### Table 1. Largest known undirected n28n01;Dn29 graphs

"... In PAGE 3: ...o construct large graphs. First of all, there are metacyclic groups, i.e. semidirect products of cyclic groups: if the multiplicative order of the unit a 2 Z n divides m,a semidirect product of Z m with Z n can be den0cned using the following multiplication: n5bx; yn5dn5bu; vn5d=n5bx+umod m; ya u + v mod nn5d: In Table1 groups of this kind are identin0ced by DH; our detailed listing in the appendix uses the symbol m n02 a n, e.g.... In PAGE 3: ... An automorphism n1b of Z n n02Z n is determined by the images of the generators n1bn28n5b1; 0n5dn29 = n5bx; yn5d and n1bn28n5b0; 1n5dn29 = n5bz;tn5d. If the order of n1b divides m we can den0cne a multiplication on Z m n02 Z n n02 Z n by: n5bc; d; en5dn5bf;g;hn5d=n5bc+f mod m; n5bd; en5d n14 x y z t n15 f +n5bg;hn5dmodnn5d: In Table1 we identify groups of this kind by DH n03 ; our detailed listing in the appendix uses the symbol m n02 n1b n 2 , e.g.... In PAGE 3: ... e.g. 8 n02 n1b 3 2 ; the action of the cyclic group is not encoded into this symbol and is specin0ced separately. A further kind of group is indicated in Table1 by DH n03n03 . These are semidirect products of G = m n02 a n with itself, where the action is by conjugation.... In PAGE 4: ... The only graph whose order din0bers from the Moore bound by 1 is the square n5b5, 25n5d. This implies that the entries n283; 3n29, n284; 2n29, and n285; 2n29 in Table1 are indeed optimal, a fact whichgoesback to Elspas n5b24n5d. More recently it has been shown that for n01 = 3, D n15 4 the Moore bound cannot be missed by 2 n5b32n5d.... In PAGE 4: ... More recently it has been shown that for n01 = 3, D n15 4 the Moore bound cannot be missed by 2 n5b32n5d. Table1 is an update of n5b9n5d and contains the orders of the largest known n28n01;Dn29 graphs with annotations indicating the nature of the graphs. Comparison of the Moore bound with the orders listed in the table shows that, mostly, the Moore bound is missed by a considerable margin.... ..."

### Table 9: Dihedral Angles for the Wedge

"... In PAGE 10: ... If the idea of line isometry is accepted, the \sloping roof quot; should make an angle with the L-shaped front face of arctan 4 3, 53:13 . The rightmost columns in Table9 show the worst and best perpendicular dihedral angles and the angle between the sloping roof and the L-shaped front face, and the mean and worst deviations from planarity. Sur- prisingly, corner orthogonality is consistently preferable to methods dihedral angles planarity ? ll 4vp small mean large mean worst JLP N N 0.... ..."

### Table 1: Possible combinations of nite subgroups of O(2): , ~ , and ^ satisfying the mutual relations (3.23). Also listed are pairs of reversing symmetry groups 0 and 1 2q arising in the return maps of surfaces of section S0 and S 1 2q (choosing the time-coordinate such that whenever 6 = ^ , 0 contains a reversing symmetry). Here, Zn and D n denote the standard representation of the cyclic and dihedral groups in R2 (i.e. the cyclic group generated by a rotation and the dihedral group generated by re ections), and p 2 Z+:

1998

"... In PAGE 15: ...xample 4.3 (Spatiotemporal symmetry groups in R2.) As an example, we examine the occurrence of nite subgroups of O(2) arising as spatiotemporal symmetry groups. In Table1 we list the possible combinations of groups , ~ , and ^ and corresponding pairs of reversing symmetry groups 0 and 1 2q (assuming whenever 6 = ^ that 0 con- tains at least one reversing symmetry). Note that pairs of nonisomorphic dual reversing k-symmetry groups only arise in the last entry of this table, when p gt; 1.... In PAGE 17: ...ime-shift symmetry). Recall in this respect from the proof of Theorem 4.1 that whenever q is odd, 0 is conjugate to 1 2q so that whenever q is odd these groups always have equivalent representations. From Table1 one obtains that in the case of O(2) in R2 there is only one situation in which one nds nonequivalent representations: when = D 2, ^ = D 1 and ~ = Id, we nd dual nonequivalent representations of the cyclic group of order two: Z2 and D 1. Example 4.... ..."

