### Table 1: 4 4 Symmetrics(S) and Skew-symmetrics(K)

"... In PAGE 8: ... For example, k j is its own conjugate, so (k j) must be symmetric; furthermore, (k j)(k j) = kk jj = 1 1, hence (k j) must also be orthogonal. Table1 shows which basis elements are symmetric and which are skew-symmetric.... In PAGE 9: ...le calculation using Eqn. (3) shows that (1 j) = J4. This is indeed fortuitous, as up to sign, B is closed under multiplication. It is easy to check that premultiplication by 1 j permutes the elements of B (again up to sign) in a simple way | the rst and third columns in Table1 are interchanged as are the second and fourth columns. Thus the Hamiltonian and skew-Hamiltonian structure of B can be quickly deduced directly from the properties of the quaternion tensor algebra: 1 i j k 1 W W H W i H H W H j H H W H k H H W H Table 2: 4 4 Hamiltonians(H) and Skew-Hamiltonians(W) Alternatively, one can use Appendix A to verify that each of the matrices in B has the structure speci ed in Table 2.... ..."

### Table 5: 2n 2n skew-symmetric Hamiltonian matrices

### Table 7: 2n 2n skew-symmetric skew-Hamiltonian matrices

### Table 5.2 Backward errors of the approximation of the eigenvalue 0 for a 30 30 random skew-symmetric skew-Hamiltonian matrix.

### TABLE I Highest Merit Factors for skew-symmetric sequences as obtained by various authors [17], [18], [19], [8], [6] and by our method.

### Table 5.4 The restarted Arnoldi approximation applied to the skew-symmetric problem of dimension 10001, for several restart lengths m (cf. Figure 5.4).

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### Table 6.1 Backward error of the eigenpair for = 0:741i of the 4 4 skew-symmetric Hamiltonian de ned by (6.1)

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### Table 5.1 Largest backward errors for the computed eigenpairs 1:2748i of the 4 4 skew-symmetric Hamiltonian de ned by (5.1)

### Table 1. Raw, low-degree solar p-mode eigenfrequency set, derived from tting a skew-symmetric model to a single, 32-month BiSON frequency spectrum. This was generated from data collected at or near the solar activity minimum. Note that these entries have not been corrected for solar cycle e ects. The listed uncertainties are the formal errors derived from the inverse of the Hessian matrix associated with each t.

"... In PAGE 4: ... However, we note that the mag- nitude of the implied bias is at a level below the formal mode-frequency uncertainties xed by the 32-month period of observation and the lifetimes of the modes themselves (see Section 3.2 and Table1 below). Clearly, estimates from individual 32-month spectra will be distributed about the mean, bias values.... In PAGE 4: ... Note that a much-less scattered linear trend is obtained from data with a 100-per-cent duty cycle. Table1 lists the skew-model- tted BiSON frequencies. Where ts at low frequencies failed the log-likelihood di er- ence test (see Section 3.... In PAGE 4: ... However, we note that the period spanned by the observations covers that near the activity minimum at the cycle 22/23 boundary. We have compared the frequencies in Table1 with those given by Toutain et al. (1998), who tted the Nigam amp; Koso- vichev skew-symmetric formalism, at 0 ` 2, to a 679-... In PAGE 6: ...termined from the frequencies in Table1 with those calcu- lated from models which include the e ects of helium and heavy-element settling. We tted a straight line to the Bi- SON spacings expressed as a function of the radial order n (Elsworth et al.... In PAGE 7: ... The left-hand [d0(n)] and right-hand [d1(n)] plots show the residuals generated by tting a straight line, as a function of radial order n, to the BiSON spacings, and then subtracting this from the three data sets shown. The data with error bars were calculated from the BiSON frequencies in Table1 ; the dotted line indicates values calculated from model `S apos; (Christensen-Dalsgaard et al. 1996), which includes heavy-element settling but neglects the e ects of turbulence; and the dashed line indicates predictions from model 4 of Christensen-Dalsgaard, Pro tt amp; Thompson (1993) which includes heavy-element settling with turbulent mixing.... ..."

### Table 2: Comparison of best solutions found genetically very di erent. The breeders call this e ect inbreeding.The above representation did not give good results, therefore we changed it slightly. The modi cation was motivated by an observation of Golay [10] . He showed that good skew-symmetric solutions of order n can be found by an interleaving of good symmetric and an- tisymmetric solutions of order n=2. We show for the case n = 13; m = 7 how this is done :

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