Confidence regions for solutions of applied boundary-value problems

We propose a practicable technique for determining the confidence regions of solutions of partial differential equations when errors are present both in the process of solution and in the process of specification (mathematical modeling) of the applied problem. The study focuses on applied boundary-value problems in the theory of shells. We elucidate the possibilities of determining the degree of confidence (both posterior and prior) of the numerical results depending on the use of various numerical schemes and mathematical models for some particular cases. Graphical and numerical data are presented demonstrating the application of the proposed technique.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 81–90, 1987.     

Nonintegrable field equations and homoclinic loops of Hamiltonian systems

Connections between soliton or self-localized states of nonlinear wave equations and special objects (homo- and heteroclinic trajectories) of dynamical systems are considered.     

Hydroxy-1-aminoindans and derivatives: preparation, stability, and reactivity

The chemical stability and reactivity of hydroxy-1-aminoindans and their N-propargyl derivatives are strongly affected by the position of the OH group and its orientation relative to that of the amino moiety. Thus, the 4- and 6-OH regioisomers were found to be stable, while the 5-OH analogues were found to be inherently unstable as the free bases. The latter, having a para orientation between the OH and the amino moieties, could be isolated only as their hydrochloride salts. 7-Hydroxy-1-aminoindans and 7-hydroxy-1-propargylaminoindans represent an intermediate case; while sufficiently stable even as free bases, they exhibit, under certain experimental conditions, unexpected reactivity. The instability of the 5- and 7-hydroxy-aminoindans is attributed to their facile conversion to the corresponding, reactive quinone methide (QM) intermediates. The o-QM obtained from 7-hydroxy-aminoindans was successfully trapped with ethyl vinyl ether via a Diels-Alder reaction to give tricyclic acetals 32a,b.     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.