Water wave radiation by a sphere submerged in water with an ice-cover

Using the multipoles method, we formulate the problems of radiation (both heave and sway) of water waves by a submerged sphere in deep as well as in uniform finite depth water with an ice-cover, with the ice-cover being modelled as an elastic plate of very small thickness. In each case this leads to an infinite system of linear equations which are solved numerically by standard techniques. The added-mass and damping coefficients for a heaving and swaying sphere are obtained and depicted graphically against the wave number for various values of the radius of the submerged sphere and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. When the flexural rigidity is taken to be zero, the numerical results for the added-mass and damping coefficient for water with a free surface are recovered.     

Numerical method and models for anti-plane strain of a system with a thin elastic wedge

The paper presents a general method to find asymptotics for a (multi-)wedge system containing a thin wedge. It employs separation of the symmetric and anti-symmetric parts of the boundary displacements and tractions of the wedge. The method is applicable when the angle of the thin wedge turns to zero. A physical interpretation of the derived equations is obtained by using power expansions of non-polynomial functions, which appear after the Mellin transform. We establish that the first term in the expansion of the symmetric part corresponds to shear, while the first term of the anti-symmetric part describes deflection of the wedge axis. Numerical experiments, performed by using a code developed on the basis of the theory, show that using only the first terms of the expansions insignificantly influence accuracy: the approximate results coincide with the exact values of roots to the third significant digit even for the wedge angle of 30°.     

Ramsey numbers for graphs with five vertices

The Ramsey numbers r(F¹, F²) are tabulated for essentially all but six pairs of graphs F¹ and F² with five vertices.     

Deuterium exchange of 4-pyrimidones and 4-pyrimidthiones

The hydrogen-deuterium exchange of H-2 in 4-pyrimidone, a number of 1- and 3-alkyl-4-pyrimidones and several of their thione analogs in deuterium oxide at moderate temperatures is reported. This reaction is apparently not susceptible to acid or base catalysis. Cations of the 4-pyrimidones were found not to exchange H-2 at all in acid media. Catalytic amounts of bases do not alter the exchange rates whereas more concentrated alkali decompose the pyrimidone. However, 1,4(3,4)-dihydro-1,3-dimethyl-4-oxopyrimidinium iodide (and its 6-methyl analog) exchange H-2 quite rapidly in neutral deuterium oxide, almost instantly when triethylamine is added as a catalyst, and very slowly in dilute solutions of deuterium chloride in deuterium oxide. Mechanisms are proposed to account for these phenomena.     

Photochemical and thermal transformations of azomethine imines

The thermal and photochemical fragmentations of a few bisazoalkenes have been investigated. 2-Phenyl-4,5-disubstituted-1, 2,3-triazoles were obtained both in the thermolysis and photolysis of 1, 2-bisphenylazo-(4, 4′-dichloro) stilbene, 1, 2-bisphenylazo(4, 4′-dimethoxy)stilbene, 1, 2-bisphenylazocyclohexene and o-(phenylazo) phenyldiazocyanide. Both 2, 3-bisphenylazo-2-butene and 1, 2-bisphenylazoethylene failed to undergo either photolysis or thermolysis in the expected manner. However, 2, 3-bisphenylazo-2-butene underwent an acid-catalysed valence isomerisation to anhydro 1-phenylimino-2-phenyl-4, 5-dimethyl-1, 2, 3-trizolium hydroxide, which on photolysis gave 2-phenyl-4, 5-dimethyl-1, 2, 3-triazole. The same iminotriazolium intermediate gave a cycloadduct, 2, 6-diphenyl-3, 3a-dimethyl-4, 5-dicarbomethoxypyrazolino [2.3.c][1.2.3] triazole, on treatment with dimethyl acetylenedicarboxylate, whereas treatment with carbon disulphide gave 2-phenyl-4, 5-dimethyl-1, 2, 3-triazole. Both photolysis and thermolysis of C-biphenylene-N^α(4-chlorophenyl)-N^β-cyanoazomethine imine gave 9-fluorenone-N- (4-chlorophenyl) anil. Photolysis of 1, 2-bisphenylazoacenaphthylene in methanol gave acenaphthenequinone monophenylhydrazone.     

Thermal addition of dibenzoylacetylene to cycloheptatriene and the photo-rearrangement of the thermal adduct

The reaction of cycloheptatriene (CHT) with dibenzoylacetylene (DBA), in refluxing toluene gave a mixture of products consisting of 6,7-dibenzoyltricyclo[3.2.2.0^2.4]nona-6,8-diene 6 and 3,4-dibenzoylbicyclo [3.2.2]nona-2,6,8-triene 7. Neat heating of a mixture of CHT and DBA around 95–100° gave besides, 6 and 7, small amounts of two dimers of DBA. Photolysis of 6 in benzene solution gave a lactonic, rearrangement product. 25. The structures of the various products have been established and reasonable mechanisms for their formation have been suggested.