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Displaying 61 – 80 of 1536

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

Let $W (m, n; \underset{̲}{ψ})$ denote the set of $ψ_{1}, ..., ψ_{n}$ –approximable points in $ℝ^{m n}$ . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underset{̲}{ψ}$ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$ . It can not be removed for $m = n = 1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m = 2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem.

A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Diego Marques, Elaine Silva (2017)

Communications in Mathematics

In this note, we prove that there is no transcendental entire function $f (z) \in ℚ [[z]]$ such that $f (ℚ) \subseteq ℚ$ and $den f (p / q) = F (q)$ , for all sufficiently large $q$ , where $F (z) \in ℤ [z]$ .

A note on two linear forms

Nikolay Moshchevitin (2014)

Acta Arithmetica

We prove a result on approximations to a real number θ by algebraic numbers of degree ≤ 2 in the case when we have certain information about the uniform Diophantine exponent ω̂ for the linear form x₀ + θx₁ + θ²x₂.

A note on zero-one laws in metrical Diophantine approximation

Victor Beresnevich, Sanju Velani (2008)

Acta Arithmetica

A note on |αp - q|

S. Srinivasan (1982)

Acta Arithmetica

A note to a bifurcation result of H. Kielhöfer for the wave equation

Otto Vejvoda, Pavel Krejčí (1991)

Mathematica Bohemica

A modification of a classical number-theorem on Diophantine approximations is used for generalizing H. kielhöfer's result on bifurcations of nontrivial periodic solutions to nonlinear wave equations.