On Center Cycles in Grid Graphs

Finding good cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP-hard in general graphs. This result holds both for the variable cardinality case (i.e., all cycles of the graph are considered) and the fixed cardinality case (i.e., only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyze rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a complete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for bounded cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.     

Comparison of linear and non-linear equations for univariate calibration

Univariate data accumulated for the purpose of calibration of chromatographic and spectroscopic methods often exhibit slight but definite curvature. In this paper the performance of a non-linear calibration equation with the capacity to account empirically for the curvature, y = a + bx(m), (m not equal to 1) is compared with the commonly used linear equation, y = a + bx, as well as the quadratic equation, y = a + bx + cx2. All equations were applied to high quality HPLC calibration data using unweighted least squares. Parameter estimates and their standard errors were calculated for each equation. Standard errors and 95% prediction intervals in analyte concentrations were estimated with the aid of the fitted equations and their respective covariance matrices. Results indicate that the non-linear and quadratic equations each provide a better fit than the linear equation to the data considered here, as judged by the Akaikes information criterion (AIC), the adjusted coefficient of multiple determination, the magnitude and scatter of residuals, standard errors in estimated analyte concentrations and lack of fit analysis of variance (ANOVA). While the difference between the equations y = a + bx + cx2 and y = a + bx(m) as judged by the same criteria is more marginal, this work suggests that the non-linear calibration equation should be considered when a curve is required to be fitted to low noise calibration data which exhibit slight curvature.     

Enantioselective [4 + 2]-annulation of chiral crotylsilanes: application to the synthesis of a C1-C22 fragment of leucascandrolide A

[reaction: see text] The asymmetric synthesis of a C1-C22 fragment (2) of leucascandrolide A is described. Synthetic highlights include the construction of the C9-C22 pyran fragment using a formal [4 + 2]-annulation of a chiral organosilane. A diastereoselctive Mukaiyama aldol was used to introduce the C9 stereocenter and complete the assembly of the macrocycle's carbon skeleton.     

Synthesis of 2,7-disubstituted-5,6-dimethylpyrrolo-[2,3-d]-1,3-oxazin-4-ones as antifungal agents

A series of novel 5,6-dimethylpyrrolo[2,3-d]-1,3-oxazin-4-ones were synthesized from 2-amino-3-tert-butoxy-carbonyl-4,5-dimethylpyrroles. Two methods were used, cyclodehydration of 2-acylamino-3-carboxypyrroles with acetic anhydride and direct conversion of the 5,6-dimethylpyrrolo[2,3-d]-1,3-oxazin-2,4-diones to the title compounds with an anhydride directly providing the 2 substituent. Molecular modeling techniques revealed that these pyrrolo[2,3-d]oxazinones were rigid analogues of the allylamine antifungals. The compounds were tested for in vitro activity against Tricophyton and Scopulariopsis sp.     

Synthesis of 1,2,4-triazol-5-ylidenes and their interaction with acetonitrile and chalcogens

Two new, more convenient methods for the synthesis of 1,2,4-triazol-5-ylidenes are described. Four new 1,2,4-triazol-5-ylidenes have been prepared using these methods: 1-(1-adamantyl)-3,4-diphenyl-1,2,4-triazol-5-ylidene (2a), 1-(1-adamantyl)-3-phenyl-4-(p-bromophenyl)-1,2,4-triazol-5-ylidene (2b), 1-(1-adamantyl)-3-phenyl-4-(alpha-naphthyl)-1,2,4-triazol-5-ylidene (2c), and 1-(1-adamantyl)-3,4-di(p-bromophenyl)-1,2,4-triazol-5-ylidene (2d). The X-ray crystal structures of 2d and the precursor salt 1-(1-adamantyl)-3,4-di(p-bromophenyl)-1,2,4-triazolium bromide (1e) are described. Compound 2a reacts with CH(3)CN via C-H insertion to form 1-(1-adamantyl)-3,4-diphenyl-5-cyanomethyl-5H-1,2,4-triazoline (3), and 2a and 2d react with elemental sulfur and elemental selenium, respectively, to form the corresponding thione (4) and selenone (5).     

