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Let P z be a doubly periodic Weierstrass function with periods 2ω1, 2ω2, and let Q be the parallelogram of periods, Q = {z ∈ C : z = 2α1ω1+2α2ω2, α1, α2 ∈ [0, 1)}. We consider a simply connected domain D, D ¯ ⊂ Q, such that its boundary ∂D contains cusps, and a function f that is analytic in D and continuous on ∂D. We assume that the modulus of continuity ω(t) satisfies the relation ∫ x 0 ω t t d t + x ∫ ∞ x ω t t 2 d t ≤ c ω x. Let Φ map conformally the domain C \ D onto C \ D with the normalization Φ(∞) = ∞,Φ′(∞) > 0. We put L1+t = {z ∈ C \ D : |Φ(z)| = 1+t}, δn(z) = dist(z, L 1 + 1 n ), z ∈ ∂D. The main result of the paper is the following statement. Theorem 1. Assume that there exists a sequence of polynomials Pn(u, v), deg Pn ≤ n, such that f z - P n P z , P ′ z ≤ C δ n r z ω δ n z , z ∈ ∂ D , C is independent of n and z. Then f ∈ Hr+ω(D). [ABSTRACT FROM AUTHOR]

Univalent polynomials form an important class of conformal maps and have deep applications in approximate construction of conformal maps and other fields. The paper reveals rather surprising features of univalent polynomials whose finite critical points are placed on the boundary of the univalence disk. This provides explicitly the extremal quasiconformal dilatation and other quasiinvariants associated to such maps. The general case admits only much weaker estimates. [ABSTRACT FROM AUTHOR]

In inverse problems, an important role is played by the following fact: the functions of the form ∑ k = 1 n f k x , y , z g k x , y , z , where fk, gk are the solutions of a second order elliptic equation in a bounded domain Ω ⊂ R 3 , constitute a dense set in L2(Ω). This paper deals with the Laplace equation. We show that the density does hold if fk and gk are harmonic polynomials, whereas the factors gk are invariant with respect to shifts or rotations. [ABSTRACT FROM AUTHOR]

Let K be a multidimensional local field with characteristic different from the characteristic of its residue field, c be a unit of K, and F(X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module F( $$ \mathfrak{M} $$ ) over the maximal ideal of the ring of integers in K. Assume that K contains the group of roots of the isogeny [ p ](X), which we denote by μ,. Let be the multiplicative group of Cartier curves and be the formal analog of the module Fc( $$ \mathfrak{M} $$ ). In the present paper, the formal symbol { ·, · }c : K()× → μ, is constructed and its basic properties are checked. This is the first step in the construction of an explicit formula for the Hilbert symbol. [ABSTRACT FROM AUTHOR]

In this paper, we examine a Cauchy-type problem for one ordinary differential equation with Riemann–Liouville fractional derivatives. For this problem, based on the Lebesgue space of functions that are summable with an arbitrarily fixed degree, we propose a family of pairs of spaces of elements and right-hand sides that provide the well-posed statement of the problem. In these pairs of spaces, we develop a generalized polynomial projection method for solving the problem considered and justify it from the functional-theoretic point of view. Based on the general results obtained, we prove the convergence of the "polynomial" Galerkin method, the collocation method, and the subdomain method for solving the corresponding Cauchy-type problem. [ABSTRACT FROM AUTHOR]

The paper continues the research of R. Kronover and J. Minlor, H.-O. Peitgen and P. H. Richter. Using mathematical methods and computer experiments, the first- and second-order framings of the Mandelbrot sets of three families of the quadratic polynomials of a complex variable are revealed. The connection between the first- and second-order framings of Mandelbrot sets of the functions f2(z) = z2 + cz, f3(z) = z2 + z + c, and f4(z) = cz2 + c with remarkable curves (cardioid, lemniscate, and circle) was established. The algorithms of constructing the framings of Mandelbrot sets of considered functions in the mathematical package MathCad and Pascal programs have been worked out. Algorithms for constructing Mandelbrot sets in Pascal programs are developed. [ABSTRACT FROM AUTHOR]

