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Vol 217, No 1 (2016)

Article

Fractional Factorials and Prime Numbers (A Remark on the Paper “On Prime Values of Some Quadratic Polynomials”)

Andrianov A.N.

Abstract

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

Journal of Mathematical Sciences. 2016;217(1):1-2
pages 1-2 views

A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

Vinogradov O.L., Gladkaya A.V.

Abstract

Let σ > 0, m, r ∈ ℕ, mr, let Sσ,m be the space of splines of order m and minimal defect with nodes \( \frac{j\pi }{\sigma } \) (j ∈ ℤ), and let Aσ,m(f)p be the best approximation of a function f by the set Sσ,m in the space Lp(ℝ). It is known that for p = 1,+∞,

\( \begin{array}{l} \sup \hfill \\ {}f\in {W}_p^{(r)}\left(\mathbb{R}\right)\hfill \end{array}\frac{A_{\sigma, m}{(f)}_p}{{\left\Vert {f}^{(r)}\right\Vert}_p}=\frac{K_r}{\sigma^r}, \)
where Kr are the Favard constants. In this paper, linear operators Xσ,r,m with values in Sσ,m such that for all p ∈ [1,+∞] and f ∈ Wp(r)(),
\( {\left\Vert f-{X}_{\sigma, r,m}(f)\right\Vert}_p\le \frac{K_r}{\sigma^r}{\left\Vert {f}^{(r)}\right\Vert}_p \)
are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.

Journal of Mathematical Sciences. 2016;217(1):3-22
pages 3-22 views

Configuration Module and Removable Sets

Demshin I.N., Shlyk V.A.

Abstract

In the paper, a new class of polycondensers, for which the capacity of a polycondenser equals its module, is introduced. It is established that the sets that are removable for the condenser’s capacity are also removable for the module of the polycondensers considered. Bibliography: 5 titles.

Journal of Mathematical Sciences. 2016;217(1):23-27
pages 23-27 views

Distortion Theorems for Circumferentially Mean P -Valent Functions

Dubinin V.N.

Abstract

Some distortion theorems for circumferentially mean p-valent functions are proved using the symmetrization method. The cases of functions with a zero of order p at the origin, functions having no zeros, and functions with Montel’s normalization are considered. The equality cases in the estimates obtained are described. Bibliography: 10 titles.

Journal of Mathematical Sciences. 2016;217(1):28-36
pages 28-36 views

A Condition of Smallness of Girth on Sub-Finsler Spaces

Dymchenko Y.V.

Abstract

In the paper, a condition of smallness of girth with respect to some curve families for removable sets on sub-Finsler spaces is established. Bibliography: 18 titles.

Journal of Mathematical Sciences. 2016;217(1):37-44
pages 37-44 views

Strong Approximation of Functions by Positive Operators

Zhuk V.V.

Abstract

The paper considers strong approximation of continuous functions by positive operators. Estimates in terms of the modulus of continuity and its convex majorant are established. Bibliography: 4 titles.

Journal of Mathematical Sciences. 2016;217(1):45-53
pages 45-53 views

Two-Dimensional Approximations by the Method of Dividing Toric Tilings

Zhuravlev V.G.

Abstract

An infinite sequence of dividing two-dimensional toric tilings is constructed by the differentiation method. The karyons of the tilings have radii tending to zero and contain points having the best approximations on the torus in certain normalized metrics, which are defined by the initial karyon. The properties of the latter metrics can be essentially different from those of the standard metrics on a torus, used in approximation problems.

Journal of Mathematical Sciences. 2016;217(1):54-64
pages 54-64 views

Dividing Toric Tilings and Bounded Remainder Sets

Zhuravlev V.G.

Abstract

With the use of differentiation of exchanged toric developments, an infinite sequence of twodimensional toric tilings is constructed. The karyon of these tilings is proved to be a bounded remainder set.

Journal of Mathematical Sciences. 2016;217(1):65-80
pages 65-80 views

Estimating the Norm of the Holomorphic Component of a Meromorphic Function in a Finitely Connected Domain

Kalmykov S., Nagy B.

Abstract

In this paper, we extend a Gonchar–Grigorjan type estimate for the norm of the holomorphic parts of meromorphic functions in finitely connected Jordan domains with C2-smooth boundary in the case where the poles are in a compact set. A uniform estimate for Cauchy type integrals is also given.

Journal of Mathematical Sciences. 2016;217(1):81-90
pages 81-90 views

Normalized Incomplete Beta Function: Log-Concavity in Parameters and Other Properties

Karp D.B.

Abstract

The logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been established by Finner and Roters in 1997 as a corollary of a rather difficult result, based on generalized reproductive property of certain distributions. In the first part of this paper, a direct analytic proof of the logarithmic concavity/convexity mentioned above is presented. These results are strengthened in the second part, where it is proved that the power series coefficients of the generalized Turán determinants formed by the parameter shifts of the normalized incomplete beta function have constant sign under some additional restrictions. The method of proof suggested also leads to various other new facts, which may be of independent interest. In particular, linearization formulas and two-sided bounds for the above-mentioned Turán determinants, and also two identities of combinatorial type, which we believe to be new, are established.

Journal of Mathematical Sciences. 2016;217(1):91-107
pages 91-107 views

Polarization and Circular Truncation of a Domain

Kuznetsov V.O.

Abstract

The paper considers the difference between the reduced module m(D) of a simply connected domain D with respect to the point z = 0 and the reduced module m(Dr) of its radial circular truncation, where Dr is the connected component of the set D ∩ {|z| < r} containing the point z = 0. It is proved that this difference does not decrease under polarization and circular symmetrization.

Journal of Mathematical Sciences. 2016;217(1):108-113
pages 108-113 views

The Module Method in a Certain General Extremal Decomposition Problem

Kuz’mina G.V.

Abstract

An extension of the method of modules of curve families to extremal decomposition problems in which the associated quadratic differentials have poles of arbitrary orders is considered.

Journal of Mathematical Sciences. 2016;217(1):114-124
pages 114-124 views

On the Mean Square of the Error Term For Dedekind Zeta Functions

Fomenko O.M.

Abstract

Let Kn be a number field of degree n over ℚ. By D(x,Kn) denote the number of all nonzero integral ideals in Kn with norm ≤ x. The Dedekind zeta function ζKn(s) is a meromorphic function with a simple pole at s = 1 and with residue, say, Λn. Define

Journal of Mathematical Sciences. 2016;217(1):125-137
pages 125-137 views