Volume 4, Issue 6

Existence results for problems with monotone nonlinear boundary conditions obtained in the previous publications by the author for functional differential equations are transferred to the case of nonconvex differential inclusions with the help of the selection theorem due to A. Bressan and G. Colombo.

When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad. Sci. Paris 301: 337–340, 1985], [Guin, J. Pure Appl. Algebra 50: 109–137, 1988] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms and .

The notion of an internal category in a left exact cosimplicial category is introduced. For any topos over sets a certain left exact cosimplicial category is constructed functorially and the category of internal categories in it is investigated. The notion of a fundamental group is defined for toposes admitting the notion of “a discrete category.”

Sufficient conditions of solvability and unique solvability of the boundary value problem u ( m ) ( t ) = f ( t , u ( τ 11 ( t )), . . . , u ( τ 1 k ( t )), . . . , u ( m –1) ( τ m 1 ( t )), . . . . . . , u ( m –1) ( τ mk ( t ))), u ( t ) = 0, for t ∉ [ a , b ], u ( i –1) ( a ) = 0 ( i = 1, . . . , m – 1), u ( m –1) ( b ) = 0, are established, where τ ij : [ a , b ] → R ( i = 1, . . . , m ; j = 1, . . . , k ) are measurable functions and the vector function f : ] a , b [× R kmn → R n is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

Sufficient conditions are found for the bifurcation of flow of a viscous heat-conducting fluid between two rotating permeable cylinders.

Using a multidimensional analogue of Vinogradov's inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in and of the system (cos j 1 π ξ 1 . . . cos j s π ξ s , 0 ≤ j 1 , . . . , j s ≤ k , j 1 + . . . + j s ≥ 1), where ( ξ 1 , . . . , ξ s ) is uniformly distributed in [0; 1] s .

Boundary properties of first-order partial derivatives of the Poisson integral are studied in the half-space .