The exact decrease rate of the best approximations of non-integer numbers by polynomials with integer coefficients of growing degrees is found on a disk in the complex plane, a cube in ℝd, and a ball in ℝd. The sup-norm is used in the first two cases, and the norm in Lp, 1 ≤ p < ∞, is applied in the third one. Detailed comments are given (two remarks at the end of the paper). [ABSTRACT FROM AUTHOR]

We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes, our formulas generalize the classical hook-length formula and the Littlewood formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its q-analogs, which were studied in previous papers of the series. [ABSTRACT FROM AUTHOR]

A family of measures on a given set of real algebraic numbers of arbitrary fixed degree is considered. The paper presents a method for constructing a sequence of measures that weakly converges to the Cauchy distribution. The method is based on the theory of random polynomials. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

A regular graph of the ring of n × n matrices over a field is a graph whose vertices are nonsingular matrices. Two different matrices are adjacent if their sum is singular. In 2009, S. Akbari, M. Jamaali, and S. Seed Fakhari found that the clique number of this graph is finite whenever the field is not of characteristic 2. The same authors asked if the chromatic number of the graph is finite (for fields of characteristic 0 this question is still open). In this paper, we introduce a concept of total and regular graph of a polynomial, generalizing the regular graph of a matrix ring. We investigate some properties of these graphs and their relationship with the above question. Several new open questions are also posed. [ABSTRACT FROM AUTHOR]

Semirings of skew polynomials such as invariant, without nilpotent elements, Abelian, and Rickart without nilpotent elements are considered in this paper. Properties and characterizations of these semirings are obtained. [ABSTRACT FROM AUTHOR]

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, µ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, µ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, µ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

An extremal value of a function is the value of the function at one of its extremum points. Each extremum point of a differentiable function of several variables is described by the system of equations expressing the vanishing of all partial derivatives. However, in the general case, one cannot obtain equations for extremal values of the function. The case of polynomials significantly differs from the general case, and in this paper we obtain an equation for extremal values of a given polynomial in several variables. [ABSTRACT FROM AUTHOR]

In this paper, estimates are obtained for the Fourier transform of smooth charges (measures) concentrated on some nonconvex hypersurfaces. The summability of the maximal Randall function is proved for a wide class of nonconvex hypersurfaces. In addition, in the three-dimensional case, estimates are obtained depending on the Varchenko height. The accuracy of the obtained estimates is proved. The proof of the estimate for oscillatory integrals is based on the Weierstrass preparatory theorem. [ABSTRACT FROM AUTHOR]

We study Bernstein polynomials for simple nonsmooth rational module functions. We show that these polynomials can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called "generalized Popoviciu expansions." To write generalized expansions, we develop a special formalism based on combinatorial calculations. Based on the formulas obtained, we propose a complete description of the character of convergence of the Bernstein polynomials studied in the complex plane. We also discuss the relationship of generalized Popoviciu expansions with the distribution of zeros of Bernstein polynomials. In the final part of the paper, we present additional new relations for Bernstein polynomials of the rational module function. [ABSTRACT FROM AUTHOR]

In this paper, a structure theorem for the symmetries of a graded flat cosymbol algebra of differential operators is proved. Together with a lemma on equivariant polynomials also proved in the paper, this theorem gives an upper bound on the dimension of the graded Lie algebra associated with the symmetries of geodesic flow on a smooth variety. Bibliography: 14 titles [ABSTRACT FROM AUTHOR]

The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized. [ABSTRACT FROM AUTHOR]

In this paper, we present a method of programmatic identification of complex roots of polynomials with complex coefficients without specifying the domain of localization of their roots. The method is based on the algorithm of stable address sorting with minimal amount of calculations. Numerical ranges of the real and imaginary parts of the roots are programmatically determined, the roots are identified without loss of significant figures of the mantissa in the format of presentation of numerical data. [ABSTRACT FROM AUTHOR]

Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric 2 × 2 system of ODEs to obtain RS 4 2 -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. The paper demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations. [ABSTRACT FROM AUTHOR]

This paper is a slightly expanded text of our talk at the PCA-2014. There we announced two recent results concerning explicit polynomial equations defining exceptional Chevalley groups in microweight or adjoint representations. One of these results is an explicit characteristic-free description of equations on the entries of a matrix from the simply connected Chevalley group G( E, R) in the 56-dimensional representation V. Before, a similar description was known for the group G( E, R) in the 27-dimensional representation, whereas for the group of type E it was only known under the simplifying assumption that 2 ϵ R. In particular, we compute the normalizer of G( E, R) in GL(56, R) and establish that, as also the normalizer of the elementary subgroup E(E, R), it coincides with the extended Chevalley group $$ \overline{G}\left({\mathrm{E}}_7,R\right) $$. The construction is based on the works of J. Lurie and the first author on the E-invariant quartic forms on V. Another major new result is a complete description of quadratic equations defining the highest weight orbit in the adjoint representations of Chevalley groups of types E, E, and E. Part of these equations not involving zero weights, the so-called square equations (or π/2-equations) were described by the second author. Recently, the first author succeeded in completing these results, explicitly listing also the equations involving zero weight coordinates linearly (the 2 π/3-equations) and quadratically (the π-equations). Also, we briefly discuss recent results by S.Garibaldi and R.M.Guralnick on octic invariants for E. [ABSTRACT FROM AUTHOR]

In this paper, we show that an overdetermined tropical linear system has a solution if and only if it contains a square subsystem having a stable solution that is a solution of the original system. This leads to a simple algorithm for solving tropical linear systems in time Onmn4, where m is the number of equations and n is the number of variables. For weakly overdetermined systems, this time is polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we consider a scalar linear differential-difference equation (LDDE) of neutral type x⋅(t) + p(t)x⋅(t − 1) = a(t)x(t − 1) + f(t). We examine the initial-value problem with an initial function in the case where the initial condition is given on an initial set. We use the method of polynomial quasisolutions based on the representation of the unknown function x(t) in the form of a polynomial of degree N. Substituting this function in the original equation we obtain the discrepancy Δ(t) = O(tN), for which an exact analytic representation is obtained. We prove that if a polynomial quasisolution of degree N is taken as an initial function, then the smoothness of the solution generated by this initial functions at connection points is no less than N. [ABSTRACT FROM AUTHOR]

The paper presents a technique for the automatic calculation of Belyi functions for trees with weighted edges. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper provides a counterexample to the hypothesis on the uniqueness of continuation for polynomials of harmonic quaternion fields in a compact domain with a nonanalytic metric. The constructed polynomial vanishes identically in a neighborhood of the boundary. A connection of this construction with the problem on resonances of the Schroedinger operator on a line is noted. [ABSTRACT FROM AUTHOR]

The paper is concerned with the computational complexity of the initial value problem (IVP) for a system of ordinary dynamical equations. A formal problem statement is given, containing a Turing machine with an oracle for getting the initial values as real numbers. It is proven that the computational complexity of the IVP for the three-body problem is not bounded by a polynomial. The proof is based on the analysis of oscillatory solutions of the Sitnikov problem, which have a complex dynamical behavior. These solutions contradict the existence of an algorithm that solves the IVP in polynomial time. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

The paper presents an explicit formula for the Hilbert pairing between the Milnor K-group of a higher-dimensional local field and a polynomial formal module. This formula generalizes similar results for the one-dimensional case and the higher-dimensional case of multiplicative group. The case where the field and its first residue field have different characteristics, is considered. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

For a rational function r ∈ R n , Wali and Shah (The Journal of Anal., 25(1):(2017), 43-53) proved the following: | r ′ (z) | ≥ 1 2 | B ′ (z) | + | c n | - | c 0 | | c n | + | c 0 | | r (z) |. In this paper, we prove Bernstein-type inequalities for rational functions with prescribed poles. The obtained results refine some inequalities on rational functions and strengthen known polynomial inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we study an irreducible decomposition structure of the D -module direct image π + (O c n) for the finite map π : C n → C n / (S n 1 × ⋯ × S n r ). We explicitly construct the simple components of π + (O C n) by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D -module decomposition of the polynomial ring localized at the discriminant of π . Furthermore, we study the action of invariant differential operators on the higher Specht polynomials. [ABSTRACT FROM AUTHOR]

