The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

On the ℂ-differentiability of mappings of Banach spaces

We study the monogeneity conditions for -differentiable mappings of domains of a Banach space and establish criteria of -differentiability of a mapping at a point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1336–1342, October, 1994.     

ℂ-differentiability of mappings of Hilbert spaces

The process of differentiation is studied for a class of mappings of regions of a Hilbert space. We introduce the concept of Cl-differentiability of a mapping at a point and establish sufficient conditions of Cl-differentiability.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 424–428, March, 1995.     

The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups

The polynomial sub-Riemannian differentiability is established for the large classes of Hölder mappings in the sub-Riemannian sense, namely, the classes of smooth mappings, their graphs, and the graphs of Lipschitz mappings in the sub-Riemannian sense defined on nilpotent graded groups. We also describe some special bases that carry the sub-Riemannian structure of the preimage to the image.     

The Hodge–Poincaré polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group G and such that ${X^{ss} \neq X_{(0)}^{s}\neq \emptyset}$ . We give formulae for the Hodge–Poincaré series of the quotient ${X_{(0)}^{s}/G}$ . We use these computations to obtain the corresponding formulae for the Hodge–Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.     

A Mergelyan Theorem for mappings to ℂ2∖ℝ2

We investigate the question whether a Mergelyan Theorem holds for mappings to ℂⁿ ∖ A. The main result is the prove of such a theorem for mappings to ℂ²∖ℝ².     

On an elementary version of I.M. Vinogradov’s method

We prove estimates for complete rational arithmetic sums of Bernoulli polynomials whose arguments are formed by the fractional parts of values of a polynomial with rational coefficients. The results are applied to the problem of finding the convergence exponent for the mean values of the sums under consideration.     

On the Distribution of the Limit of Products of I.I.D. 2×2 Random Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2×2 random stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is finite.     

The structure of irreducible matrices †

Let A be an irreducible matrix with index of imprimitivity h is shown that there exists a permutation matrix P such that PAP^t is in a superdiagonal block form with k nonzero blocks if and only if k divides h It is also shown that a matrix in a superdiagonal block form without zero rows or columns is irreducible if and only if the product of the superdiagonal nonzero blocks is irreducible.     

On π 2 almost geodesic mappings of almost Hermitian manifolds

We consider a class of almost geodesic mappings, namely, almost geodesic mappings of class π ₂, and obtain conditions under which almost Hermitian manifolds admit almost geodesic mappings of class π ₂. We prove that an almost Hermitian manifold admits a π ₂-mapping with respect to a Riemannian connection if and only if it is an NK-manifold. We obtain a condition on the defining form ψ of any nontrivial π ₂(e)-mapping under which a properNK-structure is taken to a proper NK-structure.     

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollár's results on the Łojasiewicz exponent of polynomial mappings is obtained. Oblatum 9-II-1998 & 3-VI-1998 / Published online: 14 January 1999     

Quadratic decomposition of symmetric semi-classical polynomial sequences of even class: an example from the case s = 2

In the present work, we deal with the quadratic decomposition of symmetric semi-classical polynomial sequences of even class. An example from class 2 is settled. We give an integral representation of the considered symmetric form.     

Lower bound for the area of an analytic curve inside a cube in the space ℂ n

The exact lower bound of the area of an analytic curve which lies inside a cube in the space and passes through its center is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 145–148, January, 1993.     

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

On exact solutions of Stokes second problem for a Burgers’ fluid, I. The case γ < λ2/4

In the present work, the exact solutions of Stokes second problem for a Burgers’ fluid are investigated. The expressions for the velocity field and the corresponding tangential stress are obtained when the relaxation times satisfy the condition γ < λ²/4. The solutions have been determined by means of Laplace transform. Only one initial condition is necessary for velocity and these solutions presented in the forms of simple or multiple integrals in terms of Bessel functions. The corresponding solutions for a Newtonian fluid as well as Oldroyd-B fluid appear as the limiting cases of the presented results. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. Moreover, a comparison for velocity is made with Oldroyd-B and Newtonian fluids.