On algebraic independence of some numbers

A theorem is proved about algebraic independence of values of the exponential function.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 525–532, November, 1968.     

On the independence of expansions of algebraic numbers in an integer base

Let b 2 be an integer. According to a conjecture of ÉmileBorel, the b-adic expansion of any irrational algebraic numberbehaves in some respects ‘like a random sequence’.We give a contribution to the following related problem: let and ' be irrational algebraic numbers, then prove that theirb-adic expansions either have the same tail, or behave in somerespects ‘like independent random sequences’.     

Concerning algebraic independence of some transcendental numbers

Given the three numbers of,a ^{1 2} , and Ina ₂/Ina ₁, wherea ₁ anda ₂ are algebraic numbers whose logarithms are linearly independent in a rational field and is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.Translated from Matematicheskie Zametki, Vol.3, No. 1, pp. 51–58, January, 1968.     

Linear independence of harmonic numbers over the field of algebraic numbers II

The Ramanujan Journal - In a recent article, the authors have studied the arithmetic natures of a certain class of harmonic numbers. The present paper will explore the linear independence of...     

The algebraic independence of certain numbers related by the exponential function

We give a variation of a theorem of Gelfond. One of the corollaries is Schneider's conjecture that either exp(e) or exp(e²) is transcendental. It also follows that at least one of the numbers exp(α), exp(α²), exp(α³) is transcendental for any nonzero complex number α.     

On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities

|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N     

On powers of relational and algebraic systems

We extend the well-known Birkhoff’s operation of cardinal power from partially ordered sets onto n-ary relational systems. The extended power is then studied not only for n-ary relational systems but also for some of their special cases, namely partial algebras and total algebras. It turns out that a concept of diagonality plays an important role when studying the powers.     

On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.     

On the linear independence of certain numbers

With the help of a version of Siegel’s method, we establish in a fairly general case the linear independence of values of hypergeometric functions with irrational parameters and their derivatives; the corresponding quantitative results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 383–390, September, 1997. Translated by A. M. Chebotarev     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.