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.
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}}{{lambda }^{k}}). We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.
The Baillie PSW conjecture was formulated in 1980 and was named after its authors Baillie, Pomerance, Selfridge, and Wagstaff, Jr. The conjecture is related to the problem of the existence of odd numbers (n equiv pm 2;(bmod ;5)), which are both Fermat and Lucas pseudoprimes (in short, FL‑pseudoprimes). A Fermat pseudoprime to base a is composite number n satisfying the condition ({{a}^{{n - 1}}} equiv 1)(mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying ({{F}_{{n - e(n)}}} equiv 0)(mod n), where e(n) is the Legendre symbol (e(n) = left( begin{gathered} n hfill 5 hfill end{gathered} right)) and ({{F}_{m}}) is the mth term of the Fibonacci series. According to the Baillie PSW conjecture, there are no FL pseudoprimes. If the conjecture 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;(bmod ;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 Russian Mathematics in 2012. We have established new restrictions on probable FL pseudoprimes and described new algorithms for checking FL primality, and, using them, we proved the absence of such numbers up to the boundary (B = {{10}^{{21}}}), which is more than 30 times larger than the previously known boundary 264 found by Gilchrist in 2013. An inaccuracy in the formulation of Theorem 4 in the mentioned article has also been corrected.
For the polynomial (P(z) = sumnolimits_{j = 0}^n {{c}_{j}}{{z}^{j}}) of degree n having all its zeros in ({text{|}}z{text{|}} leqslant k), (k geqslant 1), Jain in “On the derivative of a polynomial,” Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that (mathop {max }limits_{|z| = 1} {text{|}}P'(z){text{|}} geqslant nleft( {frac{{{text{|}}{{c}_{0}}{text{|}} + ,{text{|}}{{c}_{n}}{text{|}}{{k}^{{n + 1}}}}}{{{text{|}}{{c}_{0}}{text{|}}(1 + {{k}^{{n + 1}}}) + ,{text{|}}{{c}_{n}}{text{|}}({{k}^{{n + 1}}} + {{k}^{{2n}}})}}} right)mathop {max }limits_{|z| = 1} {text{|}}P(z){text{|}}.) In this paper we strengthen the above inequality and other related results for the polynomials of degree (n geqslant 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 (that is, 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 semiaxis. 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 nonstrict signs, zeros, roots, and hyperrots 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 to 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 results are theoretical in nature; they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.
In this paper, the direct and two inverse problems for a model equation of mixed parabolic-hyperbolic type are studied. In the direct problem, the Tricomi problem for this equation with a noncharacteristic line of type change is considered. The unknown of the inverse problem is the variable coefficient at the lowest derivative in the parabolic equation. To determine it, two inverse problems are studied: with respect to the solution defined in the parabolic part of the domain, the integral overdetermination condition (inverse problem 1) and one simple observation at a fixed point (inverse problem 2) are given. Theorems on the unique solvability of the formulated problems in the sense of a classical solution are proved.
In this paper we study the locally uniform convergence of homeomorphisms with bounded ((1,sigma ))-weighted ((q,p))-distortion to a limit homeomorphism. Under some additional conditions we prove that the limit homeomorphism is a mapping with bounded ((1,sigma ))-weighted ((q,p))-distortion. Moreover, we obtain the property of lower semicontinuity of distortion characteristics of homeomorphisms.
In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the Dubrovin system of equations and the first trace formula satisfies a sine-Gordon type equation with an additional term.
We construct the asymptotics of the eigenvalues for a quasidifferential Sturm–Liouville boundary value problem on eigenvalues and eigenfunctions considered on a segment (J = [a,b]), with the boundary conditions of type I on the left and right, that is, for a problem of the form (in the explicit notation) ({{p}_{{22}}}(t)left( {{{p}_{{11}}}(t)left( {{{p}_{{00}}}(t)x(t)} right){kern 1pt} '; + {{p}_{{10}}}(t)left( {{{p}_{{00}}}(t)x(t)} right)} right){kern 1pt} '; + {{p}_{{21}}}(t)left( {{{p}_{{11}}}(t)left( {{{p}_{{00}}}(t)x(t)} right){kern 1pt} '; + {{p}_{{10}}}(t)left( {{{p}_{{00}}}(t)x(t)} right)} right)) ( + ;{{p}_{{20}}}(t)left( {{{p}_{{00}}}(t)x(t)} right) = - lambda left( {{{p}_{{00}}}(t)x(t)} right);;(t in J = [a,b]),) ({{p}_{{00}}}(a)x(a) = {{p}_{{00}}}(b)x(b) = 0.) The requirements for smoothness of the coefficients (that is, functions ({{p}_{{ik}}}( cdot ):J to mathbb{R}), (k in 0:i), (i in 0:2)) in the equation are minimal, namely, these are as follows: the functions ({{p}_{{ik}}}( cdot ):J to mathbb{R}) are such that the functions ({{p}_{{00}}}( cdot )) and ({{p}_{{22}}}( cdot )) are measurable, nonnegative, almost every finite, and almost everywhere nonzero and the functions ({{p}_{{11}}}( cdot )) and ({{p}_{{21}}}( cdot )) also are nonnegative on the segment (J,) and, in addition, the functions ({{p}_{{11}}}( cdot )) and ({{p}_{{22}}}( cdot )) are essentially bounded on (J,) the functions (frac{1}{{{{p}_{{11}}}( cdot )}},;;frac{{{{p}_{{10}}}( cdot )}}{{{{p}_{{11}}}( cdot )}},;;frac{{{{p}_{{20}}}( cdot )}}{{{{p}_{{22}}}( cdot )}},;;frac{{{{p}_{{21}}}( cdot )}}{{{{p}_{{22}}}( cdot )}},;;frac{1}{{min { {{p}_{{11}}}(t){{p}_{{22}}}(t),1} }}) are summable on the segment (J.) The function ({{p}_{{20}}}( cdot )) acts as a potential. It is proved that under the condition of nonoscillation of a homogeneous quasidifferential equation of the second order on (J), the asymptotics of the eigenvalues of the boundary value problem under consideration has the form ({{lambda }_{k}} = {{(pi k)}^{2}}left( {D + O({text{1/}}{{k}^{2}})} right)) as (k to infty ,) where (D) is a real positive constant defined in some way.
In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution to some integral equation. The local solvability of this euqation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray–Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.
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