Volume 5, Issue 6

The heat equation with a small parameter, is considered, where ε ∈ (0, 1), 𝑚 < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.

The structure of a Lie superalgebra is defined on the space of multiderivations of a commutative algebra. This structure is used to define some cohomology algebra of Poisson structure. It is shown that when a commutative algebra is an algebra of 𝐶 ∞ -functions on the 𝐶 ∞ -manifold, the cohomology algebra of Poisson structure is isomorphic to an algebra of vertical cohomologies of the foliation corresponding to the Poisson structure.

Using the method of separation of variables, an exact solution is constructed for some boundary value and boundary-contact problems of thermoelastic equilibrium of one- and multilayer bodies bounded by the coordinate surfaces of generalized cylindrical coordinates ρ , α , 𝑧. ρ, α are the orthogonal coordinates on the plane and 𝑧 is the linear coordinate. The body, occupying the domain Ω = { ρ 0 < ρ < ρ 1 , α 0 < α < α 1 , 0 < 𝑧 < 𝑧 1 }, is subjected to the action of a stationary thermal field and surface disturbances (such as stresses, displacements, or their combinations) for 𝑧 = 0 and 𝑧 = 𝑧 1 . Special type homogeneous conditions are given on the remainder of the surface. The elastic body is assumed to be transversally isotropic with the plane of isotropy 𝑧 = const and nonhomogeneous along 𝑧. The same assumption is made for the layers of the multilayer body which contact along 𝑧 = const.

A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.

The necessary and sufficient conditions are found for the weight function 𝑣, which provide the boundedness and compactness of the Riemann–Liouville operator 𝑅 α from 𝐿 𝑝 to . The criteria are also established for the weight function 𝑤, which guarantee the boundedness and compactness of the Weyl operator 𝑊 α from to 𝐿 𝑞 .

We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

We give a characterization of the weights 𝑢(·) and 𝑣(·) for which the fractional maximal operator 𝑀 𝑠 is bounded from the weighted Lebesgue spaces 𝐿 𝑝 (𝑙 𝑟 , 𝑣𝑑𝑥) into 𝐿 𝑞 (𝑙 𝑟 , 𝑢𝑑𝑥) whenever 0 ≤ 𝑠 < 𝑛, 1 < 𝑝, 𝑟 < ∞, and 1 ≤ 𝑞 < ∞.