Synthesis, structures and reactions of some metallocene alcohols

E-1-Ferrocenyl-4,4-dimethylpent-2-ene-1-one has been synthesised from the Friedel-Crafts acylation of ferrocene with E-3-tert-butylacryloylchloride and converted to 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-one using ethereal hydrogen chloride. This new chloro ketone has been converted into three new ferrocene alcohols: 1-ferrocenyl-3,4-dimethyl-4-hydroxypentan-1-one, 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-ol, and 2,2,6,6-tetramethyl-3-ferrocenyl-5-chloroheptan-3-ol. A new dinuclear ferrocene derivative, E,E-2,2,9,9-tetramethyl-5,6-diferrocenyl-deca-3,7-diene, was isolated after treatment of 1-ferrocenyl-3-chloro-4,4-dimethylpentan-1-ol with acidic alumina; its structure was confirmed by X-ray crystallography, whilst electrochemistry revealed metal-metal interactions of similar magnitude to those seen for other 1,2-bis(ferrocenyl)ethane derivatives. Crystal structures have also been determined for 2,2,6,6-tetramethyl-3-ferrocenyl-5-chloroheptan-3-ol, rac-1-hydroxy[3]ferrocenophane, rac-1S,3S-1,3-diphenyl-1-hydroxy[3]ferrocenophane, and of rac-1,1^′-diphenyl-1,1^′-(1,1^′- ruthenocenediyl)dimethanol and show an intramolecular Cl?H-O hydrogen bond, a tetramer based on O?H-O hydrogen bonds, no hydrogen bonding, and a dimer with inter- and intramolecular O?H-O hydrogen bonds, respectively.     

Fluoroaryl-substituted ketiminate complexes of aluminum

The ketiminate complex AlCl[OC(Me)CHC(Me)N(p-C₆H₄F)]₂ (4) has been prepared from the β-aminoenone, OC(Me)CHC(Me)N(H)(p-C₆H₄F) (3) by lithiation of 3 with n-BuLi, followed by reaction with AlCl₃ and by the reaction of 3 with Me₂AlCl. A second compound, [AlCl₂{OC(Me)CHC(Me)N(H)(p-C₆H₄F)}₄][AlCl₄] (5), was also isolated from the AlCl₃ reaction. The structures of 4 and 5 were determined by X-ray diffraction analysis.     

Structural and bonding investigation of an unbridged donor-acceptor complex 1-(2,2′-bipyrimidine)-1-(t-butyl)-2, 3-bis(trimethylsilyl) -2,3-dicarba-1-galla-closo-heptaborane(6)

The reaction between thecloso-gallacarborane, 1-(t-C₄H₉)-1-Ga-2,3-(SiMe₃)₂-2,3-C₂B₄H₄ (1), and 2,2-bipyrimidine in a molar ratio of 11 in dry benzene produced the unbridged donor-acceptor complex 1-(2,2-C₈H₆N₄)-1-(t-C₄H₉)-1-Ga-2,3-(SiMe₃)₂-2, 3-C₂B₄H₄ (2) as a pale-yellow crystalline solid in 81% yield. The spectroscopic data of2 are almost identical to those of the bridged species 1,1-(2,2-C₈ H₆N₄)-[1-(t-C₄H₉)-1-Ga-2,3-(SiMe₃)₂-2,3-C₂ B₄H₄]₂ (3). However, the single-crystal X-ray structure of2 reveals that the complex consists of only one distorted gallacarborane unit and a bipyrimidine molecule, that acts as a bidentate ligand. The apical gallium in2 is displaced further away from the centroidal position above the carborane than it is in 1. Despite the steric hindrance imposed by the bulky trimethylsilyl groups on the cage carbons, the Ga-C_(t-buty) bond is tilted toward the cage carbon atoms. Complex2 crystallizes in the monoclinic space groupP2₁ witha=11.279 (3),b=10.060 (3),c=11.911 (3) Å,=94.94 (2)°,V=1346.5 (6) ų,Z=2,D _x=1.24g cm^–3, (MoK) F(000)= 528, andT=220 K. Full-matrix least-squares refinement for2 converged at R=0.036 andRw= 0.045 for 3032 observed reflections. Molecular orbital calculations showed that antibonding interactions between the gallium and carborane cage carbons, induced by base complexation, are a contributing factor to the gallium's slip distortion that is found in2.     

Synthesis, crystal structures and the preliminary evaluation of the new dibenzotetraaza[14]annulene-based DNA/RNA binding agents

A series of water-soluble dicationic dibenzotetraaza[14]annulenes have been prepared in order to examine their interactions with nucleic acids. Pendant water-solubilizing N-pyridinium, 4,4′-bipyridinium and N-methyl pyridinium moieties have been attached to the central core via linkers generated by direct N-alkylations and ester creating couplings, respectively. The crystal structures of derivatives equipped with 3-(N-pyridinium-1-yl)propyl and 3-(4,4′-bipyridinium-1-yl)propyl substituents have been determined. Interactions with ct-DNA have been studied and evidenced by means of spectrophotometric titrations with Scatchard analysis and thermal denaturation experiments.