Volume 4, Issue 1

The initial boundary value problems with mixed boundary conditions are considered for a system of partial differential equations of generalized electrothermodiffusion. Approximate solutions are constructed and a mathematical substantiation of the method is given.

Sufficient conditions for oscillation of a certain class of nonlinear third-order differential equations are found.

For a Köthe sequence space, the classes of Λ 0 -nuclear spaces and spaces with the Λ 0 -property are introduced and studied and the relation between them is investigated. Also, we show that, for Λ 0 ≠ c 0 , these classes of spaces are in general different from the corresponding ones for Λ 0 = c 0 , which have been extensively studied in the non-archimedean literature (see, for example, [De Grande-De Kimpe, Proc. Konink. Nederl. Akad. Wetensch A75: 113–129, 1972]–[Kiyosawa, Canad. Math. Bull. 32: 450–458, 1989])

For the torsion-free modules over noncommutative principal ideal domains von Staudt's theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruple is calculated.

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.

Translation invariant subbases of the differential basis B 2 (formed of all intervals), which differentiates the same class of all nonnegative functions as B 2 does, are described. A possibility for extending the results obtained to bases of more general type is discussed.