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in ( dd′) n and the size of the input. This work concludes a series of four papers. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

A full integral of a polynomial is defined as its integral with the property that any multiple root of the polynomial is a root of this integral. The paper investigates relationships between the existence of a full integral and the form of a polynomial. In particular, it is proved that a full integral exists if the polynomial has no more than one multiple root. On the other hand, if the number of multiple roots of a polynomial strictly exceeds the number of its simple roots increased by one, then the polynomial has no full integral. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time . This result slightly improves the result by Vorobjov, who described an algorithm with complexity bound (sd)O(k′(k−k′)) for the same problem. The complexity bound of the algorithm described in this paper has a better dependence on the number s of polynomials in the input. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

The splitting method is one of the most powerful and well-studied approaches to solving various NP-hard problems. The main idea of this method is to split the input instance of a problem into several simpler instances (further simplified by certain simplification rules) such that when the solution for each of them is found, one can construct the solution for the initial instance in polynomial time. There exists a huge number of papers describing algorithms of this type, and usually a considerable part of such a paper is devoted to case analysis. In this paper, we present a program that, given a set of simplification rules, automatically generates a proof of an upper bound on the running time of a splitting algorithm using these rules. As an example, we report the results of experiments with such a program for the SAT, MAXSAT, and (n, 3)- MAXSAT (the MAXSAT problem for the case where every variable in the formula appears at most three times) problems. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number. [ABSTRACT FROM AUTHOR]

In the present paper, we construct an invariant of knots in the thickened sphere with g handles dependent on 2 g + 3 variables. In the construction of the invariant we use the Wirtinger presentation of the knot group and the concept of parity introduced by Manturov []. In the present paper, we also consider examples of knots in the thickened torus considered in [] such that their nonequivalence is proved by using the constructed polynomial. [ABSTRACT FROM AUTHOR]

In this paper, the exact rate of decrease of best approximations of non-integer numbers by polynomials with integer coefficients is established. [ABSTRACT FROM AUTHOR]

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space L q , τ (T m) , the class of functions with bounded mixed fractional derivative W q , τ r ¯ , 1 < q , τ < ∞ , and studies the order of the best M-term approximation of a function f ∈ L p , τ (T m) by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class W q , τ 1 r ¯ in the norm of the space L p , τ 2 (T m) for various relations between the parameters p , q , τ 1 , τ 2 . [ABSTRACT FROM AUTHOR]

This paper is devoted to the application aspects of homomorphic cryptography. It provides a description of a fully homomorphic matrix polynomial-based encryption scheme. It also gives the results of practical comparison of fully homomorphic encryption schemes. We consider some special cases of homomorphic encryption allowing computations of a limited number of functions. [ABSTRACT FROM AUTHOR]

This paper is a survey of recent results of investigations on varieties of Poisson algebras. We give constructions of varieties of Poisson algebras with extremal properties, we give equivalent conditions for the polynomial codimension growth of a variety of Poisson algebras, we study varieties of Poisson algebras whose ideals of identities contain the identity {x, y} ⋅ {z, t} = 0, and we study the interrelation between such varieties and varieties of Lie algebras. [ABSTRACT FROM AUTHOR]

A semi-local smoothing spline of degree n and class C is a function defined on an interval, having p continuous derivatives on that interval, and coinciding with a polynomial of degree n on the subintervals forming its partition. The domain of each polynomial is a subinterval on which m + 1 values of the approximated function are given, but in order to construct the polynomial, it is necessary to know M ≥ m + 1 values ( m and M are determined by the class and the degree of the spline). The additional values can be borrowed from the adjacent subintervals. When constructing an S-spline in the periodic case, the problem of additional values is solved on the basis of periodicity, but in the nonperiodic case, one is expected to define the lacking values of a function beyond the domain. The present paper is aimed at nonperiodic two-sided S-splines whose construction does not require additional data. [ABSTRACT FROM AUTHOR]

