The Cauchy problem for the nonlinear Schrödinger equation in H1

We consider the initial value problem for the nonlinear Schrödinger equation in H¹(Rⁿ). We establish local existence and uniqueness for a wide class of subcritical nonlinearities. The proofs make use of a truncation argument, space-time integrability properties of the linear equation, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.Research supported by NSF grants DMS 8201639 and DMS 8703096.     

The fermat equation over quadratic fields

Kummer's method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).     

Mass spectra of lincomycin,lincomycin analogs and degradation products

The principal ions observed in the mass spectra of lincomycin hydrochloride, some of its analogs and two degradation products are discussed. Lincomycin-related compounds can be characterized by examination of the molecular ion and four other ions. Plausible mechanistic interpretations of these ions are presented.     

Synthesis of 7,8-dihydroxychlorpromazine and analogs

7,8-Dihydroxy derivatives of chlorpromazine, promazine, phenothiazine, and 7-methoxy-8-hydroxychlorpromazine and its nor₁ analog were prepared through multi-step syntheses. Structures were confirmed by chemical and spectroscopic evidence. The unusual nmr spectral shifts in several of these compounds are discussed.     

ORGANIC PHOTOCHEMISTRY—XV. APPLICATIONS OF PHOTOCYCLOADDITION REACTIONS IN NATURAL PRODUCT SYNTHESES*,†

Abstract— Photocycloaddition reactions which have been employed as one of the steps in the total/ synthesis of natural products are reviewed. The utility of the cycloadducts in these syntheses is also discussed.     

On the estimate of the stochastic layer width for a model of tracer dynamics

An analytical estimate of the width of the generated chaotic layer in a time-periodically driven stream function model for the motion of passive tracers is discussed. It is based essentially on the method of the separatrix map and the use of the Melnikov theory. Energy-time variables are used to derive lower bounds for the half width of the layer. In order to perform a comparison with numerical simulations, the results are transformed into space variables. The analytic results of the layer thickness in both parallel and perpendicular directions to the shear flow are compared with numerical computations and some systematic deviations are discussed.     

Reactions of the perfluoroalkyltriazapentadiene Ph2N3C2(C3F7)2H with acids,bases, and water

Treatment of the bis(perfluoroalkyl)triazapentadiene PhN=C(C(3)F(7))-N=C(C(3)F(7))-NHPh, 2, with bases affords salts of the conjugate base. Alkylation of the Na(+) salt with CH(3)I yields PhN=C(C(3)F(7))-N=C(C(3)F(7))-NMePh. The crystal structure of [Bu(4)N][Ph(2)N(3)C(2)(C(3)F(7))(2)] demonstrates a twisted, zigzag geometry for the anion in the solid state, but in solution, it is conformationally unstable. Both compounds are stable in aqueous methanol, but hydrolysis occurs under acidic conditions. Protonation with CF(3)SO(3)H acid occurs at the sp(2) nitrogen to give [PhN-(H) HC(C(3)F(7))-C(C(3)F(7))-NHPh][CF(3)SO(3)]. Heating 2 in CF(3)SO(3)H produces 2,4-bis(heptafluoropropyl)-1,3-quinazoline.