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### Table 2: Dn.

1997

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### Table 1: Combinatorial data for the Weyl groups. It is now straightforward to use to obtain recursive bounds for the reduced com- plexities of these chains of groups. Proof of Theorem 5.3. From the data in Table 1, Corollary 4.10, and Theorem 3.1, we obtain the recurrences tBn tBn?1 + 2n(2n ? 1) and tDn tDn?1 + 6n(n ? 1). 25

1997

"... In PAGE 25: ... In addition the minimal coset representatives for W=WJ and their minimal factorizations all occur as subwords of a minimal factorization for wSwJ, where wS is the longest element in W and wJ is the longest word in WJ. In Table1 we summarize the data required to bound the complexities for the... ..."

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### Table IV Average Values of Peptide Dihedral Anglesa

### Table 8: Dihedral Angles for the Angle Bracket

"... In PAGE 10: ... Ideally, after in ation, all dihedral angles should be either 90 or 45 (making the convex hull of the top face an equi- lateral triangle might also be a reasonable interpretation). The nal four columns of Table8 show the worst and best right-angles, the worst and best 45 angles, and the mean and worst deviations from planarity. It is clear from the right-angle columns that JLP is preferable to corner orthog- onality, both in terms of dihedral angles and face planarity, con rming the initial impression that corner orthogonality does not produce good results for non-normalons.... ..."

### Table 10: Dihedral Angles for the A-Block

"... In PAGE 10: ... If line parallelism is accepted, the inter- nal dihedral angle in the triangular through hole should be 60 . The rightmost columns of Table10 list, respectively, these three angles, the worst and best perpendicular dihe- dral angles, and the mean and worst deviations from pla- narity. This drawing was included because it illustrates a common type of problem junction (an occluding T-junction completing a triangular partial face) for which only line par- allelism generates a reasonable equation, and as expected, line parallelism is almost essential for good dihedral angles.... ..."

### Table 1: Combinatorial data for the Weyl groups. It is now straightforward to use to obtain recursive bounds for the reduced complexities of these chains of groups. Proof: [of Theorem 5.3] From the data in Table 1, Corollary 4.10, and Theorem 3.1, we obtain the recurrences tBn tBn?1 + 2n(2n ? 1) and tDn tDn?1 + 6n(n ? 1). Iterating the recurrence for tBn gives the result for that series of groups, but for tDn we need a more careful count. Let s1; : : :; sn denote the simple re ections for Dn, in the order shown in Diagram 5. Then M(si) = 2 for i 4, M(s3) = 3 and M(si) = 1 for i = 1 or i = 2. The maximal minimal coset representative for Dn=Dn?1 is sn s3s2s1s3 sn and the minimal coset representatives have the following minimal factorizations 1; sn; sn?1sn; : : :; s3 sn; s2s3 sn; s1s3 sn; s1s2s3 sn; s3s1s2s3 sn; : : :; sn s3s2s1s3 sn: (39) The number of times s3 occurs in these words is exactly equal to the number of times s1 and s2 occur in total, so the average value of M over all occurrences of symbols in the set of minimal factorizations is 2. The sum of the lengths of the minimal coset representatives of Dn=Dn?1 is 2n(n ? 1). Therefore if we let X be equal to the set of words (39), then we have

1997

"... In PAGE 20: ... In addition the minimal coset representatives for W=WJ and their minimal factorizations all occur as subwords of a minimal factorization for wSwJ, where wS is the longest element in W and wJ is the longest word in WJ. In Table1 we summarize the data required to bound the complexities for the Weyl groups. W WJ M(S) jWj NS P0 W=WJ;S(1) Sn Sn?1 2 n! 1 2n(n ? 1) 1 2n(n ? 1) Bn Bn?1 2 2nn! n2 n(2n ? 1) Dn Dn?1 3 2n?1n! n(n ? 1) 2n(n ? 1) Table 1: Combinatorial data for the Weyl groups.... ..."

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