The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representativeup−1ut−Δpu=0,tx∈0T×Ω,Ω∈ℝn,n≥1,p>0,and with the following blow-up condition for the energy:εt≔∫Ωutxp+1dx+∫0t∫Ω∇xuτxp+1dxdτ→∞ast→T,where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the conditionεt≤Fαtω0T−t−α∀t0,α>1p+1,a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T. [ABSTRACT FROM AUTHOR]

The class of γ-generating matrices and its subclasses of regular and singular γ-generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular-singular factorization. The class of generalized γ-generating matrices was introduced in [14, 20]. In the present paper, subclasses of singular and regular generalized -generating matrices are introduced and studied. As the main result, a theorem of existence of the regular-singular factorization for a rational generalized γ-generating matrix is proved. [ABSTRACT FROM AUTHOR]

The paper presents new results on varieties of commutative metabelian algebras over a field of zero characteristic. We study the structure of the multilinear part of the variety of all commutative metabelian algebras as a module of the symmetric group. Two almost nilpotent varieties are introduced and studied in this class of algebras. We prove the nonexistence of other almost nilpotent commutative metabelian varieties of subexponential growth. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the additive structure of the algebra F(5), i.e., a relatively free, associative, countably-generated algebra with the identity [x1,..., x5] = 0 over an infinite field of characteristic ≠2, 3. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions cn = dimPn/Pn∩ T(5), where Pn is the space of multilinear polynomials of degree n in F(5) and T(5) is the T-ideal generated by the long commutator [x1,..., x5]. [ABSTRACT FROM AUTHOR]

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller-Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions. [ABSTRACT FROM AUTHOR]

The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials. [ABSTRACT FROM AUTHOR]

Let Δ(1)(Y) be a set of all 2𝜋-periodic functions f that are continuous on the real axis R and change their monotonicity at various fixed points yi ∈ [-𝜋; 𝜋); i = 1; :::; 2s; s ∈ N (i.e., there is a set Y := {yi}i∈ℤ of points yi = yi+2s + 2𝜋 on R such that f are nondecreasing on [yi; yi−1] if i is even, and nonincreasing if i is odd). In the article, a function fY = f 2 C(1) ∩ Δ(1)(Y) has been constructed such that lim n → ∞ sup n E n 1 f ω 4 f ′ π / n = ∞ , where E n 1 f is the error of the best uniform approximation of the function f ∈ Δ(1)(Y) by trigonometric polynomials of order n ∈ N, which also belong to the set Δ(1)(Y), and 𝜔4(f'; ∙) is the 4-th modulus of smoothness of the function f'. So, for a certain constant c, the inequality E n 1 f ≤ c n ω 3 f ′ π / n is the best with respect to the order of the modulus of smoothness. [ABSTRACT FROM AUTHOR]

It was shown by Griggs and Wu that a graph of minimal degree 4 on n vertices has a spanning tree with at least $$ \frac{2}{5} $$ n leaves, which is asymptomatically the best possible bound for this class of graphs. In this paper, we present a polynomial time algorithm that finds in any graph with k vertices of degree greater than or equal to 4 and k′ vertices of degree 3 a spanning tree with $$ \left[ {\frac{2}{5} \cdot k + \frac{2}{{15}} \cdot k'} \right] $$ leaves. [ABSTRACT FROM AUTHOR]

In this paper, covering theorems for analytic functions related to polynomials that have curved majorants on two symmetric intervals are proved. These theorems contain and complement some new and classical results for polynomials with constraints on one and two intervals. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

complex n × n matrix A is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following assertion: Let $ A\bar{A} $ be a nonderogatory matrix with real positive spectrum. Then A can be made real by a unitary congruence transformation if and only if A and $ \bar{A} $ are unitarily congruent. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]