Correction to: Involutions of Type G2 over a Field of Characteristic Two

Algebras and Representation Theory - We would like to correct two errors in our paper Involutions of type G2 over a field of characteristic 2 [2].     

Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.     

Mixing and sweeping out

A weakly mixing transformationT and a sequence (d _n) are constructed such thatT is uniformly mixing on (d _n),T is uniformly sweeping out on ([αd _n]) for allα∈(0, 1), and for all rationalα∈(0, 1)T is not mixing on ([αd _n]).     

Ferrocene derivatives XXVIII. The metallation of phenylazoferrocene and the synthesis of phenylazocyclopentadienyl derivatives of other metals

Phenylazoferrocene undergoes nickelation and palladation exclusively on the phenyl group. Lithium phenylazocyclopentadienide has been converted into derivatives of Mo, Mn, Fe, Ru and Rh.     

The Ulam number of infinite graphs

The following is a conjecture of Ulam: In any partition of the integer lattice on the plane into uniformly bounded sets, there exists a set that is adjacent to at least six other sets. Two sets are adjacent if each contain a vertex of the same unit square. This problem is generalized as follows. Given any uniformly bounded partitionP of the vertex set of an infinite graphG with finite maximum degree, letP (G) denote the graph obtained by letting each set of the partition be a vertex ofP (G) where two vertices ofP (G) are adjacent if and only if the corresponding sets have an edge between them. The Ulam number ofG is defined as the minimum of the maximum degree ofP (G) where the minimum is taken over all uniformly bounded partitionsP. We have characterized the graphs with Ulam number 0, 1, and 2. Restricting the partitions of the vertex set to connected subsets, we obtain the connected Ulam number ofG. We have evaluated the connected Ulam numbers for several infinite graphs. For instance we have shown that the connected Ulam number is 4 ifG is an infinite grid graph. We have settled the Ulam conjecture for the connected case by proving that the connected Ulam number is 6 for an infinite triangular grid graph. The general Ulam conjecture is equivalent to proving that the Ulam number of the infinite triangular grid graph equals 6. We also describe some interesting geometric consequences of the Ulam number, mainly concerning good drawings of infinite graphs.     

Evaluating the financial performance of bank branches

We evaluate the financial performance of most of the branch offices of a large European savings bank for a recent accounting period. We employ a complementary pair of nonparametric techniques to evaluate their financial performance, in terms of their ability to conserve on the expenses they incur in building their customer bases and providing customer services. We find variation in the ability of branch offices to perform this task, and agreement on the identity of the laggard branches. We then employ parametric techniques to determine that the list of indicators on which their financial performance is evaluated can be reduced without statistically significant loss of information to bank management. Both findings suggest ways in which the bank can increase the profitability of its branch network. A previous version of this paper was presented at CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, where we received many helpful comments from the audience. We are grateful to three referees for their helpful comments as well.     

Thermal effects and band spreading in capillary electro-separation

Summary Four Capillary Electroseparation methods are distinguished. All have an ultimate efficiency limited only by axial diffusion and are in principle capable of achieving 10⁶ plates in <1 hour.The main limitation to performance arises from Ohmic heating of the electrolyte. While forced convection at 10ms^–1 is recommended to keep tubes cool, the parabolic temperature profile within the electrolyte limits the tube bore which can be used. A simple limiting expression is derived: (d_c/m)³ (E/kV m^–1)³ (c/mol dm^–3) <3.3×10⁹.Values of constants underlined     

Gibberellin metabolites from ent-kaura-2,16-dien-19-ol and its succinate in Gibberella fujikuroi

Exposure of ent-kaura-2,16-dien-19-ol (1) or its succinate (2) to resuspended mycelia of G. fujikuroi has produced a complex mixture of acids which after methylation gave the esters of two C₁₉ (24) and (30) and five C₂₀ gibberellins (4, 11, 20, 32 and 33). The triester (32) and the lactone ester (24) have been prepared before from the esters of gibberellin A₁₃ (8) and gibberellin A₄ (26) respectively. The structures of the other metabolites were assigned on spectroscopic data and by chemical transformations. Thus the lactone diester (4) has been converted to the known keto triester (6). The epoxide (11) has been related to gibberellin A₁₄ (14) and the aldehyde (33) has been related to gibberellin A₁₃ trimethyl ester (8) by way of the triol (34). Selective de-epoxidation of the 16,17-epoxy function in diepoxides has provided a route from the dienes (20 and 24) to the epoxides (11 and 30) respectively, but not from the ester of gibberellin A₅ (23) to that of gibberellin A₆ (29). On the other hand the latter can be obtained by epoxidation of gibberellin A₅ methyl ester trifluoroacetate. Backfeeding experiments carried out with the epoxy diacid (12), the diene diacid (21) and the derived diol (39) indicate pathways connecting the various metabolites. The natural gibberellins A₅ and A₆ were shown to be formed in some of the backfeeding experiments.