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.     

Studies on chromium(III) complexes of some new dithiocarbamato ligands

Summary Tris-chelates of chromium(III) have been synthesised with five new dithiocarbamates, [RR'NCS₂]^–, where R=PhCH₂ and R/t'=H, PhCH₂, Me, Et and i-Pr. Magnetic moments together with electronic, i.r. and e.s.r spectra of the complexes have been described. Various ligand-field parameters have been evaluated and discussed.     

A Sensitive HPTLC method for Estimation of Swertiamarin in Enicostemma littorale Blume,Swertia chirata (Wall) Clarke,and in formulations containing E. littorale

JPC – Journal of Planar Chromatography – Modern TLC - Swertiamarin is a secoiridoid glucoside present in members of the Gentianaceae family, including Swertia chirata (Wall) Clarke, S....     

Synthesis of a novel cis-proline-derived cyclic type VI beta-turn mimic via ring-closing metathesis

A cis-proline derived cyclic mimic of a type VI beta-turn is synthesized via a ring-closing metathesis reaction. The solution NMR conformational study indicates that the major conformer of the cyclic peptide adopts a type VIa beta-turn in CDCl(3) and a type VIb beta-turn in DMSO-d(6).     

Five-coordinate Cobalt(II), Nickel(II) and Zinc(II) Complexes Derived from 2-pyridine-2-yl-3-(pyridine-2-carboxylideneamino)-1,2-dihydroquinazolin-4(3H)-one. The Crystal Structure of the Cobalt(II) Complex

The synthesis of cobalt(II), nickel(II) and zinc(II) complexes of 2-pyridine-2-yl-3(pyridine-2-carboxylideneamino)-1,2-dihydroquinazolin-4(3H)-one is described. The ligand and metal complexes were characterized by elemental analysis, conductivity measurements, spectral (u.v.–vis., i.r., 1D n.m.r., 2D hetcor and mass) and thermal studies. The cobalt(II) complex crystallizes as pink crystals in the monoclinic crystal system, space group P21/n with a = 10.066(6) Å, b = 15.929(9) Å, c = 12.624(7) Å, α = 90.00(9)°, β = 110.850 (8)°, γ = 90.00, V = 1891.5 (18) ų and Z = 4. The geometry around the cobalt atom is distorted trigonal bipyramidal with τ = 0.83 [structural parameter, τ = (β − α)/60; where α and β are the two basal angles in a five coordinate complex].     

Reverse turn induced pi-facial selectivity during polyaniline-supported cobalt(II) salen catalyzed aerobic epoxidation of N-cinnamoyl L-proline derived peptides

A novel chemo- and diastereoselective aerobic epoxidation of the N-cinnamoyl peptides catalyzed by polyaniline-supported cobalt(II) salen (PASCOS) is described. The N-cinnamoyl proline derived peptides 1 show a high pi-facial selectivity during these epoxidations. The origin of this diastereoselectivity in 1 has been attributed to (i) the propensity of the N-cinnamoyl proline amide to exist predominantly as trans rotamer in CDCl3, DMSO-d6, and CH3CN medium and (ii) existence of these peptides as organized structures (gamma- and beta-turns) due to the presence of intramolecular hydrogen bonds. An extensive solution NMR and MD simulation study on 1d and 1f indicates that the origin of the high pi-facial selectivity is due to the well-defined gamma- and beta-turns which result in the hindrance of one face of the cinnamoyl double bond in the transition state of the epoxidation reaction.     

Enantioselective total synthesis of polyoxygenated cyclohexanoids: (+)-streptol, ent-RKTS-33 and putative ‘(+)-parasitenone’. Identity of parasitenone with (+)-epoxydon

Short, simple and enantioselective syntheses of the natural product (+)-streptol, the non-peptide apoptosis inhibitor ent-RKTS-33 and the putative structure of ‘parasitenone’ have been accomplished from the readily available chiral building block. ‘Parasitenone’ has been shown to be identical with the known natural product epoxydon.     

Oxidative transformation of ciprofloxacin by alkaline permanganate – A kinetic and mechanistic study

This spectroscopic study presents the kinetics and degradation pathways of oxidation of ciprofloxacin by permanganate in alkaline medium at constant ionic strength of 0.04 mol⁻³. Orders with respect to substrate, oxidant and alkali concentrations were determined. Effect of ionic strength and solvent polarity of the medium on the rate of the reaction was studied. The oxidation products were identified by LC-ESI-MS technique. Product characterization of ciprofloxacin reaction mixtures indicates the formation of three major products corresponding to m/z 263, 306, and 348 (corresponding to full or partial dealkylation of the piperazine ring). The piperazine moiety of ciprofloxacin is the predominant oxidative site to KMnO₄. Product analyses showed that oxidation by permanganate results in dealkylation at the piperazine moiety of ciprofloxacin, with the quinolone ring essentially intact. The reaction kinetics and product characterization point to a reaction mechanism that likely begins with formation of a complex between ciprofloxacin and the KMnO₄, followed by oxidation at the aromatic N1 atom of piperazine moiety to generate an anilinyl radical intermediate. The radical intermediates subsequently undergo N-dealkylation. Investigations of the reaction at different temperatures allowed the determination of the activation parameters with respect to the slow step of proposed mechanism. The proposed mechanism and the derived rate laws are consistent with the observed kinetics.     

Some compounds of iron(III) with bidentate bases

Summary The iron(III) compound, FeCl(OH)(MeCO₂) · MeCO₂H · 0.5 H₂O, (1), interacts with the sample bidentate ligand, 2,2-bipyridyl (bipy), to form an aduct, Fe₂Cl₂(OH)₂(MeCO₂)₂ · (bipy). By the loss of protons, anthranilic (anthH) and salicylic (salH₂) acids behave as uninegative charged ligands to give FeCl(anth)₂ · H₂O and FcCl(salH)₂ · H₂O, respectively. The former decomposes on heatingin vacuo to form Fe₂O(anth)₄ while the latter yields Fe₂O(sal)₂. Acetylacetone (acacH) and 8-hydroxyquinoline (quinH) give FeCl(acac)₂ and FeCl(quin)₂ · (quinH). The latter desolvates at 140°/10^–4 torr to form FeCl(quin)₂. l.r. spectra, thermal decomposition, molar conductance and magnetic susceptibility studies at room temperature have been used to characterize these compounds.     

Reactions of titanium(IV) isopropoxide with aliphatic and aromatic hydroxy esters,part I. 5-, 6-, 7- and 8-coordinate titanium(IV) complexes

Summary Reactions of titanium(IV) isopropoxide with ethyl 1-hydroxycyclohexylacetate, ethyl 2-hydroxy-2-arylpropionate, ethyl 2-hydroxy-2-arylbutyrate and ethyl 2-hydroxy-2-arylhydrocinnamate have been studied in dry benzene in different molar ratios under strictly anhydrous conditions. The Ti(OPr-i)₄-_nL_n type compounds, where L = hydroxy ester and n = 1 to 4, have been isolated and characterised by elemental analysis, molecular weight, i.r. and n.m.r. spectral studies. These products may be distilled unchanged under reduced pressure.Reprints of this are not available.