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A subgroup H of a finite group G is said to be F(G)-subnormal if it is subnormal in HF(G), where F(G) is the Fitting subgroup of G. In the paper, the problem of whether or not a formation β contains products of F(G)-subnormal β-subgroups of finite solvable groups is studied. In particular, solvable saturated formations β with this property are described. Formation properties of groups having three solvable F(G)-subnormal subgroups with pairwise coprime indices are studied. The supersolvability of any group G having three supersolvable F(G)-subnormal subgroups whose indices in G are pairwise coprime is proved.

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

A linear homogeneous congruence ay ≡ bY (mod q) is considered and an order-sharp upper bound for the number of its solutions is proved. Here a, b, and q are given jointly coprime numbers and y and Y are coprime variables in a given closed interval such that the number y/Y can be expanded in a continued fraction with partial quotients from some alphabet A ⊆ ℕ. For A = ℕ (and without the assumption that y and Y are coprime), a similar problem was solved by N. M. Korobov.

The paper is devoted to the study of finite groups factorizable by generalized subnormal subgroups of coprime indices.

Let G be a group of finite generic rank, φ an injective endomorphism of the group G, and G(φ) the descending HNN-extension of G corresponding to the endomorphism φ. Let the index of the subgroup Gφ in G be finite and equal to n. It is proved that, if the group G is almost residually π-finite for some set π of primes coprime to n, then the group G(φ) is residually finite. This generalizes a series of known results, including the Wise-Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.

Suppose that $${g\left( n \right)}$$ is an additive real-valued function, W(N) = 4+ $$\mathop {\min }\limits_\lambda $$ ( λ2 + $$\sum\limits_{p < N} {\frac{1}{2}} $$ min (1, ( g(p) - λlog p)2), E(N) = 4+1 $$\sum\limits_{\mathop {p < N,}\limits_{g(p) \ne 0} } {\frac{1}{p}.} $$ In this paper, we prove the existence of constants C1, C2 such that the following inequalities hold: \(\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) \in [a,a + 1)} \right\}} \right| \leqslant \frac{{C_1 N}}{{\sqrt {W\left( N \right)} }},\)\(\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) = a} \right\}} \right| \leqslant \frac{{C_2 N}}{{\sqrt {E\left( N \right)} }},\). The obtained estimates are order-sharp.

Suppose that g(n) is a real-valued additive function and τ(n) is the number of divisors of n. In this paper, we prove that there exists a constant C such that \(\sup \limits_a \sum\limits_{n

The Frobenius problem is to find a method ( $$ =$$ algorithm) for calculating the largest “sum of money” that cannot be given by coins whose values $$b_0 ,{\text{ }}b_1 ,{\text{ }} \ldots {\text{, }}b_w$$ are coprime integers. As admissible solutions (algorithms), it is common practice to study polynomial algorithms, which owe their name to the form of the dependence of time expenditure on the length of the original information. The difficulty of the Frobenius problem is apparent from the fact that already for $$w = 3$$ the existence of a polynomial solution is still an open problem. In the present paper, we distinguish some classes of input data for which the problem can be solved polynomially; nevertheless, argumentation in the spirit of complexity theory of algorithms is kept to a minimum.

Alpern proved that the phase space of an aperiodic measure preserving automorphismT can be decomposed into Rokhlin-Halmos towers of given heightsh i and weightsm i whenever the numbersh i are relatively prime. In this paper an extension of the Alpern theorem to the case of free ℤ d -actions is given. Namely, we investigate the decomposability of the action phase space into towers of rectangular, form and present conditions on the configuration (the set of tower forms) sufficient for the existence of such a decomposition. The proof of the main result uses the technique of tilings.

Let Λ be an arbitrary set of positive integers andSn(Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |Sn(Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |Sn(Λ)| stands for the number of elements in the finite setSn(Λ).

LetM be a continuous real-valued function on ℂn,n≥1, and letE(M) be the class of entire functions f on ℂn such that log ¦f¦≤M. We give a dual statement for the problem of the description of zero sets of functions inE(M) and the possibility of representing functions f meromorphic on ℂn by ratios f=g/h, whereg andh are entire functions belonging toE(M) and coprime at each pointz∈ℂn. The dual approach suggested in the paper is new even for the casen=1.