- Journal Details
- Format
- Journal
- eISSN
- 2309-5377
- First Published
- 30 Dec 2013
- Publication timeframe
- 2 times per year
- Languages
- English

#### Search

Open Access

#### On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

Page range: 1 - 40

#### Abstract

In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.

#### Keywords

- quantum integer
- quantum algebra
- -series
- semigroup
- polynomial functional equation
- cyclotomy

#### MSC 2010

- 81R50
- 11R18
- 11T22
- 11B13
- 11C08
- 39B05

Open Access

#### Extreme Values of Euler-Kronecker Constants

Page range: 41 - 52

#### Abstract

In a family of _{n}_{K}_{K}_{K}_{K}_{K}

#### Keywords

- Euler-Kronecker constants
- Dedekind zeta functions
- Logarithmic derivatives of -functions

#### MSC 2010

- Primary: 11R42
- Secondary: 11M41

#### Abstract

We provide an algorithm to approximate a finitely supported discrete measure _{N}_{N}^{d }_{N}^{−}^{1}. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure

#### Keywords

- star-discrepancy
- low-discrepancy sequences
- Borel Measures
- total variation

#### MSC 2010

- 11K36
- 11K38
- 11J71
- 11K06

Open Access

#### The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ _{ℬs}

Page range: 71 - 92

#### Abstract

In the present paper the author uses the function system Γ_{ℬs}constructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.

#### Keywords

- function system Γ constructed in Cantor bases
- extreme and star discrepancy
- inequalities of Erd˝ os-Tur´ an-Koksma

#### MSC 2010

- 11K06
- 11K31
- 11K36
- 65C05

Open Access

#### Uniform Distribution of the Weighted Sum-of-Digits Functions

Page range: 93 - 126

#### Abstract

The higher-dimensional generalization of the weighted _{q,γ}_{q,γ}_{1}_{q, γ}_{2}_{q,γ}_{1} and _{2} are integers such that _{1} + _{2} ≠ 0 and that the akin two-dimensional sequence _{q,γ}_{q,γ}_{q,γ}_{q,γ}

#### Keywords

- uniform distribution
- van der Corput sequence
- higher-dimensional weighted sum-of-digits function
- discrepancy
- distribution function

#### MSC 2010

- Primary 11K06
- Secondary 11K31, 11J71