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The paper refines the currently known estimates of sums of sine series with monotone coefficients, as well as the upper bound for the sum of the cosine series with convex coefficients.

We refine the classical boundedness criterion for sums of sine series with monotone coefficients $$b_k$$ : the sum of a series is bounded on $$\mathbb R$$ if and only if the sequence $${\{kb_k\}}$$ is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a series via a special norm of the sequence $$\{kb_k\}$$ . The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most $$0.2$$ .

New estimates of the sums of sine series with monotone coefficients of special classes in terms of the Salem majorant are obtained. The asymptotic sharpness of the obtained estimates for sequences of coefficients from the classes under consideration is proved.

In this paper, the extremal problem of finding the infimum of the positive roots of the sum of a sine series with monotone coefficients for special subclasses of such series is solved.

It is well known that the least possible type under the order ρ from the class of entire functions with upper root density 1 (for the exponent ρ) is 1/(eρ). The author proves that if all the roots of entire functions lie on one ray, then the situation is different: the least type for such a class on the set of orders (1,+∞) ∖ ℕ is distinct from zero and is bounded above.

For a real measure with variation V(x) satisfying the estimate V(x) ≤ c 0 exp(Cx) and for the Laplace transform holomorphic in the disk {¦ -C¦ ≤ C} and having at least one pole of order m, we obtain lower bounds for the positive and negative parts of the measure V ± (x) > cx m , x > x 0. We establish lower bounds for V +- (x) on “short” intervals. Applications to number theory of the results obtained are considered.

In this paper, we obtain two-sided estimates with sharp constants of the sums of sine series with a convex sequence of coefficients tending to zero.

An example of a ring variety without an independent basis is constructed. It is proved that this variety is the intersection of two independently based varieties.

An example of a series of varieties of rings\(\mathfrak{X}_p \) with the finite basis property is constructed for which the word problem in the relatively free ring\(F_n \mathfrak{X}_p \) of rankn in the variety\(\mathfrak{X}_p \) is decidable if and only ifn

On the basis of operator extension theory, a model of the quantum point is constructed as a resonator with semitransparent boundary. It is proved that the model operator is the limit (in the resolvent sense) of short-range Hamiltonians.

The existence of a finitely based variety of anticommutative rings (in the sense of the identityx 2=0) with unsolvable equational theory is proved.

A result illustrating the complexity of describing the varieties of rings with undecidable equational theory is obtained.

In this paper we consider an extremum problem for even periodic functions having small intervals as their supports and subject to restrictions on the sum of the absolute values of their Fourier coefficients.

Function classes are described in which the moments problem $$\int_0^{ + \infty } {t^{\alpha _n } f(t)dt = c_n } $$ is solvable for any right-hand sides from a sequence {c n }. Constructive solutions are given. The results obtained generalize and supplement some theorems proved earlier by other mathematicians.

For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a hole and on a semi-ellipse connected by a hole with a half-plane, we justify the choice of extension that guarantees the coincidence of the model solution with the solution of the “actual” problem in the far zone with a high degree of accuracy.

Lower bounds for the absolute values of the functions\(M(x) = \sum\nolimits_{n \leqslant x} {\mu (n)} \) and\(\Delta (x) = \left( {\sum\nolimits_{n \leqslant x} {\Lambda (n)} } \right) - x\), whereμ is the Mobius function and Λ is the Manholdt function, are obtained.