Notes on the non-autonomous Lyness equation

We prove that the solutions of the non-autonomous Lyness equation x_n+1=(x_n+b_n)/x_n−1, b_n>0, x₀>0, x₁>0, are bounded away from zero and infinity if the sequence b_n is monotonic. We also give an example to show that the solutions need not be bounded even when the sequence b_n attains only two values.     

Synthesis of protected derivatives of O-phosphotyrosine incorporation in a heptapeptide

The solid phase synthesis of the phosphopeptide Leu-Arg-Arg-Ala-Tyr(P)-Leu-Gly is reported via the stepwise incorporation of the protected phosphoamino acid Nα -tert-butyloxycarbonyl-O-dimethyl-phosphono-L-tyrosine.     

X-ray and mössbauer studies of tricyclohexyltin(IV) halides. The crystal structures of (cyclo-C6H11)3SnX (X = F,Br and I)

The structures of tricyclohexyltin fluoride (I), bromide (II) and iodide (III) have been determined by X-ray analysis. Compound I crystallizes in the space group P2₁/m with a = 10.422(6), b = 17.238(9), c = 5.769(3) Å, β = 104.6(1)° and Z = 2. Compounds II and III crystallize in the space group Pcmn with a = 10.427(6), b = 16.914(9), c = 11.366(6) Å, Z = 4; and a = 10.400(6), b = 16.900(10), c = 11.400(4) Å, Z = 4, respectively. All three structures consist of discrete tetrahedral (cyclo-C₆H₁₁)₃SnX units.The temperature dependence of the Mössbauer resonance areas has been examined in order to obtain information about the relationship between chemical structure and lattice dynamics.     

Long-range dispersion coefficients for like centrosymmetric linear molecules and an application to H2-H2

The real spherical tensor theory of long-range intermolecular coefficients developed in previous papers is applied to derive explicit formulae for the first three dispersion coefficients for like centrosymmetric linear molecules. The expansion of angle-dependent coefficients in associated Legendre polynomials allows one to identify the isotropic and anisotropic components of the dispersion interaction in terms of London dispersion constants, the treatment of higher coefficients being simplified by the coupling of the elementary (l, l′)-polarizations to resultant angular momenta L_A and L_B onto each molecule. The contributions from all coupling schemes are given explicitly for C₆, C₈, C₁₀, and numerical results are presented for H₂-H₂ using two-term reduced spectra values from the Kaiserlautern group.     

On the supposed equilibrium Ph6Pb2Ph4Pb+Ph2Pb in solution

The reaction between pentadeuteriophenyllithium and hexaphenyldilead combined with new measurements of isotopic exchange between Ph₆Pb₂ (labelled with RaD) and Ph₄Pb, have shown that the dilead compound is undissociated in solution. These results cast doubt upon the previously reported equilibrium: Ph₆Pb₂Ph₄Pb+Ph₂Pb.     

Synthesis of some 2-(heteroarylamino)benzimidazoles and their cyclization with phosgene

2-(Benzimidazol-2-ylamino)pyridine (4a) , 2-(benzimidazol-2-ylamino)pyrazine (4b) , and 2-(benzimidazol-2-ylamino)thiazole (4c) underwent a ring-closure reaction on treatment with phosgene affording 6H-pyrimido-[1′,2′:5,4][1,3,5]triazino[1,2-a]benzimidazol-6-one (1a) , 6H-pyrazino[1′,2′:5,4][1,3,5]triazino[1,2-a]benzimidazol-6-one (1b) , and 5H-thiazolo[2′,3′:4,5][1,3,5]triazino[1,2-a]benzimidazol-5-one (1c) respectively. The structure of these hitherto unknown heterocyclic systems was confirmed by their ir and mass spectra.     

Synthesis of new 1,2,3-triazolo[1,2-a]benzotriazole derivatives or substituted 2,3-benzo-1,3a,6,6a-tetraazapentalenes. II

This paper continues the synthesis of new 1,2,3-triazolo[1,2-a]benzotriazoles or 2,3-benzo-1,3a,6,6a-tetrazapentalenes to submit to biological assays. The derivatives were obtained by deoxycyclization reactions of appropriate nitrophenyl-1,2,3-triazole derivatives and by thermal decomposition of appropriate azidophenyl-1,2,3-triazoles (Schemes 1 and 2). Some attempts to extend these synthetic routes to the preparation of 1,2,4-triazolo[1,2-a]benzotriazoles (Scheme 3) and 1,2,3-triazolo[1,2-b]-4H-1,2,3-benzo-triazines (Scheme 4) completely failed.     

A generalised London formula for dispersion coefficients

A generalised London formula is presented for the one-centre calculation of dispersion coefficients of spherically symmetric atoms in S-states in terms of the static multipole polarisabilities and the related orbital excitation energies of the individual atoms. The results obtained for the C₆, C₈, C₁₀ coefficients for H₂ and for the C₆ coefficients for He₂, Li₂, Be₂ and their mixtures are highly encouraging.     

New 1,2,3‐triazolo[4,5‐e]1,2,4‐triazolo[4,3‐c]pyrimidine derivatives II

The 7‐chloro‐3‐(2‐chlorobenzyl)‐ and 7‐chloro‐3‐(2‐fluorobenzyl)‐1,2,3‐triazolo[4,5‐d]pyrimidines ( 1 and 4 ), by nucleophilic replacement with some hydrazides, gave the corresponding 7‐hydrazidoderivatives ( 2a‐e and 5a‐e ). These, by heating in Dowtherm, underwent an intramolecular cyclization to form the new tricyclic 7‐substituted‐3‐(2‐chlorobenzyl)‐ and 3‐(2‐fluorobenzyl)‐1,2,3‐triazolo[4,5‐e]1,2,4‐triazolo[4,3‐c]pyrimidines ( 3a‐d and 6a‐d ). The 7‐hydrazino‐3‐(2‐chlorobenzyl)‐ and 7‐hydrazino‐3‐(2‐fluorobenzyl)‐triazolo‐pyrimidines ( 9a and 9b ) were also prepared via the corresponding mercapto ( 7a and 7b ) and thiomethyl ( 8a and 8b ) derivatives.