The complexity of finding generalized paths in tournaments

Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation υ₁υ₂…υ_n of the vertices, such that the results of the games υ₁υ₂, υ₂υ₃,…, υ_n−1υ_n match a prescribed pattern. If the pattern requires all arcs to go forwrd, i.e., υ₁ → υ₂, υ₂ → υ₃,…, υ_n−1 → υ_n, and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cn lg n, and we make the observation that O(n lg n) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-forward-backward, etc., admits an algorithm for which O(n) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn lg^αn for any α between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 4].     

Tuning the Surface Structure and Optical Properties of CdSe Clusters Using Coordination Chemistry

A series of nonstoichiometric CdSe clusters with lowest energy electronic absorptions between 409 - 420 nm has been prepared from cadmium 1-naphthoate, 2-naphthoate, 4-thiomethyl-1-naphthaote, and 1-naphthalene thiolate complexes and diphenylphosphine selenide (DPPSe). Pair distribution function analysis of X-ray diffraction data, ligand exchange experiments, and NMR molecular weight analyses suggest the nanocrystal core changes minimally among these clusters despite significant changes to their absorption and luminescence spectra. Photoluminescence excitation spectra obtained at 77 K reveal an energy transfer process between the surface-trapped excited state and the naphthalene-containing ligands that leads to ligand phosphorescence. A Dexter energy transfer mechanism is proposed to explain the observation of ligand phosphorescence on excitation of the cluster. These compounds demonstrate that cluster absorption and trap luminescence can be controlled with surface coordination chemistry.     

Alternative Syntheses of (l→3)-β-D-Galacto-Oligosaccharides

ABSTRACT

Galactosyl halides bearing different substituents at O-3 [i.e. acetyl (15), benzoyl (14), benzyl (3), bromoacetyl (12), and the 2, 3, 4, 6-tetra-O-benzoyl-β-D-galactopyranosyl group (17)] have been prepared, and used to study the stereoselectivity of the coupling reaction to position O-3 of different galactose derivatives [i.e. methyl 2, 4, 6-tri-O-acetyl-(9) and 2, 4, 6-tri-O-benzoyl-β-D-galactopyranoside (7), l, 2, 4, 6-tetra-O-benzoyl-β-D-galactose (6) and O-(2, 4, 6-tri-O-benzoyl-β-D-galactopyranosyl)-(1→3)-β-D-galactose (33)], as well as to benzoic acid. In more polar solvents, using silver trifluoro-methanesulfonate as the promoter, a higher proportion of β-linked products was formed, whereas with silver perchlorate as the promoter the α-linked product predominated. Under basic conditions, applied to prevent anomerisation of 1-O-benzoylated nucleophiles 6 and 33, no orthoesters were found as end products. Under those conditions, a better overall yield of the β-(l→3)-linked galactotriose 31 was obtained by condensation of die disaccharide glycosyl donor 17 and the monosaccharide glycosyl acceptor 6 than by condensation of 14 and 33. The disaccharide glycosyl chloride 17 was obtained in 75% yield by the cleavage of the corresponding methyl glycoside with dichloromethyl methyl ether.     

Synthesis of Specifically Fluorinated Methyl β-Glycosides of (1→6)-β-D-Galactooligosaccharides II. Methyl 4-Deoxy-4-fluoro-6-O-(β-D-galactopyranosyl)-β-D-galactopyranoside

Abstract

Condensation of methyl 2,3-di-O-benzyl-4-deoxy-4-fluoro-β-D-galactopyranoside with 2,3,4,6-tetra-O-acetyl-α-D-galactopyranosyl bromide in the presence of mercuric cyanide in benzene afforded, in excellent yield, the β-linked product. Its deblocking, effected by hydrogenolytic cleavage of the benzyl groups followed by deacecylation or, alternatively, via a pathway where the sequence of the deblocking reactions was reversed, gave crystalline title disaccharide 10. The structures of the compounds involved in the synthesis were confirmed by C NMR spectroscopy.     

