The paper is devoted to the problem of describing unary algebras whose congruence lattices have a given property. By now this problem has been solved for algebras with one unary operation. In the paper it is shown that this problem is much more difficult for arbitrary commutative unary algebras. We give some necessary conditions for such lattices to be distributive or modular. Besides, it is proved here that a lattice of all subsets of a set is isomorphic to the congruence lattice of a suitable connected commutative unary algebra.
This article gives an overview of several actual problems of optimal coefficients method. This overview was done on September 12, 2013 on XI internation conference «Algebra and number theory: modern problems and applications» in Saratov city.
An asymptotic formula for the number of representations of a positive integer N in the form q1 + q2 + [®q3] is obtained, where q1, q2, q3 are squarefree numbers and ® > 1 is a fixed irrational algebraic number.
The graded version ofWedderburn–Artin theorem is obtained. It gives description of semisimpleG-graded ring for arbitrary groupG. Homological classification of graded semisimple rings is given.
In this paper was described a class of Artin’s L-functions, each of which is meromorphic, their poles lays on the critical line Re s = 1/2 and coincides with zeroes of Dedekind’s Z-functions of some fields.
The inverse problem of recovering Sturm–Liouville operators on the half-line with a nonintegrable Bessel-type singularity in an interior point from the given Weyl function is studied. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided. Necessary and sufficient conditions of the solvability of the inverse problem are obtained.
In this paper we obtain necessary and sufficient conditions for a function to belong to the Besov–Potapov classes. Using functions with Fourier coefficients with respect to multiplicative systems from the class GM, we show the sharpness of some these results.
