The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.

Let \phi \in S with \int \phi (x) dx = 1, and define \phi t(x) = 1 tn \phi \Bigl( x t \Bigr) , and denote the function family \{ \phi t\ast f(x)\} t>0 by \Phi \ast f(x). Let \scrJ be a subset of \BbbR (or more generally an ordered index set), and suppose that there exists a constant C1 such that \sum t\in \scrJ | \^\phi t(x)| 2 < C1 for all x \in \BbbR n. Then i) There exists a constant C2 > 0 such that \| V2(\Phi \ast f)\| Lp \leq C2\| f\| Hp, n n + 1 < p \leq 1 for all f \in Hp(\BbbR n), n n + 1 < p \leq 1. ii) The \lambda -jump operator N\lambda (\Phi \ast f) satisfies \| \lambda [N\lambda (\Phi \ast f)]1/2\| Lp \leq C3\| f\| Hp, n n + 1 < p \leq 1, uniformly in \lambda > 0 for some constant C3 > 0.

With the help of the formula for the general solution of a difference equation with constant coefficients, it is shown that the set of solutions to this equation contains classical solutions of the type k^m\λ^k. We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.

We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.

For the polynomial P(z) = n \sum j=0 cjzj of degree n having all its zeros in | z| \leq k, k \geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P \prime (z)| \geq n \biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) \biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n \geq 2.

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

The paper investigates sufficient conditions for the absolute convergence of trigonometric Fourier series of almost-periodic functions in the sense of Besikovitch in the case when the Fourier exponents have a single limiting point at infinity. A higher-order modulus of continuity is used as a structural characteristic of the function under consideration.

The Baillie PSW hypothesis was formulated in 1980 and was named after the authors R. Baillie, C. Pomerance, J. Selfridge and S. Wagstaff. The hypothesis is related to the problem of the existence of odd numbers n \equiv \pm 2 (mod 5), which are both Fermat and Lucas-pseudoprimes (in short, FL-pseudoprimes). A Fermat pseudoprime to base a is a composite number n satisfying the condition an - 1 \equiv 1 (mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying Fn - e(n) \equiv 0 (mod n), where n(e) is the Legendre symbol e(n) = \bigl( n 5 \bigr) , Fm the mth term of the Fibonacci series. According to Baillie’s PSW conjecture, there are no FL-pseudoprimes. If the hypothesis is true, the combined primality test checking Fermat and Lucas conditions for odd numbers not divisible by 5 gives the correct answer for all numbers of the form n \equiv \pm 2 (mod 5), which generates a new deterministic polynomial primality test detecting the primality of 60 percent of all odd numbers in just two checks. In this work, we continue the study of FL-pseudoprimes, started in our article "On a combined primality test" published in the journal "Izvestia VUZov.Matematika" No. 12 in 2022. We have established new restrictions on probable FL-pseudoprimes and described new algorithms for checking FL-primality, and with the help of them we proved the absence of such numbers up to the boundary B = 1021, which is more than 30 times larger than the previously known boundary 264 found by J. Gilchrist in 2013. An inaccuracy in the formulation of theorem 4 in the mentioned article has also been corrected.

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.