Beta Sheets with a Twist: The Conformation of Helical Polyisocyanopeptides Determined by Using Vibrational Circular Dichroism

Detailed information on the architecture of polyisocyanopeptides based on vibrational circular dichroism (VCD) spectroscopy in combination with DFT calculations is presented. It is demonstrated that the screw sense of the helical polyisocyanides can be determined directly from the C?N‐stretch vibrational region of the VCD spectrum. Analysis of the VCD signals associated with the amide I and amide II modes provides detailed information on the peptide side‐chain arrangement in the polymer and indicates the presence of a helical β‐sheet architecture, in which the dihedral angles are slightly different to those of natural β‐sheet helices.     

The photolysis in polymer matrices of dyes containing a benzothioxanthene chromophore linked with a hindered amine

Novel dyes based on a benzothioxantheneimide chromophore covalently linked with a sterically hindered amine (HAS) were prepared and their light stability was tested in polymer matrices. The following dyes: 2-(2,2,6,6-tetramethyl-4-piperidyl)-thioxantheno[2,1,9-dej]isoquinoline-1,3-dione (BTXINH) and N-alkoxy derivative 2-(1-(1′-phenylethoxy)-2,2,6,6-tetramethyl-4-piperidyl)-thioxantheno[2,1,9-dej] isoquinoline-1,3-dione (BTXINOR) were prepared. For comparison the parent dye without HAS structural unit benzothioxanthene-3,4-dicarboxylic anhydride and the N-alkyl derivative 2-(1-dodecyl)-thioxantheno[2,1,9-dej]isoquinoline-1,3-dione (BTXID) and the stable nitroxyl radical 2-(1-oxo-2,2,6,6-tetramethyl-4-piperidyl)-thioxantheno[2,1,9-dej]isoquinoline-1,3-dione (BTXINO) were also tested. Their spectral properties, absorption and fluorescence have been examined in polystyrene (PS), poly(methyl methacrylate) (PMMA), poly(vinyl chloride) (PVC) and in isotactic polypropylene (PP). The light stability of these dyes and model compounds were examined in thin polymer films. The photolysis rate was monitored by UV spectroscopy and for all additives under study it was in the range 10⁻⁴–10⁻³ h⁻¹. The rate of decomposition was the lowest for the parent amine BTXINH in PMMA, PS and PVC. The rate constants of photolysis are about 10 times higher for all adducts and the lowest rate of decomposition in PP matrix was observed for BTXINOR. A distinct stabilization effect of HAS structural unit on the dye decomposition was not observed. The light stability of the dyes was more influenced by the selection of the polymer. The photolysis proceeds rather fast in PP, and moderately in PS and PVC compared to PMMA.     

Kinetics of hypericin association with low-density lipoproteins

Steady‐state and time‐resolved fluorescence spectroscopy have been used for the study of the incorporation kinetics of hypericin (Hyp) into low‐density lipoproteins (LDL). Biphasic kinetics of Hyp association with LDL was observed when solutions of Hyp and LDL were mixed at various concentration ratios. The rapid phase of Hyp incorporation is completed within seconds, while the slow phase lasts several minutes. The relative contributions of the individual phases show that a higher amount of Hyp molecules (65%) are incorporated into LDL in the second phase. The kinetics of the incorporation of Hyp into LDL particles preloaded with Hyp (Hyp/LDL = 25:1) was also investigated. The decreased intensity of Hyp fluorescence is a sign of the formation of Hyp aggregates after penetration of additional Hyp molecules into Hyp/LDL = 25:1 complex. The time dependence of Hyp fluorescence was measured after mixing the complex Hyp/LDL = 200:1 with appropriate amounts of free LDL molecules. For each final Hyp/LDL ratio, an increase in the intensity and lifetime of Hyp fluorescence was observed, suggesting a monomerization of Hyp aggregates. The half‐time of Hyp transfer from Hyp/LDL complex to LDL particles is similar to the half‐time of the slow phase of Hyp incorporation into free LDL particles.