to the breaking of cubic symmetry.... ..."

### Table 1: Evolution of scheduling algorithms with parallel and distributed computing systems

2006

"... In PAGE 4: ... Looking back at such efforts, we find that scheduling algorithms are evolving with the architecture of parallel and distributed systems. Table1 captures some important features ... ..."