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]).     

Allocating energy in a first-to-n match

This paper models a decision where a player must allocate limitedenergy over a contest of uncertain length. The motivating exampleis a squash match where one of the players is not as fit asthe other. Should a player's energy be concentrated in the earlygames of the match? Should it be spread evenly over all possiblegames? Or should it be conserved for the end of the match? Wemodel this as a decision problem where, in each game, the decision-makermust determine how much energy to expend. We assume that thereare only a small number of discrete energy choices for eachgame and that the more energy the decision-maker expends, themore likely he is to win that game. We solve for the optimaldecision with dynamic programming. With only two possible energychoices for each game, we show that it does not matter how energyis expended. In the case where there are three or more energychoices, we show how to take advantage of the structure of theproblem to determine the optimal sequence of decisions. As forpractical advice, the model suggests that when the decision-makerfalls behind in a match, he ought to switch to a more conservativeapproach by dividing his remaining energy evenly among all thepossible remaining games. Received 14 May 2003. Revised 5 January 2004.     

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.     

Influence of surface energy and relative humidity on AFM nanomechanical contact stiffness

The effects of surface functionality and relative humidity (RH) on nanomechanical contact stiffness were investigated using atomic force acoustic microscopy (AFAM), a contact scanned-probe microscopy (SPM) technique. Self-assembled monolayers (SAMs) with controlled surface energy were studied systematically in a controlled-humidity chamber. AFAM amplitude images of a micropatterned, graded-surface-energy SAM sample revealed that image contrast depended on both ambient humidity and surface energy. Quantitative AFAM point measurements indicated that the contact stiffness remained roughly constant for the hydrophobic SAM but increased monotonically for the hydrophilic SAM. To correct for this unphysical behavior, a viscoelastic damping term representing capillary forces between the tip and the SAM was added to the data analysis model. The contact stiffness calculated with this revised model remained constant with RH, while the damping term increased strongly with RH for the hydrophilic SAM. The observed behavior is consistent with previous studies of surface energy and RH behavior using AFM pull-off forces. Our results show that surface and environmental conditions can influence accurate measurements of nanomechanical properties with SPM methods such as AFAM.     

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).