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]).     

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.     

Squaric acid N-hydroxylamides: synthesis, structure, and properties of vinylogous hydroxamic acid analogues

The synthesis of squaric acid N-hydroxylamide esters 5 and amides 6 from dimethyl squarate 2a is described. These derivatives are analogues of the naturally occurring iron(III) chelator hydroxamic acid. On the basis of a comparative reactivity study, a concerted retro-Cope mechanism for the formation of the N-hydroxylamide esters 5 by reaction of dimethyl squarate with hydroxylamines is proposed. A preliminary iron(III) binding study of these hydroxamic acid analogues is presented, demonstrating binding of iron(III) to amides 6 in aqueous solutions, while the esters 5 did not show any sign of metal ion binding. 13C NMR spectroscopic data (chemical shift and spin-lattice relaxation time determination) of these and related derivatives delineate the resonance structures predominant in these molecules. The resonance structures of the derivatives rationalize their spectroscopic data, chemical reactivity, and iron(III) binding properties. Single-crystal X-ray structure analyses of squaric acid N-hydroxylamide ester 5b and squaric acid N-hydroxylamide amide 6c confirm their connectivity and provide structural evidence supporting the spectroscopically derived conclusions. The squaric acid N-hydroxylamides are potentially useful in the construction of chemosensors for iron(III).     

Reactivity of (bicyclo[5.1.0]octadienyl)iron(1+) cations: application to the synthesis of cis-2-(2'-carboxycyclopropyl)glycines

The addition of carbon and heteroatom nucleophiles to (bicyclo[5.1.0]octadienyl)Fe(CO)(2)L(+) cations 5 or 8 (L = CO, PPh(3)) generally proceeds via attack at the dienyl terminus on the face of the ligand opposite to iron to generate 6-substituted (bicyclo[5.1.0]octa-2,4-diene)iron complexes (11 or 13). In certain cases, these products are unstable with respect to elimination of a proton and the nucleophilic substituent to afford (cyclooctatetraene)Fe(CO)(2)L (4 or 7). Decomplexation of 13f, arising from addition of phthalimide to 8, gave N-(bicyclo[5.1.0]octa-3,5-dien-2-yl)phthalimide (19). Oxidative cleavage of 19 (RuCl(3)/NaIO(4)) followed by esterification gave the cyclopropane diester 22, which upon hydrolysis gave cis-2-(2'-carboxycyclopropyl)glycine (CCG-III, 18) (eight steps from 4, 43% overall yield). This methodology was also utilized for preparation of stereospecifically deuterated CCG-III (d-18) and optically enriched (-)-18. Deprotonation of 22 resulted in cyclopropane ring opening to afford the benzoindolizidine (23).     

An improved algorithm to locate critical points in a 3D scalar field as implemented in the program MORPHY

A new algorithm for location of the critical points in general scalar fields is described. The new method has been developed as part of an on-going process to exploit the topologic analysis of general 3D scalar fields. Part of this process involves the use of topologic information to seed the critical point search algorithm. The continuing move away from topologic studies of just the electron density requires more general algorithms and the ability to easily "plug in" new functions, for example, the Laplacian of the electron density ( triangle down (2)rho), the Electron Localisation Function (ELF), the Localised Orbital Locator (LOL), the Lennard-Jones function (LJF), as well as any new functions that may be proposed in the future. Another important aspect of the current algorithm is the retention of all possible intermediate information, for example, the paths describing the connectivity of critical points, as well as an ability to restart searches, something that becomes increasingly important when analysing larger systems. This new algorithm represents a core part of a new local version of the MORPHY code. We distinguish nine universal types of gradient paths.     

Fluorescent signaling based on control of excited state dynamics. Biarylacetylene fluorescent chemosensors

We have previously reported that metal ion binding could restrict the excited state rotation of a biaryl chromophore, suppressing intersystem crossing and leading to increased emission. We have now applied the restriction of excited state dynamics to suppression of the other fundamental nonradiative decay pathway, internal conversion, in biarylacetylenes. This indicates that both nonradiative decay pathways are subject to conformational control, and that this signaling pathway should be generally accessible in simple flexible fluorophores.     

Structures and cytotoxicities of fascaplysin and related alkaloids from two marine phyla—Fascaplysinopsis sponges and Didemnum tunicates

The structural variations and bioactivity properties of the alkaloids in the fascaplysin (1) and the reticulatine (3) families were examined. Four organisms were analyzed consisting of two collections of the sponge Fascaplysinopsis reticulata and two collections of the tunicate Didemnum sp. Reported are the isolation of three new compounds: 3-bromofascaplysin (2), 14-bromoreticulatine (4), and 14-bromoreticulatate (6) along with reticulatate (5) previously known as a semi-synthetic product of 1. Compounds 1 and 5 showed selectivity in a cell based cytotoxicity assay.