We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have smooth inertial manifold in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.
By means Stepanov's method the bound of cardinality of the intersection of additive shifts of several subgroups of multiplicative group of the finite field was obtained. This bound apply to some question of additive decomposition of subgroups.
In this paper we prove that there exists function f from L sqrt ( ln+ L ) such that Vilenkin-Fourier serie of diverges almost everywhere
The vertices of the commuting graph of a semigroup S are the noncentral elements of this semigroup, and its edges join all pairs of elements g, h that satisfy the relation gh = hg. The paper presents a proof of the fact that the diameter of the commuting graph of the semigroup of real matrices of order n ≥ 3 is equal to 4. A survey of results in that subject matter is presented, and several open problems are formulated.